Multi-Unit Common Value Auctions: Theory and Experiments

Multi-unit common value auctions: Theory and experiments

Dedikation Till Kicki, Lova och Laban

Örebro Studies in Economics 22

JOAKIM AHLBERG

Multi-unit common value auctions: Theory and experiments

Title: Multi-unit common value auctions: Theory and experiments. Publisher: Örebro University 2012 www.publications.oru.se [email protected]

Print: Örebro University, Repro 10/2012

ISSN 1651-8896 ISBN 978-91-7668-893-9 Abstract

Research on auctions that involve more than one identical item for sale was almost non-existing in the 90’s, but has since then been getting increasing attention. External incentives for this research have come from the US spectrum sales, the European 3G mobile-phone auctions, and Internet auctions. The policy relevance and the huge amount of money involved in many of them have helped the theory and experimental research advance. But in auctions where values are equal across bidders, common value auctions, that is, when the value depends on some outside parameter, equal to all bidders, the research is still embryonic.

This thesis contributes to the topic with three studies. The ﬁrst uses a Bayesian game to model a simple multi-unit common value auction, the task being to compare equilibrium strategies and the seller’s revenue from three auction formats; the discriminatory, the uniform and the Vickrey auction. The second study conducts an economic laboratory experiment on basis of the ﬁrst study. The third study comprises an experiment on the multi-unit common value uniform auction and compares the dynamic and he static environments of this format.

The most salient result in both experiments is that subjects overbid. They are victims of the winner’s curse and bid above the expected value, thus earning a negative profit. There is some learning, but most bidders continue to earn a negative profit also in later rounds. The competitive effect when participating in an auction seems to be stronger than the rationality concerns. In the first experiment, subjects in the Vickrey auction do somewhat better in small groups than subjects in the other auction types and, in the second experiment, subjects in the dynamic auction format perform much better than subjects in the static auction format; but still, they overbid.

Due to this overbidding, the theoretical (but not the behavioral) prediction that the dynamic auction should render more revenue than the static fails in the second experiment. Nonetheless, the higher revenue of the static auction comes at a cost; half of the auctions yield negative proﬁts to the bidders, and the winner’s curse is more severely widespread in this format. Besides, only a minority of the bidders use the equilibrium bidding strategy.

The bottom line is that the choice between the open and sealed-bid formats may be more important than the choice of price mechanism, especially in common value settings.

Keywords: Multi-Unit Auction; Common Value Auction; Laboratory Exper- iment; Game Theory

Acknowledgement

At the beginning of time, I started as a research assistant at VTI. Time passed, and I enrolled in the PhD program at Orebro¨ University. Time passed again, and again. Finally, quite late in time, I am writing this. There is an end after all!

Along this never-ending journey, I have had the opportunity to meet and discuss my work, and the work of others, with many inspiring and clever colleagues at the Department of Economics of VTI in both Stockholm and Borl¨ange. This thesis is the oﬀspring of seminars and discussions with all of you. Thanks! * I would especially like to thank my two supervisors Lars Hultkrantz and Jan-Eric Nilsson for sharp comments and straightforward cri- tique. And of course Gunnar Lindberg for his patience during this eon of time.

My thanks to Svante Mandell and Jan-Erik Sw¨ard for making my workday enjoyable, willingly reading my manuscripts (Svante), and helping me with econometrics (Jan-Erik).

I gratefully acknowledge that this work would not have been possible without ﬁnancial support from The Centre of Transport Studies (CTS).

During the timespan of this thesis, I engaged in a procreation process with the love of my life, Kicki Fjell, not only once but twice! I am indebted to Kicki, Lova and Laban for being there; you could not do more for me!

La Jaille Yvon, August 2012 Joakim Ahlberg

Contents

1 Introduction 10

2 Auction theory vs. laboratory experiments 11

2.1 Private value auctions 11

2.2 Common value auctions 12

2.3 Interdependent value auctions 13

2.4 Multi-unit auctions 14

3 Essays in the thesis 16

3.1 Essay I - Analysis of discrete multi-unit, common value auctions: A study of three sealed-bid mechanisms 16

3.2 Essay II - Multi-unit common value auctions: A laboratory experiment with three sealed-bid mechanisms 17

3.3 Essay III - Multi-unit common value auctions: An experimental comparison between the static and the dynamic uniform auction 18

4 Lessons learned, policy implications and future research 20

Bibliography 22

Essay 1

Essay 2

Essay 3 1 Introduction

When governments started to realize that well-designed auctions were most likely a better method to allocate resources than the beauty contest,where bureaucrats had to rely on business plans which made the process susceptible to corruption, the incentives for testing auction theory increased. This is especially true for multi-unit auctions, the category into which many of the new auction markets fall.

But, still, there is a vast knowledge gap in this ﬁeld, particularly in cases where values are equal across bidders; that is, when the value depends on some outside parameter, common to all bidders. Such auctions are referred to as common value auctions.

Examples of such auctions are (CO2) allowance auctions, electricity auctions and bond auctions. All three can be modeled as common value auctions. In the allowance auction case, the value is a proxy for the social abatement cost, in the electricity auction the value comes from the electricity price, and in the bond case the value is driven by the interest rate.

The complexity of this type of auctions stems from, ﬁrst, bidders demanding more than one unit and, second, the value being common to all bidders. This makes closed-form equilibrium analysis diﬃcult.

The following sections provide a unifying framework and a summary of the papers of this thesis. Section 2 oﬀers a condensed survey with, ﬁrst, single-unit auctions and, second, multi-unit auctions, where theory is related to experiments. The section also serves as the theoretical scope of the thesis. Then follows a brief summary of the essays in section 3, and last, section 4, on some lessons learnt, concludes this summary.

10 2 Auction theory vs. laboratory experiments

2.1 Private value auctions

Even though auctions have been used since antiquity for the sale of a variety of objects, the modern analysis of auctions as games of incomplete information started with the work of William Vickrey (1961). He introduced the independent private value model, where each bidder’s valuation is independent of the information held by her opponents, and derived equilibrium strategies for the ﬁrst-price auction as well as the open, and the sealed-bid, second-price auctions, when values come from a uniform distribution. 1 The second-price auction, introduced in the same paper, was shown to be an auction form with truthful bidding and eﬃcient outcomes.

Vickrey also recognized that the expected revenues in ﬁrst- and second-price auctions were the same, particularly under arbitrary distributions. This was formally proved in a subsequent paper, Vickrey (1962). 2 The ﬁnding consisted of some special cases of the (today) celebrated Revenue Equivalence Theorem developed, roughly at the same time, by Riley and Samuelson (1981) and Myerson (1981) The theorem states:

Assume each of a given number of risk-neutral potential buyers of an object has a privately known signal independently drawn from a common, strictly increasing, atomless distribution. Then any auction mechanism in which (i) the object always goes to the buyer with the highest signal, and (ii) any bidder with the lowest-feasibly signal expects zero surplus, yields the same expected revenue.

It is a quite remarkable result, that, on average, all auctions with these two properties generate the same revenue to the seller. This result does not only apply to (independent) private value (PV) models, but to more general common value models, provided that the bidders’ signals are independent.

Surprisingly, this result does not hold in the laboratory. Specifically, there should be strategic equivalence between the two most common first-price auctions; the (sealed-bid) first-price auction and the Dutch auction 3 ,aswellas

1 In a ﬁrst-price auction, you pay what you bid (if you win), whereas, in a second- price auction, you pay the second highest bid (if you win). 2 Vickery was awarded the Sveriges Riksbank Prize in Economic Sciences in Mem- ory of Alfred Nobel 1996, to a certain extent because of the two mentioned papers. 3 In the Dutch auction, the auctioneer begins with a (suﬃciently) high price. The price is gradually reduced until a bidder indicates her interest. The object is then sold to this bidder at the given price.

11 the two most common second-price auctions: the (sealed-bid) second-price auction, and the English auction. 4 But Coppinger et al. (1980) ﬁnd that there is no strategic equivalence in ﬁrst-price and Dutch auctions, and that the Dutch auction yields lower prices. Kagel et al. (1987) report failures in the strategic equivalence of second-price and English auctions, with lower prices in the English auction. The English auction is the only auction in which bidding con- verges to the dominant strategic prediction. The rationale for this seems to be the transparency of the dynamic, or open, (ascending) auction, and subjects learn how to behave quickly enough.

2.2 Common value auctions

At the other end of the value spectrum, we have the (pure) common value. In the common value (CV) model, all bidders assign the same value to the unit for sale, but bidders have different private information about the (unknown) value at the time that they bid. Thus, even though the ex-post value is common for everyone, it is unknown at the time of bidding. The CV model was first introduced by Wilson (1969). He also developed the first closed-form equilibrium analysis of the winner’s curse in that article. The core of the winner’s curse (WC) is that a bidder must not ignore the (adverse selection) effect inherent in winning the auction, in order not to pay more than the estimated worth of the object. (Even if a bidder’s estimate is an unbiased estimator of the value, the largest of all the estimators is not. The max function is convex and thus overestimates the value.)

The presence of the WC as an empirical fact was pointed out by Capen et al. (1971) in the context of bidding for oﬀshore oil drilling leases. They claimed that the oil companies suﬀered unexpectedly low returns in oil lease auctions. An oil lease auction is a typical CV auction, since the value of the oil in the ground is essentially the same for all bidders. This type of auction spurred the development of the pure CV model (or the mineral rights model).

A seminal experiment by Bazerman and Samuelson (1983) on the WC showed that, in general, subjects were pretty inclined to overbid, since they overestimated the value and, as a consequence, fell prey to the WC. Subsequent experiments show that the WC is prevalent in many different settings, and that if there is some learning, it is context specific. Even if subjects over time learn to avoid getting a negative profit from the WC in an experimental set-up, this does not carry over to another set-up. For example, in the open

4 In the English auction, the auctioneer starts with a low price and gradually raises it as long as there are at least two interested buyers. The auction stops when only interested buyers remain. This might be the most common auction form used in practice.

12 variant of the second-price auction, the English auction, subjects with higher estimates of the true value get information of the value as a consequence of lower valued bidders dropping out. This alleviates, but does not eliminate, the WC. However, subjects do not transfer this experience to the static, or sealed-bid, variant of the same auction.

2.3 Interdependent value auctions

The PV and the CV models are extreme cases, at either end of the spectrum of the interdependent value model. The value in this model can, but must not, depend on other bidders’ information, or signals. Milgrom and Weber (1982) developed the symmetric model with interdependent values (IV) and affiliated signals, encompassing both the PV and CV models. Affiliated signals play a part in IV models, meaning that bidders’ signals are correlated, positively or negatively. 5 When the signals are affiliated, the revenue ranking theorem (RET) above ceases to hold, since independent signals are assumed.

Their article derives the general revenue ranking, as well as other equilibrium characterizations, of the three most common auction formats; the ﬁrst price, the sealed-bid second price, and the open second price, i.e. the English auction. The main results are that the English auction leads to a higher expected revenue than sealed-bid second-price auctions which, in turn, lead to a higher revenue than ﬁrst-price auctions.

The intuition for this is that, in the open auction with interdependent values, there is a great deal of information revelation as bidders drop out when the price increases. This implies that the price paid will not only depend on private information, but also on all other bidders’ information. The more the price depends on other bidders’ information, the higher will be the price (revenue); that is, the winner’s private information becomes less valuable. (In the second- price auction, the price depends on one other bidder besides the winner, but in the ﬁrst-price auction the price only depends on the winner.)

However, when the revenue ranking from Milgrom and Weber (1982)’s article is experimentally tested in laboratory settings, focusing on pure CV models only, the conclusion is that, just like the RET, it does not hold either. The main reason seems to be that subjects in the experiments often fall prey to the WC, and therefore earn a negative proﬁt. This happens for both inexperienced bidders and those with varying degrees of experience. The overbidding is alleviated in the English auction, thereby giving the subjects a greater proﬁt,

5 If the signals are positively aﬃliated, it roughly means that, if a subset of the signals is all large, then this makes it more likely that the remaining signals are also large.

13 which translates into less revenue. This is contrary to the revenue ranking which implies that the English auction creates a greater revenue. But as the number of bidders increases, the diﬀerence in revenue between auction formats decreases. (A comparison between the English and the ﬁrst-price auction can be found in Levin et al. (1996).)

For the three value structures described above, the subjects generally seem to understand the English auction best in experiments. It is in this auction format that the behavior matches the predicted theory. The rationale is the inherent price discovery mechanism in the English auction, which is absent in sealed-bid formats. This mechanism is important in IV structures, since bidders learn how aggregate demand changes as the auction proceeds. That is, the transparency of the dynamic auction seems to make it relatively easy for subjects to understand that they should not bid above their value, which is the dominating equilibrium in the PV case. (In the PV case, where all subjects know their value, and where this value does not depend on other bidders’ values, the price discovery mechanism has no bearing.)

2.4 Multi-unit auctions

When the US government started selling radio spectrum licenses, and when the design of Internet auctions became important, the theory on multi-unit auctions started to advance. In a multi-unit auction, the implicit meaning is that bidders demand multiple units; nothing drastic happens in the theory when selling multiple units, as long as individuals demand a single unit. The commodities for sale could be either homogenous or complements; the focus here is on homogenous units.

Vickrey (1961) also pioneered in this setting, as he described an efficient mechanism in multi-unit settings in the PV environment, nowadays called the Vick- rey auction. Ausubel (2004) then came up with an open format that has the same outcome as the multi-unit Vickrey auction in a PV setting, and also continues to be efficient in an affiliated value environment, in contrast to the static Vickrey auction.

Manelli et al. (2006) experimentally compare the static Vickrey auction with the Ausubel auction, also known as the dynamic Vickrey auction, in both a PV setting and an IV setting, in which the values are affiliated. They conclude that due to overbidding being slightly more frequent in the Vickrey auction, the revenue from the Vickrey auction is greater, while the efficiency is lower in the Ausubel auction. In the IV setting, they observe less overbidding and a trade-off between efficiency and revenue; the Vickrey auction is more efficient while the revenue is higher in the Ausubel auction.

14 A seminal (game theoretic) article on common value multi-unit auctions is that by Wilson (1979). He found that, in an auction of shares, there existed equilibria with prices lower than if the item were sold as an indivisible unit. Later, Ausubel and Crampton (2002) showed that the eﬃciency of the second- price, multi-unit auction might break down due to demand reduction in an IV model. Demand reduction, which is the phenomenon that bidders reduce demand (on marginal units) in favor of a lower market-clearing price, has since then been shown in several experiments in PV, IV and CV models.

15 3 Essays in the thesis

Three self-contained and self-authored essays, which comprise the thesis, are summarized below and related in the following way. In essay I, a multi-unit CV model is constructed, and three diﬀerent sealed-bid auction mechanisms are analyzed. In essay II, the model from paper I is scrutinized through a laboratory experiment. All hypotheses tested are derived from the analysis of essay I. Essay III also conducts a laboratory experiment, but the analysis dis- tinguishes between the open and the sealed-bid uniform auction mechanisms. In this experiment, many of the hypotheses also come from the analysis in essay I; but also from behaviors and anomalies in the ﬁrst experiment, essay II.

3.1 Essay I - Analysis of discrete multi-unit, common value auctions: A study of three sealed-bid mechanisms

In this paper, we suggest and evaluate a simplified common value model, which is also discrete in both values and bidding. The common value is generated as the sum of two integers, where one integer is independently displayed for each bidder and, first, serves as a signal for the bidder and, second, operates as the bidder’s type in the Bayesian game. Both the value and the signals are thus affiliated. The model has a two-unit demand environment, and the number of bidders is two, three or four. Three formats are considered, the discriminatory, the uniform, and the Vickrey auction.

Since the value of each of the two items is the same for every player, there is no efficiency issue; the relevant task for the study is rather to identify equilibrium strategies and compare the revenues of the three auction formats. In the uniform and the Vickrey auctions, there are no unique strategies, but if bidders use the payoff maximizing strategies we find the (theoretical) prediction that, of the three auction formats with two players, the discriminatory auction gives the highest revenue to the seller, and that the uniform and the Vickrey auction both give zero revenue. The zero revenue equilibria are a consequence of extreme demand reduction. Nonetheless, considering the latter two, the Vickrey auction may be placed above the uniform auction in revenue ranking because of the existence of the demand revealing equilibrium in the Vickrey auction. We also find that, in equilibrium, bidders bid the same amount on both items in the discriminatory auction; a phenomenon we do not notice in either of the other auction formats.

Using these results, we make a contribution to the ongoing discussion of which of the two formats, the uniform or the discriminatory, gives the most revenue.

16 The paper suggests that the discriminatory auction is a better alternative than the uniform auction, especially when there are only two bidders. One reason is the potentially extreme demand reduction in the uniform auction. Even with a slight demand reduction, though, the uniform auction gives less revenue in our model than the discriminatory auction.

3.2 Essay II - Multi-unit common value auctions: A laboratory experiment with three sealed-bid mechanisms

This study features a discrete, in the sense that the value of the unit and bidding are only allowed in integer numbers, common value auction with independent (one-dimensional) private signals, where the seller offers two identical units and the buyers demand both. All three auction formats are tested and subjected to a variation in the number of bidders; 2, 3 or 4 buyer-groups are employed, as are repeating bid rounds (15 - 20 rounds) for each subject. Five main questions are scrutinized. (i) which auction format gives the greatest revenue?; (ii) how does the number of bidders affect revenue?; (iii) is there demand reduction in the uniform and Vickrey auctions?; (iv) what are the implications of repeating the auction several rounds for the subjects, do we see any learning effects?; and (v) is there a winner’s curse; that is, do bidders ignore the informational content inherent in winning, and bid too high? For the first four questions, we have hypotheses derived from the predictions in essay I; for the fifth question there is a behavioral hypothesis of the winner’s curse since the winner’s curse not arise in optimum play, which we search for in essay I.

Starting with revenue, we find that the Vickrey auction always gives the least revenue, regardless of group size. The uniform and the discriminatory auctions run a close race, but due to the non-expected result in 2-player groups, the uniform auction is weakly better. (The hypothesis for the uniform auction was that, in 2-player groups, the subjects would play more according to the demand reduction theory. But, generally, they did not.) Still, for larger group sizes, the difference between them is small. The answer to the second question is without doubt in this setting; the more bidders in an auction, the larger the seller’s revenue. All formats support this statement. Third, we see demand reduction, but we hardly see an extreme demand reduction, that is, zero bidding on the second unit. Fourth, we find that subjects do learn to play equilibrium strategies in the course of the play, at least in the discriminatory auction. Moreover, they continue to learn until the final rounds.

For the last question, we ﬁnd that the winner’s curse is highly present, mostly in the uniform and discriminatory auctions, but also in the Vickrey auction. We distinguish between bidding above the naive, conventional expected value

17 and between the conditioning expected value of winning up to the naive expected value, and ﬁnd that it is twice as common to bid in this ﬁrst interval. This indicates that subjects have a problem understanding the winner’s curse.

From the results, we first notice that as the number of players increases, the pricing rules converge in collecting revenue. When there were only two bidders in the auction, all formats were significantly different in revenue raising; but when there were four bidders, the difference became insignificant. Thus, attracting bidders, or ensuring competition, could be much more important than the selected auction form.

There was especially one odd result in the experiment, namely the high revenue for 2-player groups in the uniform auction. It was rather unexpected because of the anticipated low revenue equilibria outcome in this group. One possible explanation is the competitive element; subjects did not play the theoretical equilibrium at all; they wanted to win the object(s), no matter the costs. Holt and Sherman (1994) explain this as the joy of winning phenomenon in their study. In the present study, it was not only encountered in this particular group size, but was also pretty common in all group sizes in all auction formats.

3.3 Essay III - Multi-unit common value auctions: An experimental comparison between the static and the dynamic uniform auction

It is still an open question whether the open or static format should be used in multi-unit settings, in a uniform price auction. In the private value case, a couple of ﬁeld and laboratory experiments have shown that one must be cautious about using open formats.

The present study conducts an economic experiment in a common value environment, and both the static and dynamic formats are used in two group sizes: 3 and 6-player groups. In letting the larger groups’ conﬁgurations (in own demand) be exactly two times the smaller groups, and letting the supply be equal in both groups, it eﬀectively is as comparing a loose and a tight cap at the same time (if bidding does not adapt to the increasing number of bidders). 1 The loose cap, represented by the 3-player groups, has the relation 2 of supply (numerator) and aggregated demand (denominator), whereas the tight cap, or S 1 the 6-player groups, has the relation D = 4 . Moreover, the two group sizes 1 always have the relation 2 between the large demander (numerator) and the small demander (denominator). The tight cap resembles the European Union Emission Trading Scheme auctions conducted in Great Britain (but open for participants throughout the EU).

The main results from the experiments are;

18 • Subjects’ bids do not decrease in response to an increased number of bidders, contrary to the predictions of theory. Since the bids do not increase either, we conclude that doubling the number of players is equivalent to halving the supply. • Seller revenue is significantly greater in the sealed-bid format. But it comes at a cost in terms of a considerably more negative profit for buyers, and nearly half of the auctions ended with a negative profit for the subjects. • In line with this is the considerably less WC in the open format, both bidding in the winner’s curse interval (see essay II above) and experiencing a negative profit. There is also a notable number of bids above the conventional, naive, expected value, especially in the static format. • The more bidders (or, the tighter the market), the greater the revenue. • None of the formats seem to result in high bids that coincide with individual rationality. Subjects in the open format do perform somewhat better, but not well enough since just 1/5 of all subjects’ first unit bid/dropout is at, or below, the expected value of the unit in the dynamic format. • The demand reduction, measured as the bid spread, is significantly lower in the dynamic format.

Thus, we conclude that in deciding which of the two auction formats of the uniform price auction that is preferred in CV environments, we have to determine if (i) collecting the most revenue or (ii) avoiding the most negative bidder proﬁt is the most important criterion in the choice process. The dynamic auction seems to be a better choice, especially if players are without experience. It facilitates price discovery, thereby alleviating the overly aggressive bidding.

The bottom line is that the choice between the open and sealed-bid formats may be more important than the choice of price mechanism, especially in CV settings.

19 4 Lessons learned, policy implications and future research

Many of the problems discussed in section 2 between theory and experiment, and encountered in a single-unit environment, are also detected in experiments in the multi-unit settings of this thesis. That is, all theoretical predictions from the first essay were not fulfilled in the second essay. And, likewise, not all hypotheses in the third essay were confirmed in the experiment. For example, there was a great deal of overbidding in both experiments and in all auction formats.

The reason for the overbidding is not easy to pinpoint. One rationale is the complexity of the game at hand; the concept of common value is not intuitive for most subjects, or so it seems. Both experiments used inexperienced players. Even though they had several dry-rounds before the experiment began, and had time to learn during these rounds, they had probably never been faced with some of the auction types. But the phenomenon of being a victim of the WC is not unique to inexperienced bidders, Kagel and Levin (2002) have shown that overbidding is a robust feature, also prevalent among professionals. This is consistent with the results in the two experiments in the thesis, i.e. that subjects continue to suﬀer from overbidding in later rounds. The joy of winning is another rationale; the competitive eﬀect takes over the rationality.

Even though the latter two essays diﬀer in their laboratory set-up; for example, in the number of bidders, the demand per bidder, how the value is generated, open and sealed-bid formats etc, the overbidding is constantly there. In the last essay, subjects in the open uniform auction were less prone to overbid as compared to the sealed-bid counterpart, which is best rationalized through the price discovery mechanism this format possesses, inherent from the transparency of the same.

Thus, in conducting a multi-unit CV auction (procurement), the auctioneer should, ﬁrst, attract potential bidders to participate; the number of bidders may be the most important variable when it comes to enhancing revenue. Second, given the number of bidders, the auction designer should take into consideration the choice between the open and sealed-bid formats, the latter producing more revenue, and the former reducing the bidding errors. Reducing bidding errors is paramount in the long run since buyers are not likely to come back if they lose money; hence the two cases mentioned here are intertwined.

Thus, the dynamic uniform auction is a much preferred mechanism over the static counterpart when it comes to many multi-unit auctions. For example, when selling CO2 allowances, the government must look after the industry as a whole, or when selling bonds, take careful account of the welfare in the nation. Here, the dynamic auction oﬀers a solution by means of cushioning

20 the overbidding and helps find the right price for the units for sale. Moreover, concerning procurement, the more risk there is in a sealed-bid tender, the more the final contract tends to be at cost-plus pricing. Thus, since risk is a CV, there could potentially be huge efficiency gains by more often using the dynamic auction in risky procurements.

There is a need for much more theoretical and experimental research on multi- unit auctions in general and CV auctions in particular. Since the WC is by now well known and anticipated in experiments, the next step should be to design experiments in such a way that the overbidding is neutralized; that is, ﬁnd a way of abstracting the joy of winning (or whatever it may be called) from the bids.

21 References

Ausubel, L. M.: 2004, ‘An Efficient Ascending-Bid Auction for Multiple Ob- jects’. The American Economic Review 94(5), 1452–1475. Ausubel, L. M. and P. C. Crampton: 2002, ‘Demand Reduction and Ineffi- ciency in Multi-Unit Auctions’. Mimeographed, Department of Economics, University of Maryland. Bazerman, M. H. and W. F. Samuelson: 1983, ‘I Won the Auction but Don’t Want the Prize’. Journal of Conflict Resolution 27(4), 618–634. Capen, E., R. Clapp, and W. Campbell: 1971, ‘Competitive Bidding in High- Risk Situations’. Journal of Petroleum Technology 23(6), 641–653. Coppinger, V. M., V. L. Smith, and J. A. Titus: 1980, ‘Incentives and Behavior in English, Dutch and Sealed-Bid Auctions’. Economic Inquiry 18(1), 1–22. Holt, C. A. and R. Sherman: 1994, ‘The Loser’s Curse’. The American Eco- nomic Review 84(3), 642–653. Kagel, J. H., R. M. Harstad, and D. Levin: 1987, ‘Information Impact and Allocation Rules in Auctions with Affiliated Private Values: A Laboratory Study’. Econometrica 55(6), 1275–1304. Kagel, J. H. and D. Levin: 2002, Common Value Auctions and the Winner’s Curse. Princton University Press. Levin, D., J. H. Kagel, and J.-F. Richard: 1996, ‘Revenue Effects and In- formation Processing in English Common Value Auctions’. The American Economic Review 86(3), 442–460. Manelli, A. M., M. Sefton, and B. S. Wilner: 2006, ‘Multi-Unit Auctions: A Comparison of Static and Dynamic Mechanisms’. Journal of Economic Behavior & Organization 61(2), 304–323. Milgrom, P. and R. Weber: 1982, ‘A Theory of Auctions and Competitive Bidding’. Econometrica 50(5), 1089–1122. Myerson, R.: 1981, ‘Optimal auction design’. Mathematics of Operations Re- search 6(1), 58–73. Riley, J. G. and W. F. Samuelson: 1981, ‘Optimal Auctions’. The American Economic Review 71(3), 381–393. Vickrey, W.: 1961, ‘Counterspeculation, Auctions, and Competitive Sealed Tenders’. The Journal of Finance 16(1), 8–37. Vickrey, W.: 1962, ‘Auctions and Bidding Games’. in Recent Adwances in Game Theory, Princeton Conference Series 29, 15–27. Wilson, R.: 1979, ‘Auctions of Shares’. Quarterly Journal of Economics 93, 675–689. Wilson, R. B.: 1969, ‘Competitive Bidding with Disparate Information’. Man- agement Science 15(7), 446–448.

22 ESSAY I Analysis of discrete multi-unit, common value auctions: A study of three sealed-bid mechanisms

Joakim Ahlberg

VTI - Swedish National Road and Transport Research Institute, P.O. Box 55685, SE-102 15 Stockholm, Sweden Tel: +46 8 555 770 23. E-mail address: [email protected]

Abstract

This paper proposes a discrete bidding model for both quantities and pricing. It has a two-unit demand environment where subjects bid for contracts with an unknown redemption value, common to all bidders. Prior to bidding, the bidders receive private signals of information on the (common) value. The value and the signals are drawn from a known discrete aﬃliated joint distribution. The relevant task for the paper is to compare the equilibrium strategies and the seller’s revenue of three auction formats. We ﬁnd that, of the three auction formats below with two players, the discriminatory auction always gives the largest revenue to the seller; both the uniform and the Vickrey auction have zero revenue equilibrium strategies that put them further down in the revenue ranking. In equilibrium, bidders bid the same amount on both items in the discriminatory auction; a phenomenon not noted in either of the other auction formats.

Keywords: Laboratory Experiment; Multi-Unit Auction; Common Value Auc- tion

JEL codes: C91; C72; D44

1 Introduction

The research on auctions that involve more than one identical item for sale has recently been getting increasing attention, which is not strange considering the huge amount of money involved in many multi-unit auctions. Auctions for treasury bills, spectrum rights, procurement, and emission permits are just a few examples where more than one identical item is sold at the same time for billions of dollars.

An analytical problem, so far, when these sales are scrutinized is the lack of closed form solutions for the equilibrium bidding strategies. Not only do we

1 ESSAY I [email protected] : 1 E-mail address Joakim Ahlberg 55685, SE-102 15 Stockholm, Sweden C91; C72; D44 Laboratory Experiment; Multi-Unit Auction; Common Value Auc- three sealed-bid mechanisms Analysis of discrete multi-unit, Tel: +46 8 555 770 23. common value auctions: A study of VTI - Swedish National Road and Transport Research Institute, P.O. Box 1 Introduction The research on auctionshas recently that been getting involve increasing more attention,the which than huge is amount one not of strange identical money considering treasury involved item in bills, for many spectrum multi-unit sale rights, auctions.few procurement, Auctions examples and for where emission more permitsbillions than are one of just identical dollars. item a is sold at theAn same analytical time problem, for soclosed far, form when solutions these for sales the are equilibrium scrutinized bidding is strategies. the Not lack only of do we Abstract This paper proposes aIt discrete has bidding a model two-unit demand for environmentunknown both where redemption subjects quantities bid value, and for common pricing. contracts toders with all an receive bidders. private Prior signals toand of bidding, the information the signals on bid- are theThe drawn (common) relevant from value. task The a for value knownthe the seller’s discrete paper revenue affiliated of is three jointformats to auction distribution. formats. compare below We the find with that, equilibrium twolargest of strategies players, the revenue the three and to auction discriminatory thezero auction seller; revenue always equilibrium both gives strategies the that the ranking. uniform put them and In further the equilibrium, down Vickrey bidders indiscriminatory auction auction; the bid a revenue have the phenomenon not sameformats. noted amount in either on of both the other items auction inKeywords: the tion JEL codes: face the problem with multiple units, we also have units that may have the some insights for auctions with many more objects for sale, such as bond or same (common) value for all bidders. For example, in the treasury auction, emission permit auctions, which are both common value auctions. Riksgälden the value is driven by the interest rate, which is common to all bidders, while (the Swedish National Debt Office) has recently communicated that it will in emission permit auctions, the value is a proxy for the social abatement cost. stick to the discriminatory auction, Riksgälden (2007). It also states that it is harder to reject the Vickrey auction, but since this mechanism is not used In many settings, the continuous bidding paradigm is also in doubt, since anywhere in the world, presumably due to its complicated nature, a switch to many auctions are conducted through discrete bids. This is especially true it cannot be recommended. in some Internet auctions. For example at Tradera, a Swedish-based Internet site, the bid-increments for items depend on their value. For values between Since the value of each of the two items is the same for every player, there is SEK 1 (approx 0.15, 0.11) and SEK 10000 (approx 1500, 1075), the no efficiency issue; the relevant task for the study is rather to identify equilib- increment could be as large as 100 percent of the value down to 1 percent rium strategies and compare the revenues of the three auction formats. In the of the value. (It oscillates between these values because different sections of uniform and the Vickrey auction, there are no unique equilibrium strategies, values have different increments, see www.tradera.com.) but if bidders use the payoff maximizing strategies we find that, of the three auction formats with two players, the discriminatory auction gives the high- In this paper, we analyze a discrete (in both values and bidding) common est revenue to the seller, and that the uniform and the Vickrey auction both value model. To focus the attention, the model has a two-unit demand envi- give zero revenue. The zero revenue equilibrium is a consequence of extreme ronment where subjects bid for contracts with an unknown redemption value, demand reduction. We also find that, in equilibrium, bidders bid the same common to all bidders. Prior to bidding, the bidders receive private signals of amount on both items in the discriminatory auction; a phenomenon we do information on the common value. Both the value and the signals are drawn not note in either of the other auction formats. from a known discrete affiliated joint distribution. Our results make a contribution to the ongoing discussion of which of the two The specific set-up used in this paper emanates from an unpublished wind- formats, the uniform or the discriminatory, gives more revenue, suggesting tunnel experiment by Lindén et al. (1996). There, the idea was to find a that the discriminatory auction is a better alternative than the uniform auc- different way of representing the common value and, especially, make it easier tion, especially when there are only two bidders. One reason is the potential for the subjects to understand the winner’s curse problem inherent in com- extreme demand reduction in the uniform auction. Still, even with a slight de- mon value auctions. 1 In the experiment of Lindén, three dice were rolled in mand reduction, the uniform auction gives less revenue in our model than the each round and the common value was generated as the sum of the three dice. discriminatory auction. Related to the first paragraph above, we also see that One of the dice was then shown independently to each player. In this way, many government securities auctions worldwide still use the discriminatory a common value environment with independent private signals was created. mechanism. The main objective was to compare the revenues of the uniform auction, the discriminatory auction and the Vickrey auction in a two-unit demand envi- This research is mainly related to the literature of multi-unit demand auctions ronment. with interdependent values. 2 There is little earlier theoretical work on multi- unit demand, common value auctions against which to directly compare our In the present model, a Bayesian game is constructed for the analysis. The results, except for the theoretical article from Alvares´ and Mazón (2010). They common value is generated as the sum of two integers, where one integer is have a theoretical model similar to this in a continuous setting. Like the present independently displayed for each bidder and, first, serves as a signal for the paper, they find that the discriminatory auction has an equilibrium where bidder and, second, operates as the bidder’s type in the game. Three auc- bidders bid the same price for both units, whereas the uniform auction does tion formats are considered, the discriminatory, the uniform, and the Vickrey not. Moreover, they show that the comparison of the seller’s expected revenue auction. across auction formats only depends on the ratio of the precision of private information to the precision of public information. Even though the model is simple, with only two units for sale, it may help give 2 The earlier work on optimal auctions and revenue comparison by Vickrey (1961), 1 The winner’s curse is defined as the failure to understand that the announcement Myerson (1981) and Riley and Samuelson (1981), which focused on independent of winning leads to bad news, if not accounted for when bidding. That is, the private value settings, and the seminal paper on interdependent values and affiliated possibility that, upon winning, one pays more than the estimated worth of the signals by Milgrom and Weber (1982), as well as the first common value paper by object. Wilson (1969), generally assume that the buyers just demand one unit each.

2 3 ESSAY I alden on (2010). They which focused on independent , ´ Alvares and Maz´ alden (2007). It also states that it 3 There is little earlier theoretical work on multi- 2 The earlier work on optimal auctions and revenue comparison by Vickrey (1961), have a theoretical model similar topaper, this in they a continuous setting. find Likebidders the that present bid the the discriminatory samenot. price Moreover, auction they for has show both that anacross units, the auction comparison equilibrium whereas of formats the where the only uniforminformation seller’s to expected auction depends revenue the does on precision the of ratio public of information. the precision of private 2 some insights for auctionsemission with permit many auctions, more which are objects both for common sale, value auctions. such Riksg¨ as bond or (the Swedish Nationalstick Debt to Office) the has discriminatoryis recently auction, harder communicated to Riksg¨ that rejectanywhere the it in Vickrey the will auction, world, presumably butit due since cannot to this be its mechanism recommended. complicated is nature, not a used switchSince to the value ofno each efficiency of issue; the the two relevantrium items task strategies is for and the the compare study same theuniform is for revenues and rather of every the to the player, identify Vickrey three therebut equilib- auction, auction is if formats. there bidders In are use the noauction the unique formats payoff equilibrium with maximizing strategies, two strategiesest players, we revenue the find to discriminatory that, thegive auction of seller, zero gives the and revenue. the three that The high- thedemand zero uniform reduction. revenue and equilibrium We the is alsoamount Vickrey a on auction find consequence both both that, ofnot items in extreme note in equilibrium, in the bidders either of discriminatory bid the auction; the other a same Our auction results phenomenon formats. make we a contribution do formats, to the the ongoing uniform discussionthat or of the which the discriminatory of the auction discriminatory,tion, two is gives especially a more when better revenue, there alternativeextreme are suggesting demand than reduction only the in two uniform themand bidders. auc- uniform reduction, One auction. the reason Still, uniform even is auctiondiscriminatory with the gives auction. a less potential slight Related revenue de- to in themany our first government model paragraph securities than above, the auctions wemechanism. also worldwide see still that use the discriminatory This research is mainly relatedwith to interdependent the values. literature of multi-unit demand auctions Myerson (1981) and Riley and Samuelson (1981) private value settings, and the seminalsignals paper by on Milgrom interdependent values and andWilson Weber affiliated (1969), (1982), generally as assume well that as the the buyers first just common demand one value paper unit each. by unit demand, common valueresults, auctions except for against the which theoretical article to from directly compare our 1075), the 1500, en, three dice were rolled in 2 en et al. (1996). There, the idea was to find a 11) and SEK 10000 (approx 0. In the experiment of Lind´ 1 15, . 0 The winner’s curse is defined as the failure to understand that the announcement increment could beof as the large value. as (Itvalues 100 have oscillates different percent between increments, of these see the values www.tradera.com.) because value different downIn sections to this of 1 paper, percent value we model. analyze To focus aronment the discrete where attention, subjects (in the bid both model forcommon contracts to has values with all a and an bidders. two-unit unknown bidding)information Prior demand redemption to on common envi- value, bidding, the the commonfrom bidders value. receive a Both private known the signals discrete of value affiliated and joint the distribution. signalsThe are specific drawn set-up usedtunnel in experiment this by paperdifferent Lind´ way emanates of from representing the anfor common unpublished the value and, wind- subjects especially,mon make to it value understand easier auctions. the winner’s curse problem inherent in com- 1 face the problem withsame multiple (common) units, value we forthe also value all is have bidders. driven units For byin that the example, emission may permit interest in auctions, rate, have the the which the value is treasury is common a auction, to proxyIn for all the bidders, many social while abatement settings, cost. many the auctions continuous are biddingin conducted paradigm some through Internet is discrete auctions. alsosite, For bids. example the in This at bid-increments doubt, Tradera, is forSEK a since items especially 1 Swedish-based depend Internet true (approx on their value. For values between of winning leadspossibility to that, bad upon news,object. winning, if one not pays accounted more for than when the bidding. estimated That worth is, of the the each round and the commonOne value was of generated the as thea dice sum common was of value then the three environmentThe shown dice. with main independently independent objective to was privatediscriminatory each to signals auction player. compare was In and the created. revenues the thisronment. of Vickrey way, the auction uniform in auction, a the two-unitIn demand the envi- present model,common value a is Bayesian generated gameindependently as is displayed the for constructed sum each forbidder of bidder the and, two and, analysis. second, integers, first, The tion where operates serves formats one as as are integer a considered, theauction. is the signal bidder’s discriminatory, for the type the uniform, in and the the Vickrey game.Even Three though the auc- model is simple, with only two units for sale, it may help give Ausubel and Crampton (2002) generalize earlier multi-unit demand models we analytically solve for equilibrium bid strategies, by using the symmetric by allowing each individual to demand an arbitrary number of units and by Bayesian Nash equilibrium concept, which has otherwise been evasive. How- allowing the valuations to be correlated. They show that demand reduction ever, what we will see in a companion paper, where we experimentally test is prevalent in the uniform auction and that, in many cases, the discrimina- hypotheses excerpted from this model, the way the common value is modeled tory auction outperforms the uniform-price auction, even though they also may be important for the understanding of the same. show that the revenue ranking of the three formats (uniform, discriminatory and Vickrey) is ambiguous. Moreover, since there is a greater chance that the The rest of the article is organized as follows. Section 2 presents the model, stronger bidders’ bids become the market clearing price than the weaker bid- section 3 describes the equilibrium, section 4 contains the results and section ders’ bids, strong bidders shade their bids relatively more than weak bidders. 5 concludes the paper. This, in turn, may cause the stronger bidders to lose units to otherwise weaker bidders.

In their analysis, they assume an infinitely divisible good rather than discrete goods to simplify the calculations, an approach proposed by Wilson (1979). Wilson showed, by means of examples, that the uniform auction is unfavor- 2 The model able for the seller in terms of revenue when compared to the discriminatory auction since it had collusive equilibria. Two other papers that also use the divisible-good are Back and Zender (1993) and Wang and Zender (2002) which 2.1 Preliminaries show that the discriminatory auction yields a unique equilibrium with greater expected revenues than the uniform auction. They also make a point of the fact that insights into their single unit framework cannot be directly repli- The seller has an inelastic supply of two homogenous items to sell and the cated in their multi-unit counterpart. Thus, all these papers are in line with seller’s valuation for both items is zero. There are n bidders in the game. All the present paper. bidders assign the same value to each item, i.e. bidders have flat demand, and this (common) value is generated by means of an integer generator. More For discrete units, there are two papers by Engelbrecht-Wiggans and Kahn, precisely, this is done by making two independent draws from the uniform both dealing with the independent private value case. Engelbrecht-Wiggans distribution with support D = 1, 2,...,6 and adding the two numbers to- { } and Kahn (1998a) consider a uniform price auction of two items and find that, gether. 3 in equilibrium, there is a positive probability of demand reduction, manifested as a bid-shading for each bidder’s lower-value items in such a way that the Thus, the distribution of the value is the sum of two uniform distributions, bids are strictly below their valuation for these items. This can also be seen in V = D1 + D2 . The bidders are not fully informed about this value, however. the interdependent value model in our study. Engelbrecht-Wiggans and Kahn In particular, each bidder i receives a private signal of the valuation by ob- (1998b) present a similar model but with a discriminatory format where M 2 serving one of the integer numbers (draws), before the bids are submitted. ≥ items are for sale and each bidder has a demand for two items. They establish There is equal probability for bidders to see either outcome of the two draws that there is a positive probability that a bidder bids the same for both items, that constitute the value. Moreover, different bidders may (but must not) see even though the bidder values the items differently. This is in accordance with different integer numbers. The signal will be interpreted as the player’s type. our model where the bidders always bid the same on both items when faced Each bidder submits two (discrete) sealed bids, specifying a price, but not a with the discriminatory format. particular unit. 4

Our paper extends the existing literature since it presents a different common value, a multiple demand model with private information. First, there are not many multi-unit models that have a pure common value environment and, 3 This set could be generalized to t, where t Z+, in the below analysis. But ∈ second, the way we generate the common value and the signals is novel and, since we are making use of this smaller set in the equilibrium analysis, we sacrifice third, we model three different (sealed bid) auction formats, in which both the generality for simplicity. bids (prices) and the quantities are discrete rather than continuous sets. Not 4 Like in footnote 3, we restrict the set of bids, but it could be generalized to the all real world auctions are continuous. And, for the two-player environment, real line in the below analysis.

4 5 ESSAY I bidders in the game. All n , in the below analysis. But + Z ∈ and adding the two numbers to- t } 6 5 2,..., t, where , {1 = receives a private signal of the valuation by ob- i D 4 . The bidders are not fully informed about this value, however. 2 D + 3 1 D = This set could be generalized to Like in footnote 3, we restrict the set of bids, but it could be generalized to the gether. 4 3 Thus, the distribution ofV the value is the sum of two uniform distributions, 2 The model 2.1 Preliminaries The seller hasseller’s an valuation inelastic for supply bothbidders of items is assign two zero. homogenous the Thereand items same are this to value (common) sell value to isprecisely, and each generated this the by item, means is i.e. ofdistribution done an bidders with by integer support generator. have making More flat two demand, independent draws from the uniform we analytically solve forBayesian Nash equilibrium equilibrium bid concept, strategies,ever, which what by has using we otherwise will thehypotheses been excerpted see symmetric evasive. from How- in this amay model, be companion the important way paper, the for where common the value we understanding is experimentally of modeled the test The same. rest of thesection article 3 describes is the organized5 equilibrium, as concludes section follows. the 4 Section paper. contains 2 the presents results the and section model, real line in the below analysis. since we are makinggenerality use for of simplicity. this smaller set in the equilibrium analysis, we sacrifice serving one ofThere the is integer equal numbers probabilitythat for (draws), constitute bidders before the to value. the see Moreover,different bids either different integer bidders outcome are numbers. may of The submitted. (but theEach signal must two bidder will not) draws submits be see two interpretedparticular (discrete) as unit. the sealed player’s bids, type. specifying a price, but not a In particular, each bidder 2 ≥ M 4 Ausubel and Cramptonby (2002) allowing generalize each earlier individualallowing multi-unit to the demand demand valuations models an tois arbitrary be prevalent number in correlated. of the They unitstory uniform show and auction auction that by outperforms and demandshow that, the reduction that in uniform-price the many auction, revenue cases,and ranking even Vickrey) the of is though discrimina- the ambiguous. they Moreover, threestronger since formats bidders’ also there bids (uniform, is become discriminatory aders’ the greater bids, market chance strong clearing that bidders price the This, shade than in their the turn, bids may weaker relatively cause bid- bidders. the more stronger than bidders weak to bidders. lose units to otherwise weaker In their analysis, they assumegoods an to infinitely simplify divisible goodWilson the rather showed, calculations, than by an discrete means approachable of proposed for examples, by the that Wilson sellerauction the (1979). in since uniform terms it auction of haddivisible-good is revenue are collusive unfavor- Back when equilibria. and compared Zender Twoshow (1993) to other that and the Wang the papers and discriminatory discriminatory Zender that auction (2002)expected also yields which revenues a use than unique the equilibrium thefact with uniform that greater auction. insights Theycated into also in their make their single a multi-unitthe unit point counterpart. present framework of Thus, paper. all the cannot these be papers directly are repli- inFor line discrete with units, thereboth are dealing two with papersand the by Kahn independent Engelbrecht-Wiggans (1998a) consider and private a value Kahn, in uniform equilibrium, case. price there auction Engelbrecht-Wiggans is of a twoas positive items a probability and of find bid-shading demand that, reduction, forbids manifested are each strictly bidder’s below their lower-valuethe valuation items interdependent for value in these model items. such in(1998b) This a present our can a study. way similar also Engelbrecht-Wiggans model be that and butitems seen Kahn with the are in a discriminatory for format sale where andthat each there bidder is has a a positiveeven demand probability though for that the two a bidder items. bidder valuesour They the bids establish model items the differently. same where This for is thewith both in bidders items, the accordance always discriminatory with bid format. the same on bothOur items paper extends when the faced existingvalue, a literature multiple since demand it model presentsmany with a multi-unit private different information. common models First,second, that there the are have not way a wethird, pure generate we the model common common three value different valuebids (sealed environment and (prices) bid) and, the auction and formats, signals the inall is quantities which real novel are both and, world the discrete auctions rather are than continuous continuous. sets. And, Not for the two-player environment, 2.2 The game of the payoff function.

Since players are not certain of the characteristics of the other players, viz. the signals, we have an incomplete information game. Hence, we model the 2.3 The payoﬀ game as a Bayesian game.

Such a game consists of: The value for each bidder is the realization of the two random variables drawn from D; hence V = d1 + d2. Moreover, denote by c i the two-vector of com- − A nonempty, finite set of players N = 1,...,n , where n is the number of peting bids facing player i. This is obtained by rearranging the 2(N 1) bids • { } − bidders. (The seller has no active part in the game.) aj of players j = i N in decreasing order and selecting the first two of � ∈ A nonempty, finite set Ω of possible states of nature, each of which is a these. Then, the number of units that player i wins is just the number of • description of the relevant characteristics of all players. If we denote the competing bids she defeats. (Ties will be randomly broken. This will be the outcome of the two random variables as d =(d ,d ) 1,...,6 2 and, for rule henceforth for all kinds of ties, such as the highest losing bid etc.) 1 2 ∈{ } each player i N, a specification of the observed integer as oi 1, 2 , the set of states consists∈ of: ∈{ } The price paid in the each auction is a function mapping actions and auction forms into a price, i.e. pl : A R where l U, D, V ; U stands for 2 → ∈{ } Ω= ω =(d, o):d =(d1,d2) 1,...,6 , the uniform, D for the discriminatory and V for the Vickrey pricing rule, { ∈{ } o =(o ,...,o ) 1, 2 n respectively. 1 n ∈{ } } and for each player i N there exists: In the discriminatory auction, each player pays an amount equal to the sum ∈ of her bids that are deemed to be winning - that is, the sum of her bids that 2 (1) An action set ai =(ai,1,ai,2) Ai = Z+ consisting of bids, rearranged so ∈ are among the two highest of the N 2 bids submitted in total. Formally, if that it satisfies ai,1 ai,2, for the two items being auctioned. × exactly ki of player i’s two bids, bi,1 and bi,2, are among the two highest of all (2) A set of signals/types≥ T = D = 1,...,6 and a signal function/type { } bids received, then player i pays function τi :Ω Ti assigning a signal to each state of the nature. Thus, if the state of nature→ is ω =(d, o), player i gets to see the signal o 1, 2 i ∈{ } which has the outcome doi 1,...,6 . Consequently, the signal function ki ∈{ } D is defined, for each ω Ω, by τi(ω)=doi , the outcome of the observed pi (b)= bi,j. (1) ∈ j=1 integer. Thus, τi has full range. (3) A probability measure, which is common knowledge and equal for all play- In all three payment rules, we adopt the convention that if ki is equal to zero, ers, μ :Ω [0, 1] over the states of the nature. Since the integers are gen- the sum will be zero. erated independently→ and the outcome observed is independently drawn, all states of nature have the same probability: Both objects are sold at the market clearing price in the uniform auction. This price is defined to be equal to the highest losing bid, that is 1 2 1 n ω =(d, o) Ω: μ(ω)= . ∀ ∈ 6 · 2 U pi (b) = max bi,2,c i,3 ki ki (2) { − − } (4) A utility function, or payoff function ui : A Ω R, where A = A1 × → × where c i,3 is set to zero. (This happens when ki = 0, but then the player gets An, assigning an associated payoff to each profile of actions and each − ···×state of the nature. zero units and is therefore not supposed to pay anything anyway.)

A pure strategy for player i assigns a bid to each of his or her signals, that In the Vickrey auction, a player who wins ki units pays the ki highest losing is a function bi : Ti Ai mapping signals into actions, or, more speciﬁcally, bids of the other players - that is, the ki highest bids not including her own. 2 → Hence, the winner is asked to pay an amount equal to the externality she bi : 1,...,6 Z+. { }→ exerts on other competing bidders. Thus, if player i wins ki units, then the The game is now fully speciﬁed and we continue with a more careful description amount she pays is

6 7 ESSAY I (1) (2) 1) bids − stands for N U highest losing units, then the i }; is equal to zero, k i i k k the two-vector of com- U, D, V wins i i −i c ∈{ wins is just the number of }k i l = 0, but then the player gets for the Vickrey pricing rule, k i i k units pays the 3− . V i, i i,j , are among the two highest of all − k b 2 where i, highest bids not including her own. ,c i b 2 =1 k i i, j R 2 bids submitted in total. Formally, if k 7 × and → 1 b)= N ( i, A b D i : p l p b) = max{b pays . Moreover, denote by ( 2 in decreasing order and selecting the first two of i U i d i. This is obtained by rearranging the 2( p N + 1 ∈ d i’s two bids, i = for the discriminatory and �= V j D is set to zero. (This happens when of player 3 i k −i, ; hence c D of players j a Hence, the winnerexerts is on asked other to competing pay bidders. an Thus, amount if equal player to the externality she peting bids facing player In all three payment rules, we adopt the convention that if 2.3 The payoff The value for each bidderfrom is the realization of the two random variables drawn of the payoff function. amount she pays is zero units and is therefore not supposed toIn pay the anything Vickrey anyway.) auction, a player who wins bids of the other players - that is, the bids received, then player the sum will be zero. Both objects are sold atprice the is market clearing defined price to in be the equal uniform auction. to This the highest losing bid, that is where these. Then, thecompeting number bids of she units defeats.rule that (Ties henceforth for player will all be kinds randomly of broken. ties, This suchThe will as price the be paid highest the tion losing in bid forms the etc.) into each a auction price, is i.e. a function mapping actions and auc- the uniform, exactly respectively. In the discriminatory auction,of each her player bids pays that anare are amount among deemed equal the to to two be highest the winning of sum - the that is, the sum of her bids that × 2} 1 1, A 2}, the = ∈{ and, for , 1 i 2 A o 6} ∈{ i o . is the number of n 1,..., n R, where signal function/type 1 2 ∈{ · → ) 2 2 Ω ,d and a 1 6 1 }, where × , d 2 , the outcome of the observed 6} A } gets to see the signal i 6 : o =( consisting of bids, rearranged so i )= d i d 2 + ω u ,..., Z 6}. Consequently, the signal function states of, nature each of which is a {1,...,n μ( )= {1 1,..., = ω = i 6 ( i = ), player A } τ ∈{ N n 1,..., Ω: ) ∈ } D 2 d, o 2 ∈ ) 2 ∈{ ,d = ) 1, i, 1 , for the two items being auctioned. i =( Ω, by assigns a bid to each of his or her signals, that o d 2 ,a T i, d i d, o ω 1 ∈{ ∈ a payoff function mapping signals into actions, or, more specifically, there exists: i, =( assigning a signal to each state of the nature. Thus, ) a i ω n ≥ =( i N A T 1 ω has full range. i, =( ∈ ):d a i i 1] over the states of the nature. Since the integers are gen- ∀ → . i , a specification of the observed integer as → τ , a i 2 + ,...,o N d, o [0 T 1 :Ω o : ∈ i → =( i signals/types i τ b =( , assigning an associated payoff to each profile of actions and each n o 6}→Z :Ω A μ action set utility function, or probability, measure which is common knowledge and equal for all play- Ω={ω ,..., ers, erated independently and theall outcome observed states is of independently nature drawn, have the same probability: that it satisfies state of the nature. integer. Thus, ···× is defined, for each which has the outcome if the state of nature is function {1 A nonempty, finite set of players set of states consists of: each player bidders. (The seller hasA no active nonempty, part finite in setdescription the of Ω game.) the ofoutcome relevant possible of characteristics the of two all random variables players. as If we denote the : i (3) A (4) A (1) An (2) A set of is a function A pure strategy for player and for each player 2.2 The game Since players are notthe certain signals, of we the havegame an as characteristics a of incomplete Bayesian the information game. other game. players, Hence,Such viz. we a model game the consists of: The game is now fully specified and we continue with a more careful description b • • kj = 1, the rest, i =j N have ki = 0. Hence, for all i N ki � ∈ ∈ V pi (b)= c i,2 ki+j. (3) − − 0 if b = 0, j=1 i,1 ui(b, ω)=⎧ ⎨⎪ d1 + d2 if bi,1 > 0. A crucial variable in the payment rule above and in the utility function below ⎩⎪ is the number of item(s) won, which is a function of the bids k = k (b). It (3) B1(b) B2(b) >B3(b) 0. Now both items are sold at the price B3(b). i i ≥ ≥ takes the value of zero, one or two depending on whether the player wins zero, Hence, for all i N ∈ one or two items, respectively. 0 if bi,1 c i,2 and bi,2 c i,1. − l l ⎪ − ui(b, ω)=ki(b)[d1 + d2] pi(b). (4) ⎪ − ⎩⎪ Note that B3(b) = maxj N bj,k +1 , the same operator as in the pricing ∈ { i } l rule. If a bidder achieves zero items, then ki is zero and hence, pi becomes zero; thus she also obtains zero payoff. (4) B1(b) >B2(b)=B3(b) > 0. Here, we have a clear winner for one of the items, but there is a tie for the second item. Let m be the number of bids that is equal to B (b). Then, for all i N 2 ∈ 2.4 Example: Payoff in the uniform price auction 0 if bi,1 c i,1 c i,2 >bi,2 ⎪ − − ≥ − For simplicity, we will adopt the rule that if a player bids zero on one or both ⎪ 1 ui(b, ω)=⎪ [(d1 + d2) B3(b)] if c i,1 >bi,1 c i,2 >bi,2, of her bids, the interpretation will be that she does not want those items. ⎪ m − − ⎨⎪ − ≥ Thus, if all bid zero, the seller will not sell the items. But a zero payment is 2 m [(d1 + d2) B3(b)] if c i,1 >bi,1 = bi,2 c i,2, 5 ⎪ − − ≥ − not ruled out. ⎪ ⎪ m+1 [(d + d ) B (b)] if b >b c c . ⎪ m 1 2 3 i,1 i,2 i,1 i,2 ⎪ − ≥ − ≥ − 2 ⎪ Denote by B(b) R+ the vector of all bids in decreasing order, where B1(b) is ⎩⎪ ∈ (5) B1(b)=B2(b)=B3(b) > 0. Once more, there are multiple potential the highest bid and B2n(b) the lowest. According to equation 2 above, we have U winners for the items. Hence, if m denotes the number of those, we have, the following payment rule for the uniform auction: pi = max bi,2,c i,3 ki ki. { − − } for all i N ∈ There are five different cases that can arise: 0 if bi,1 bi,2, ∈ ⎪ − ≥ − ≥ − no one gets (wants) the items, i.e. ki = 0 for all i N, and, therefore, ⎨ 2 ∈ [(d1 + d2) 2B3(b)] if bi,1 = bi,2 c i,1 c i,2. he/she pays nothing. Hence, ui(b, ω) = 0 for all i N. m − − ∈ ⎪ − ≥ ≥ (2) B (b) > 0, B (b) = 0. Then, only one item is sold, and its price, the ⎪ 1 2 In this example, where⎩⎪ B (b) is the price to pay for each item, one can see the highest losing bid, is zero. The bidder, say j, who wins the object has 3 rationale for demand reduction which means that a bidder has incentives to shade the bid for the second unit. 6 In a uniform price auction, the shading 5 That is, the bids will count when finding the price for the units, but not as an of bids for units other than the first results from the fact that, with positive active bid. For example, in the Vickrey auction, where the winner always pays a probability, the bid on the second unit may determine the price paid for the price that comes from a bid that someone else has placed, it could be a zero payment first unit (if it becomes the market-clearing price). if the price setting bid is zero. A price setting bid equal to zero is also possible in the uniform auction, but not in the discriminatory auction, since the winner pays 6 The issue of demand reduction has been studied in more detail by Ausubel and her own bid in this auction, and a zero bid is defined as not wanting the unit. Crampton (2002).

8 9 ESSAY I b). ( 3 B . , , , , . , 2 2 2 2 2 2 1 i, i, i, −i, −i, −i, −i, c c c >b >b >b ≥ ≥ 2 2 b >b −i, −i, −i, = 0, > c >b 1 1 c >c − − i, i, i, b b i, i, i, b b c c b b b b 1 1 2 if i, i, i, if b b b if N 2 d if ∈ b)] if b)] if ( i + ( b)] if 3 b) if = 0. Hence, for all b)] if b)] if 3 ( ( denotes the number of those, we have, 1 ( ( }, the same operator as in the pricing i 3 B 3 3 3 d 0 if B b) if 9 k b) if In a uniform price auction, the shading ( 2 B +1 B m ( B B − i 3 ⎧ ⎪ ⎨ ⎪ ⎩ 3 2 6 ) − − B j,k − − 2 B 0. Once more, there are multiple potential ) ) − d ) ) 2 2 have − )= {b 2 2 ) − d d 0. Here, we have a clear winner for one of the 2 0. Now both items are sold at the price ) d d + > ) 2 d N ∈N 2 1 + + j b, ω d > + + ≥ d ( + b) 1 1 ∈ i 1 1 b) is the price to pay for each item, one can see the ( [(d + ( 1 b) + b) u 3 j 3 ( ( 1 [(d [(d b). Then, for all 1 [(d [(d 3 +1 3 B ( d m B 2 1 d �= m m 2 1 2 2(d 0 ( 0 N m m m ( 0 i B ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ∈ ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ >B b) = max i ( b)= 3 b)=B b) ( )= )= ( ( )= 2 B 2 2 N B B b, ω b, ω ∈ b, ω ( ( ( i i i >B i ≥ u u u b)= b) b) = 1, the rest, ( ( ( 1 1 1 j winners for the items. Hence,for if all k B B rule. B Hence, for all that is equal to items, but there is a tie for the second item. Let Note that The issue of demand reduction has been studied in more detail by Ausubel and (3) (5) (4) In this example, where of bids for unitsprobability, other the than bid the onfirst first the unit results second (if from unit it the may becomes fact determine the that, the market-clearing with price price). positive paid for the 6 Crampton (2002). rationale for demand reductionshade which the means bid that for a the bidder second has unit. incentives to , . i N (4) (3) }k b) is i ( b). It ∈ k 1 ( i i 3− B k −i, = ,c i 2 k i, becomes zero; l i {b , and, therefore, p 0) for all , N . N ∈ i = max = (0 ∈ . i i U i b b) p ( , who wins the object has l i j p . − +j i ] 2 d = 0 for all 2−k i + is zero and hence, −i, k ) = 0 for all 1 c i k i 8 =1 k b, ω j b)[d ( i ( i u k b)= ( )= V i p b, ω ( b) the lowest. According to equation 2 above, we have l i ( u the vector of all bids in decreasing order, where 2n b) = 0. Then, only one item is sold, and its price, the 2 + B ( 2 R B ∈ 5 0, b) ( > B b) = 0. Everybody bids zero. Then, since b) ( ( 1 1 highest losing bid, is zero. The bidder, say no one gets (wants) the items, i.e. B B he/she pays nothing. Hence, That is, the bids will count when finding the price for the units, but not as an (1) (2) the following payment rule for the uniform auction: takes the value of zero,one one or or two two depending items, on respectively. whether the player winsThus, zero, the payoff isitems, the minus number the of price items the won winner multiplied must by pay the for value them. of these Denote by 5 active bid. For example,price that in comes the from a Vickreyif bid auction, that the someone where price else has the settingthe placed, bid winner it uniform could is always auction, be zero. pays a buther A zero a not own payment price in bid setting in the bid this discriminatory equal auction, auction, to and since zero a the is zero winner bid also pays is possible defined in as not wanting the unit. There are five different cases that can arise: the highest bid and thus she also obtains zero payoff. 2.4 Example: Payoff in the uniform price auction For simplicity, we will adoptof the rule her that bids, ifThus, a the if player interpretation bids all will zero bidnot on zero, be ruled one the out. that or seller both she will does not sell not the want items. those But items. a zero payment is If a bidder achieves zero items, then A crucial variable in theis payment the rule above number and of in item(s) the utility won, function which below is a function of the bids 1 In other words, a bidder’s own bid influences the price paid for all units. The τi− (ti), thus: demand reduction property may also result in a bidder not winning (or bidding μ(ω) μi(ω ti)=μ(ω ti)= 1 . on) her second object, even though she could have made a positive profit on | | μ(τi− (ti)) that object. This complicates the analysis and can, as we will see, give rise to This is identical across players, only the types matter. some extreme equilibrium strategies. Thus, upon observing her own type, a player can also update her beliefs about the opponents’ types by Bayes’ rule. When i observes ti, the application of Bayes’ rule yields the conditional probability of the opponents’ types, given 3 Equilibrium i’s type, i’s posterior belief μi, by

ω n t μ(ω) A strategy profile b =(b1,...,bn)isaBayesian(-Nash) equilibrium if, for each ∈∩j=1 j μi(t i ti)=μ(t i ti)= player i N and for each of her types ti Ti, the bid bi(ti) is optimal in the − | − | � ω t μ(ω) ∈ ∈ ∈ i sense that it maximizes her expected payoff, given the bidding functions of n 1 1 2 − 1 1 � n 1 1+ n −1 all types in t i are equal to ti, her opponents and her updated beliefs conditional on her signal. 2 − 2 − 6 ⎧ · · − n 1 1 1 = ⎪ − n 1 At least one type in t 1 is different from ti. ⎪ x · 2 − · 6 − Let the vector of strategies of all players except player i be denoted by b i = ⎨⎪ − � 1 � 1 b ,...,b ,b ,...,b . Let also the vector of all players’ type except player 2n 1 6 all types in t i are different from ti. 1 i 1 i+1 n ⎪ − · − { − } ⎪ i’s type be denoted by t i =(t1,...,ti 1,ti+1,...,tn). Then, since all types ⎪ − − where x⎩ 0,...,n 1 is defined as the number of players of the same type have positive probabilities, due to the full range of the type function τi, the as player∈{i. The first− equation} in the second row works as follows: If all other bid/strategy bi is a (pure strategy) Bayesian equilibrium if player i maximizes players, except player i, see the same value as t , two things can occur. Either her expected utility conditional on ti for each ti: i 1 they all see the same integer as player i, which happens with probability n 1 , 2 − or at least one of the others sees a different integer with the same value as ti, n 1 6 6 6 6 2 − 1 1 which happens with probability n −1 . (The first term is the probability l 2 − 6 π (bi(ti)) = μi(t i ti)E[ui(b, v)]. (5) · ··· ··· − | that at least one player sees a different integer and the second term is the t1=1 ti 1=1 ti+1=1 tn=1 − probability of that integer having the same value as ti.) where μi(t i ti) is the posterior belief, see below, about the other types con- − ditional on the| player’s own type, E is the expectation operator, and the The second equation says that if one of the non-i players sees a different value 7 n 1 1 1 summation is over each of player i’s opponents. than player i, the belief for player i becomes − n 1 . In the last row, all x · 2 − · 6 non-i players see a different value than player� i,� and, since we only have two 1 1 integers, they all see the same integer. This occurs with probability n 1 . 2 − · 6 3.1 Posterior beliefs 3.2 Conditional expected value functions 1 Since for all i, τi− (ti) ti Ti is a partition of Ω, we can identify each ti Ti with ∈ the cell of the{ partition} of Ω where the signal t is received. The interpretation∈ i Since players only get to see one integer, i. e. their signal, they will have to we follow is that once a player i receives the signal ti Ti, she deduces that 1 ∈ use the expected value when calculating their value, which is v(ti)=ti + the state is in the set τ − (t )= ω Ω τ (ω)=t . Her posterior belief about i i i i 1 6 t = t + 7 , that is, the value of her signal plus the expected value of the state that has been realized{ can∈ then| be updated} by Bayes’ rule; that 6 j=1 j i 2 1 the other integer. But in a Bayesian game, they will also need to calculate their is, she assigns to each state ω t− (t ) the probability of ω conditional on � ∈ i i competitors’ value, given their own signal. This conditional expected value for the other players is dependent on how many players there are in the auction. 7 There is also a restriction on the summations due to the fact that only two integers are used. That is, all players together can never have more than two different values The fact that induces this is that they can all see the same integer or dif- for their signals. ferent integers. The more signals (players), the more accurate becomes the

10 11 ESSAY I , , i 1 + . t for 1 6 1 i n− · 2 . i 1 t − 1 n 2 )=t i t ( . v i t , i t . In the last row, all , the application of i 1 6 t · is different from 1 .) . i − 1 t n )) 2 −1 i t t · ) ( players sees a different value � ω conditional expected value i , two things can occur. Either −1 i observes i i, and, since we only have two x −1 μ( t τ n are different from are equal to i � i i μ( − − . (The first term is the probability ) t t 1 6 ω ) · )= ω i μ( i, which happens with probability 1 t | j 11 1 − t μ( 1 − ω becomes i n =1 2 i n j n− ∈t μ( 2 ω ∈∩ ω � , by all types in At least one type in all types in )= i i � t 1 6 μ | · ω is defined as the number of players of the same type ( )= i 1 i −1 1 6 t μ 1 1} | i, see the same value as n− · − 2 n −i 1 − 2 t 1 n− , that is, the value of her signal plus the expected value of 2 μ( 7 2 1 6 · 1+ · · + � )= ,...,n 1 1 i i x t −1 i, the belief for player 0 − − t 1 1 n | n n i 2 2 � i. The first equation in the second row works as follows: If all other = − i’s posterior belief ∈{ t j ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ( t i x players see a different value than player ), thus: μ = i =1 i t 6 j ( � −1 i 1 6 the other players is dependent on how manyThe players fact there that are inferent induces the integers. this auction. The is more that signals they (players), can the all more see accurate the becomes same the integer or dif- This is identical across players, only the typesThus, upon matter. observing her ownthe type, a opponents’ player types can also by update Bayes’ her rule. beliefs about When that at leastprobability one of player that sees integer having a the different same integer value as and the second term is the the other integer. But in acompetitors’ Bayesian value, game, given they their will also own need signal. to This calculate their integers, they all see the same integer. This occurs with probability non- 3.2 Conditional expected value functions Since players only getuse to the see expected one value integer, i. when e. calculating their their signal, value, they which will is have to where τ they all see the same integer as player or at least onewhich of happens the others with sees probability a different integer with the same value as The second equation says that if one of the non- Bayes’ rule yields thei’s conditional type, probability of the opponents’ types, given than player players, except player as player = (5) i − with , the i b i about ; that τ T ∈ maximizes if, for each i t i )]. conditional on b, v ( ω i Bayes’ rule ) is optimal in the , she deduces that u i i [ t T be denoted by ( E i posterior belief i ) b i ∈ t ). Then, since all types | i n t −i t ( }. Her i i 7 : μ t is received. The interpretation i ,...,t t i , the bid t =1 i 6 +1 n i T )= t ω ,t ∈ ( i ··· i ) the probability of −1 is the expectation operator, and the τ t i i t =1 for each ( 10 Ω| 6 E i +1 t i −1 i ∈ t t i’s opponents. receives the signal ,...,t =1 1 Bayesian(-Nash))isa equilibrium ∈ i {ω 1 t 6 n − i ω t =( is a partition of Ω, we can identify each )= i i i t ··· ( − ,...,b ∈T 1 t }. Let also the vector of all players’ type except player 1 i =1 6 t b − n 1 i t } τ ) i =( t ( b )) = ,...,b i −1 i t is a (pure strategy) Bayesian equilibrium if player ( ) is the posterior belief, see below, about the other types con- +1 i i i {τ i and for each of her types b b t | ( i, ,b l N −i π −1 t i ( ∈ i i μ ,...,b 1 There is also a restriction on the summations due to the fact that only two integers player i’s type be denoted by the state is in the set sense that ither maximizes opponents her and expected her payoff, updated given beliefs the conditionalLet on bidding the her functions vector signal. of of strategies of all players except player {b we follow is that once a player ditional on thesummation player’s is own over each type, of player her expected utility conditional on 3.1 Posterior beliefs Since for all 7 In other words, ademand bidder’s reduction own property bid may also influences resulton) the in her price a bidder second paid not for object, winningthat (or all even object. bidding units. though This The she complicatessome could the extreme analysis have equilibrium and made strategies. can, a as positive we profit will on see, give rise to 3 Equilibrium A strategy profile are used. That is, allfor players their together signals. can never have more than two different values the cell of the partition of Ω where the signal bid/strategy have positive probabilities, due to the full range of the type function where the state that has been realized can then be updated by is, she assigns to each state l conditional expected value. That is, with many bidders, we approach the true π (bi(ti)) = (9) value. This is an application of information aggregation, studied by Wilson 6 6 6 6 l (1977). The conditional expected value is defined as: μ(t i ti)[v(t i ti)ki(bi� ,b i) pi(bi� ,b i)]. ··· ··· − | − | − − − t1=1 ti 1=1 ti+1=1 tn=1 − The summation is over each of player i’s opponents. 8 vi(t i ti)=v(t i ti) − | − | n 1 1 7 2 − 1 n 1 (ti + )+ n −1 2ti all tj = ti, 2 − 2 2 − · = ⎧ (6) 4 Results ⎨⎪ ti + tj (where tj t i) some tj = ti, ∈ − � ⎪ 2n 1 1 7 ⎩ n−1 ti + n 1 all tj = ti, 2 − 2 − 2 = ⎧ (7) Since the value function is symmetrical and since we have a symmetrical joint ⎨⎪ ti + tj (where tj t i) some tj = ti. distribution, only types will matter when bidding; thus, we will look for a ∈ − � symmetrical equilibrium. We start with a two-player game. ⎩⎪ The first row in equation (6) says that if the non-i players are of the same type, ti, as player i, two things can happen. Either they see the same integer 1 4.1 Two players as player i, which occurs with probability 2n 1 , or at least one of them sees a − 7 different integer. The value for the former becomes ti + 2 for player i, while the value for the latter becomes t + t =2t . i i i For the case with two players, the objective function (eq. 9) to optimize for player i N, for each ti, and for auction formats l D,U,V , is In the second row of the same equation, we see the value if one player, or ∈ ∈{ } both, is of another type than player i. Then, since there are only two distinct integers, the value becomes the sum of the integer values. Equation (7) is just 6 l 1 a simplification. π (bi(ti)) = [(ti + tj)ki(bi(ti),bj(tj)) pi(bi(ti),bj(tj))] t =1,t =t 12 − j j � i 7 3 7 The above term expectation means the expectation of the possible outcomes + ti + ki(bi(ti),bj(ti)) pi(bi(ti),bj(ti)) (10) of the integer values. From player i’s perspective, if we also take expectations 12 2 4 − over all t i, we get the expected value for a competitor given player i’s type. − where bj(ti)=bi(ti) for all ti Ti by the symmetry conjecture. That is, given That is, we must combine the posterior beliefs with the conditional expected equal signals, different players∈ choose the same strategy. value to get the expected value for any competitor. Hence, the expected value for a competitor for player i is defined as Since the pricing rule is different for the three auction formats, which implies different strategies for each auction, we have to separate the analyzes of the the three formats. We will start with the discriminatory auction.

Et i [v(t i ti)] = μ(t i ti)v(t i ti). (8) − − − − | t i | | �− 4.1.1 The discriminatory auction If we also take the expectation of all t , the terms will sum up to 7 as they i In this auction, conditional on winning, for each item won, every bidder pays should. But for any given t , the expected value for a competitor will not be i the price of her bid on that item (cf. equation (1) on page 7). ti +7/2 since we have to take into account that the competitor may get the signal from the same integer as player i. Proposition 1 The unique pure strategy Bayesian equilibrium proﬁle of this game is Equation 5 can then be stated as, for l D,U,V , the bid b is a (pure ∈{ } i strategy) Bayesian equilibrium, if player i maximizes her expected utility con- 8 With the restriction that all players together can never have more than two ditional on ti for each ti: diﬀerent values for their signals.

12 13 ESSAY I (9) (10) )]. i − ,b ))] � i j b t ( ( l i j }, is p )) ,b i t ) − i ( t ) j ( i −i ,b b ) ( D,U,V i i ,b t � i p ( 8 b i ( ∈{ b i − ( l k i ) p )) i j t t | − ( j −i t )) i ( ,b t ) ( i t j )[v ( i i t ,b i’s opponents. | b ) i 13 ( i i − t t k ( i ) by the symmetry conjecture. That is, given b j μ( i ( t i T k =1 6 + n ∈ i t i 7 4 t [(t ··· + 1 12 , and for auction formats i i =1 t i t t 6 3 2 +1 �= i j t 6 ) for all i =1 t =1,t 1 6 The unique pure strategy Bayesian equilibrium profile of this 7 j ( 12 t − i i b t + , for each )) = i ··· )) = i )= t N i t ( t i ( =1 6 i ( 1 b ∈ t j b ( i l b ( l π π With the restriction that all players together can never have more than two game is equal signals, different players choose the same strategy. Since the pricing ruledifferent is strategies different for for the eachthe three auction, three auction we formats. formats, have We which will to implies start separate with the the analyzes discriminatory of auction. the 4.1.1 The discriminatory auction In this auction, conditionalthe on price winning, of for her each bid item on won, every that bidderProposition item pays (cf. 1 equation (1) on page 7). where 8 4 Results Since the value function isdistribution, symmetrical only and since types wesymmetrical have will equilibrium. a matter symmetrical We start joint when with bidding; a two-player thus, game. we will look4.1 for a Two players For the case withplayer two players, the objective function (eq. 9) to optimize for different values for their signals. The summation is over each of player (8) (6) (7) i, while i’s type. is a (pure i b for player 7 2 , . i i t t + , , i i players are of the same i }, the bid �= �= t t t i j j t t = = j j t t , or at least one of them sees a D,U,V 1 all all . 1 i n− ) some ) some t 2 ∈{ i maximizes her expected utility con- −i −i i t t l . 2t , the terms will sum up to 7 as they ) i =2 · i. i t ∈ ∈ i t | 12 t j j i. Then, since there are only two distinct 1 −1 t t 1 −i − + t n i’s perspective, if we also take expectations − i ( 2 n t 2 v ) 7 2 i t 1 | is defined as )+ (where (where 1 , the expected value for a competitor will not be n− i i 7 2 −i 2 t t + μ( + : i i i i j j t ) t t t t i ( − t t � 1 1 | − i −1 + + 1 n n− n i i − i, two things can happen. Either they see the same integer 2 2 2 t t t ( )] = i v ⎧ ⎪ ⎨ ⎪ ⎩ ⎧ ⎪ ⎨ ⎪ ⎩ t | for each = = i −i )= t i t , we get the expected value for a competitor given player t ( | i, which occurs with probability v −i [ t −i i , as player t − 2 since we have to take into account that the competitor may get the i ( t t i / v E +7 i the value for the latter becomes strategy) Bayesian equilibrium, if player ditional on t In the secondboth, row is of of the anotherintegers, type same the than equation, value player becomes we thea see sum simplification. of the the value integer values. if Equation one (7) is player, just The or above term expectationof means the the integer expectation values. Fromover of player the all possible outcomes If we also take the expectation of all as player conditional expected value. That is,value. with This many is bidders, we an(1977). approach The application the conditional true of expected information value aggregation, is defined studied as: by Wilson should. But for any given The first row intype, equation (6) says that if the non- different integer. The value for the former becomes signal from the same integer as player Equation 5 can then be stated as, for That is, we mustvalue combine to the get posterior the beliefs expectedfor with value a for the competitor any conditional for competitor. expected player Hence, the expected value ti ti Proposition 2 b∗(t )=(t + ,t + ), (11) i i � 3 � i � 3 �

b∗(ti)=(b1∗,b2∗)=( Et i [v(t i ti)] , 0). where x is a ceiling function which maps x to the smallest following integer, � − − | � � � i.e. x = min n Z n x . Proof 2 Suppose that player j utilizes b∗(tj). Any attempt to win 2 units � � { ∈ | ≥ } for player i would either make her second unit bid, or the first unit bid of Proof 1 See proof 8 in the Appendix on page 23 player j, set the price, and since the first unit bid from player j is b∗(t ) 1 j ≥ Et j [v(t j 1)] =4, player i must bid at least 5 to win. The payoff for using � − − | � If we then assume that there is equal probability of getting either signal, from b∗(ti) is the expected value minus the price paid, which is zero, hence π∗ = equation 16 (in section 7.1.1), we have that a bidder’s expected payoff for a ti +7/2, while the expected value for using the alternative strategy would be game with two players is: π� 2(t +7/2 5). Then, we have that π� >π∗ implies (at best) 2(t +7/2 5) > ≤ i − i − (ti +7/2) ti > 6, which is impossible. 6 ⇒ D 1 25 Eπ (b∗)= π = 1.39. i 6 i 18 ≈ Thus, from the seller’s point of view, uniform price auctions may have unde- i=1 sirable properties.

But other equilibria also exist. First, any bid on the first unit above the con- 4.1.2 The Uniform price auction ditional expected value is an equilibrium bid 9 . Second, if both bidders bid 1, 2 or 3 on the second unit, irrespective of ti, it is also an equilibrium bid. The uniform auction differs from the discriminatory auction described above But, since it is highly unclear on which of these equilibria the bidders would by the pricing rule, which is defined as the highest losing bid of all bidders coordinate, the zero-bid on the second unit is clearly focal on behalf of its (cf. equation (2) on page 7). This dissimilarity makes a major difference in payoff-dominating equilibrium in undominated strategies. strategies, and, hence, equilibria. Using the focal equilibrium strategy, we have that a bidder’s expected payoff We now have a multitude of equilibria, but they have one thing in common; for a game with two players is namely, the bid for the first unit is always the same for each type of player. Still, this comes as no surprise since bidders have a weak incentive to bid their 1 6 EπU = =7 full value on the first unit in independent private value auctions, cf. Noussair i πi , 6 i=1 (1995). This is not entirely true in this setting due to the constraint of integer bids and the fact that the value of the objects is the same for all players. Here, the expected value of the two integers. the first unit bid is the smallest following integer of the conditional expected value, given the bidder’s signal. Hence, from equation 8, we have: If the players instead played the best the seller could hope for, i.e. 3, on the second unit, a bidder’s expected payoff would be

6 U 1 b1∗(ti)= Et i [v(t i ti)] . Eπ = π =4. � − − | � i 6 i i=1 However, bidders have incentives to shade their bids for additional units since, with positive probability, the bid on the second unit may determine the price 4.1.3 The Vickrey auction paid on all units. In the Vickrey auction, the number of units won for a bidder is equal to the The worst that can happen for the seller is what is known as extreme demand number of competing bids that she defeats. Likewise, the prices that she pays reduction, where all bidders act as a single-unit demander and bid zero on the are determined by the competing bids she defeats. second unit, which gives the seller zero revenue. In this setting, the zero bid is indeed an equilibrium and, since it also gives the most payoff to the players, it 9 Levin (2005) has showed that, in an IPV setting with a reservation price equal to may also count as the most preferable equilibrium for them. Hence, we argue zero, any bid on the first unit weakly above the endpoints of the value-distribution that no other equilibrium payoff dominates is an equilibrium bid, as long as there are as many bidders as units for sale.

14 15 ESSAY I > = ≥ ) ∗ j 5) π t ( − ∗ 1 b is +7/2 i j 2(t to win. The payoff for using , . . Second, if both bidders bid 5 , it is also an equilibrium bid. . Any attempt to win 2 units 9 i implies (at best) ) t j ∗ =7 =4 t ( i i ∗ π π b >π 0). 6 6 � =1 =1 i i π 15 6 6 1 1 )]�, i t | = = utilizes −i U U i i t j ( must bid at least v Eπ Eπ [ i i − t �E , which is impossible. , player 6 )=( ∗ 2 > =4 ,b 5). Then, we have that i ∗ 1 t − b would either make her second unit bid, or the first unit bid of 1)]� i | ⇒ Suppose that player −j )=( +7/2 i , set the price, and since the first unit bid from player t , while the expected value for using the alternative strategy would be i 2) t ( j 2 ( / is the expected value minus the price paid, which is zero, hence v / ∗ [ 2(t b j ) i − +7 t t ≤ Levin (2005) has showed that, in an IPV setting with a reservation price equal to +7 ( i � ∗ t i π ( t 1, 2 or 3 on the second unit, irrespective of Proof 2 9 Proposition 2 zero, any bid onis the an first equilibrium unit bid, weakly above as the long endpoints as of there the are value-distribution as many bidders as units for sale. Thus, from the seller’ssirable point properties. of view, uniform price auctions mayBut have other unde- equilibria alsoditional exist. expected First, any value bid is on an the equilibrium first unit bid above the con- for player player But, since it iscoordinate, highly the unclear on zero-bidpayoff-dominating which on equilibrium of the in these undominated second equilibria strategies. the unit bidders is would Using clearly the focal focal on equilibriumfor behalf strategy, a we of game have with its that two a players bidder’s is expected payoff the expected value of the two integers. If the players insteadsecond played unit, the a best bidder’s the expected seller payoff could would be hope for, i.e. 3, on the 4.1.3 The Vickrey auction In the Vickrey auction,number the of number competing of bids units thatare she determined won defeats. by for Likewise, the a the competing bidder prices bids is that she she equal defeats. pays to the �E b (11) 39. 1. ≈ 25 18 = i π 14 6 =1 i 1 6 , �) )= i ∗ 3 t b � ( x}. D )]�. i + i ≥ t i | Eπ n −i | in the Appendix on page 23 �,t t i Z ( 8 3 t v � [ ∈ i − + t i t �E is a ceiling function which maps x to the smallest following integer, min{n See proof )= )=( i i = t t ( ( �x� ∗ 1 ∗ b b �x� If we then assume thatequation there 16 is (in equal probability sectiongame of 7.1.1), with getting two we either players signal, have is: from that a bidder’s expected payoff for a where i.e. 4.1.2 The Uniform price auction The uniform auction differsby from the the pricing discriminatory rule, auction(cf. described which equation above is (2) definedstrategies, on as and, page hence, the 7). equilibria. highest This losing dissimilarity bid makes ofWe a now all major have bidders a differencenamely, multitude in the of bid equilibria, for butStill, the they this comes first have as one unit nofull thing is surprise value in since always on bidders common; the the have same(1995). a first weak This for unit incentive is each in to not independent type bid entirelybids their private of true and value in player. the auctions, this fact cf. setting thatthe Noussair the due first value to of unit the the bid constraint objectsvalue, of is given is integer the the the smallest same bidder’s for following signal. all integer Hence, players. of Here, from the equation conditional 8, expected we have: However, bidders have incentives to shadewith their positive bids probability, for the additional bidpaid units on since, on the all second units. unit may determine theThe price worst that can happenreduction, for where the all seller bidders is actsecond what as unit, is a which known single-unit gives as demander theindeed extreme and seller an demand bid zero equilibrium zero revenue. and, on In sincemay the this it also setting, also count the gives as zero thethat bid the most is no most payoff to other preferable the equilibrium equilibrium players, payoff for it dominates them. Hence, we argue Proof 1 The Vickrey auction has (no more than) two equilibria, which are the two For this equilibrium, as in the uniform auction, we have that any bid on the extreme strategies, on each side of the market. First there is the equilibrium first unit above the conditional expected value is an equilibrium bid. most preferable to the seller, the demand-revealing equilibrium and, on the other side, the most preferable equilibrium for the buyer, the extreme demand- Consequently, using the payoff-dominating equilibrium strategy, the expected reduction equilibrium. payoff for a game with two players is

Proposition 3 The demand-revealing equilibrium is 1 6 EπV = π =7, i 6 i i=1 ti ti b∗(t )=(b∗,b∗)=(t + +2,t + + 1), i 1 2 i � 2 � i � 2 � the same as for the uniform auction, where is deﬁned as above. whereas, using the demand-revealing equilibrium strategy gives �·�

The reason for diﬀerent bids on the two units is purely technical and has 6 V 1 85 its origins in the restriction to only bid in integers. The conditional expected Eπi = πi = 1.18. 6 i=1 72 ≈ value is (for all signals) between bid 1 and bid 2, i.e. b1∗ >vi(t i ti) >b2∗. Thus, − it has nothing to do with demand reduction. |

Proof 3 Suppose that bidder i wins ki units when both bidders follow the 4.1.4 Revenues proposed strategy. That is, exactly ki of her bids are among the two highest bids of both players. If the type-ti bidder bids less on one or both units, then In a (pure) common value auction, the revenue is strongly negatively correlated the number of units that she wins is at most what she would win by bidding with the profit. Above, we saw that both the uniform and the Vickrey auctions b∗(ti). For any of the units won, the prices will be the same as before, but she had an equilibrium that gave all surplus to the buyer, which was the same as would forgo some expected surplus for units that she did not win. the expected value of the two integers, i.e. 7. This translates into zero revenue to the seller. The discriminatory auction, on the other hand, had a unique If she instead bids higher on one or both of her units, then she wins at least as equilibrium that gave the expected profit of 1.39 to the buyers. many units as before. The prices for the first kti units will remain the same as if she bids b∗(ti). For any additional units, however, the price paid will be To find the expected revenue in the discriminatory auction, we calculate the too high, since for k>ki the price is greater than the expected value of the probability for each set of possible joint signals between the players. We make item(s); which is seen by b1∗ >vi(t i ti) >b2∗. use of the strategies implicitly inherent in the signals to compute the price − | paid for each possible set of joint signals. Then, we have the expected revenue Proposition 4 The extreme demand-reduction equilibrium is as the product of the intersection of the signals and the realized price in that outcome. For two players, player i and player j, it becomes:

b∗(ti)=( Et i [v(t i ti)] , 0). (12) � − − | � E[R]=P (t t )p(b ,b ), (13) Proof 4 The proof is as in the uniform auction, hence it is omitted. i ∩ j i j

There is a much weaker equilibrium strategy in the Vickrey auction than in where p(bi,bj) is the price paid. If there are three players, we instead calculate the uniform auction because, if player i bids the above strategy in the uniform P (ti tj tk)p(bi,bj,bk), and so on. Then, we get that the expected revenue auction, player j’s best response is to bid the same. That is not entirely true is 2.09∩ in∩ the discriminatory auction. in the Vickrey auction, since you never pay what you bid, but what the other player bids. Hence, in the Vickrey auction, player j can bid any number below Consequently, the discriminatory auction delivers the greatest revenue to the her conditional expected value for the second unit and still be an equilibrium seller. It is a close game between the uniform and the Vickrey auction. Nonethe- strategy. And, by the same token, any bid below the conditional expected less, because of the existence of the Vickrey auction’s demand-revealing equi- value is an equilibrium bid. librium, the Vickrey auction may be ranked higher than the uniform auction.

16 17 ESSAY I (13) 18. 1. , , it becomes: 39 to the buyers. j . ≈ =7 i 72 85 π = 6 =1 i i π 17 6 1 6 and player =1 i = i 6 1 V i = , Eπ ) j V i ,b i Eπ b ), and so on. Then, we get that the expected revenue k p( ) ,b j j t ,b ∩ i i b t ( p( ) is the price paid. If there are three players, we instead calculate ) P j k t ,b i ∩ ]= b j R t p( [ ∩ E 09 in the discriminatory auction. i t ( To find the expectedprobability revenue for in each the set of discriminatoryuse possible auction, of joint we signals the calculate betweenpaid strategies the the for players. implicitly each We make inherent possible setas in of the the joint product signals. signals of Then,outcome. to the we For intersection have compute two the of players, the expected the player price revenue signals and the realized price in that 4.1.4 Revenues In a (pure) common value auction,with the the revenue is profit. strongly Above, negatively we correlated sawhad that an both equilibrium the that uniformthe and gave the expected all Vickrey value surplus auctions of to theto the two buyer, the integers, which i.e. seller. 7. wasequilibrium This the The that translates same discriminatory into gave as zero auction, the revenue expected on profit the of other 1 hand, had a unique where the same as for the uniform auction, whereas, using the demand-revealing equilibrium strategy gives For this equilibrium, asfirst in unit the above uniform the auction, conditional we expected have value that isConsequently, any an using bid equilibrium the on payoff-dominating bid. equilibrium the payoff strategy, for the a expected game with two players is is 2. Consequently, the discriminatory auction deliversseller. It the is greatest a close revenue gameless, to between the because the uniform of and the the Vickreylibrium, existence auction. the Nonethe- of Vickrey the auction Vickrey may auction’s be demand-revealing ranked equi- higher than the uniform auction. P (12) . Thus, ∗ 2 >b ) i t | −i t ( i >v ∗ 1 b can bid any number below units will remain the same j i t k + 1), � . i ∗ 2 2 t units when both bidders follow the � i >b bids the above strategy in the uniform + k of her bids are among the two highest ) i i i 16 i t | ,t k bidder bids less on one or both units, then −i t i wins ( +2 t i i � 0). i 2 t >v � the price is greater than the expected value of the ∗ 1 )]�, i b + i t | i i t − t ( v k>k . For any additional units, however, the price paid will be [ )=( ) i i ∗ 2 − t ’s best response is to bid the same. That is not entirely true The demand-revealing equilibrium is The extreme demand-reduction equilibrium is t ( j ,b ∗ ∗ 1 b �E b is defined as above. The proof is as in the uniform auction, hence it is omitted. Suppose that bidder )=( )=( i i t t ( ( �·� ∗ ∗ . For any of the units won, the prices will be the same as before, but she b b ) i t ( ∗ The reason forits different origins bids in the onvalue restriction is the to (for all two only signals) bid units between in bid is integers. 1 and The purely bid conditional technical 2, expected i.e. and has Proof 4 There is a muchthe weaker uniform equilibrium auction because, strategy ifauction, in player player the Vickrey auctionin the than Vickrey in auction, sinceplayer bids. you Hence, never in pay the whather Vickrey you conditional auction, bid, expected player but value whatstrategy. for the the And, other second by unitvalue the and is still same an be equilibrium token, an bid. any equilibrium bid below the conditional expected would forgo some expected surplus for unitsIf that she she instead did bids not highermany win. on units one as or before. both The of her prices units, for then the she first wins at least as proposed strategy. That is, exactly it has nothing to do with demandProof reduction. 3 where the number of unitsb that she wins is at most what she would win by bidding The Vickrey auctionextreme has strategies, (no on more eachmost than) side preferable of two to the equilibria,other the market. which side, First seller, the are there most the the preferable isreduction equilibrium demand-revealing the two for equilibrium. equilibrium the equilibrium buyer, and, the on extreme demand- the Proposition 3 too high, since for as if she bids Proposition 4 bids of both players. If the type- item(s); which is seen by 4.2 Three or more players to that, but it is hard to say in what way bids will change without doing a numerical calculation.

In the discriminatory auction, when we increase the bidders by one, all bidders When there are more than two players in the uniform auction, two things but the type-6 player bid the same as in a two-player game (see section 7.1.2). occur. First, we have to correct downwards instead of upwards, because now The type-6 players raise their bids on both units by one increment unit due there is a chance that someone’s first-unit bid may become the price setting to more fierce competition. bid. And, second, as a result of the first, the zero bid on the second unit is no longer an equilibrium. This is due to the fact there are now at least three bidders and two units, and all three bidders have a weak incentive to bid the ti ti (ti + 3 ,ti + 3 ) ti 1, 2, 3, 4, 5 , true (expected) value of the first unit. ui(b, ω)=⎧ � � � � ∈{ } ⎨⎪ (9, 9) ti = 6. Conjecture 5 (More than two players) When there are more than two ⎩⎪ bidders in the auction, the pure Bayesian equilibrium strategy is to bid the The expected payoff decreases for all players which, in turn, raises the revenue following on the first unit: for the seller.

In a four-player game, we get the same increase in the bids of the type-6 b1∗(ti)= Et i [v(t i ti)] . (15) � − − | � players as in the three-player game (see section 7.1.3), but a decrease in the where x is deﬁned as above. bids of the type-1 players. The decrease is one increment unit. The optimal � � strategy can then be written as: Proof 5 First, note that to bid more than b1∗ will incur an expected loss if the

bid is above both Et i [v(t i ti)] and the price. That is, suppose that player � − − | � ti ti i bids b1� > Et i [v(t i ti)] . Then, if b1� >p> Et i [v(t i ti)] , a loss of b∗(ti)=(ti + ,ti + ), (14) � − − | � � − − | � � 2 � � 2 � p Et i [v(t i ti)] will be realized on that unit. −� − − | � where x is a floor function which maps x to the largest previous integer, i.e. Second, suppose that the bid is below the equilibrium bid b b1� . Then, nothing would change Thus, when the number of players increases, the bids can either increase or ≥� − − | � if the player were to raise the bid to Et i [v(t i ti)] . Next, if the bid is be- decrease but the expected revenue always increases with the number of par- � − − | � low Et i [v(t i ti)] and above the price, Et i [v(t i ti)] >b1� >p. Nothing ticipants in the auction. For a game with three players, the expected revenue � − − | � � − − | � would change here either if the bid were increased to Et i [v(t i ti)] . The last − − is 2.69, and for a game with four players, it is 2.82 (in the two-player game, it case is if the value is greater than the price and the� price is| weakly� greater was 2.11). than the bid, Et i [v(t i ti)] >p b1� . Now, if the player raised the bid to � − − | � ≥ Et i [v(t i ti)] , she would win a unit at a profitable price. Thus, to bid the − − Thus, we see two counteracting effects on equilibrium bidding behavior. The �proposed equilibrium| � bid on the first unit is (weakly) dominant in expectation. increasing bids reflect the competitive effect; the more bidders, the greater the competition. The decreasing bids reveal, on the other hand, the winner’s Now, by the last conjecture, when there are more than two players, the bid on curse effect; the more bidders, the greater the potential winner’s curse. It is the second unit will be weakly bounded from below by the first-unit bid from also readily visible that the high-value-bidders belong to the former category the low type player. Thus and the low-value-bidders to the latter category. And for the seller it is better to have more bidders than a few, as the competitive effect dominates the Conjecture 6 (More than two players) The second unit bid is weakly bounded winner’s curse effect. by 4, i.e. b2(ti)∗ b1∗(1) = Et i [v(t i 1)] =4. ≥ � − − | � For more bidders, the calculations become more complicated, but one can see Proof 6 If player i, say, bids below 4, she will win at most one unit and get from the model that increasing the number of bidders even more would induce the payoff: π� =(t +7/2 p)k�, where k� 1. While, if the player bids 4, i i − i ≤ the low-value-bidders to drop out of the auction, due to a negative expected the payoff will be: πi∗ =(ti +7/2 p)ki∗, where ki∗ ki� since the bid b2∗(ti) payoff. High-value-bidders would, of course, anticipate this and bid according now competes against the other bids− which the zero-bid≥ (in 2-player games)

18 19 ESSAY I ) i 4, t ( (15) ∗ 2 b . Nothing )]�. The last i . Then, three )]�, a loss of >p t ∗ 1 i | t � 1 | −i t −i b since the bid v � 1 ( [ b v i � i [ − i k t )]� − i t t ≥ | �E i )]�. Next, if the bid is be- ∗ i − i t t k ( | v [ −i 1. While, if the player bids i . Then, nothing would change t . will incur an expected loss if the � 1 − ( The second unit bid is weakly bounded When there are more than two t ≤ v ∗ 1 [ b >p>�E i � =4 �E − >b k , where t � 1 ∗ i b . Now, if the player raised the bid to k � 1 �E 1)]� b )]� | 19 p) i 4, she will win at most one unit and get t | −i t − and the price. That is, suppose that player ( −i , where 2 t v � i [ ( / i k v )]� >p≥ − [ i t i p) t | − +7 i t �E i )]� − − i t E )]�. Then, if )]�. t t i i 2 ( | t t / | v | [ −i =( i t ≥� −i −i − ( t (1) = t and above the price, t ∗ i will be realized on that unit. +7 ( ( ∗ 1 v p [ i, say, bids below π i b v v i [ t [ �E i i − )]� )]� t i ≥ − − i t t t t | ∗ | =( ) �E i �E −i � i �E )]�, she would win a unit at a profitable price. Thus, to bid the −i t i t t π ( t ( ( | is defined as above. 2 v i > v If player First, note that to bid more than b [ [ )= − i i i t � 1 − t − ( t b t ( �x� v ∗ 1 [ E b i �E , i.e. − 4 t bids −� the payoff will be: Proof 6 the payoff: by proposed equilibrium bid on the first unit isNow, (weakly) by dominant the last in conjecture, expectation. the when second there unit are will morethe be than low two weakly players, type bounded the player. from bid Thus below on by the first-unitConjecture bid 6 from (More than two players) �E to that, but itnumerical calculation. is hard to say inWhen what way there bids are willoccur. more change First, than without we two doing havethere to players a is correct in a downwards the chance insteadbid. that of uniform And, someone’s upwards, auction, second, because first-unit two asno now bid things longer a may an result become equilibrium. of thebidders This the and price is two first, setting due units, the totrue and zero the (expected) all fact bid value three there of on bidders the are have the first now a second at unit. weak unitConjecture incentive least to is 5 three bid (More the than two players) bidders in thefollowing auction, on the the pure first unit: Bayesian equilibrium strategy is to bid the now competes against the other bids which the zero-bid (in 2-player games) would change here either if the bid were increased to low Second, suppose that the bid is below the equilibrium bid p where Proof 5 bid is above both cases appear, first iflow the the bid price, is i.e. below the value which, in turn, is weakly be- if the player were to raise the bid to case is ifthan the the value bid, is greater than the price and the price is weakly greater i (14) }, 5 4, , 3 2, , 82 (in the two-player game, it 1 ∈{ = 6. i i t t 18 �) i 3 t , � �) i + 2 t i � + �,t x}. i i 3 t � ≤ �,t i 9) + 2 t m , i | � t (9 ( Z + ⎧ ⎪ ⎨ ⎪ ⎩ ∈ i t )= is a floor function which maps x to the largest previous integer, i.e. )=( i b, ω t �x� ( ( max{m 11). i ∗ . b u = 69, and for a game with four players, it is 2. The expected payoff decreases forfor all the players which, seller. in turn, raises the revenue In a four-playerplayers game, as we in get thebids three-player the of game same the (see type-1 increasestrategy section can players. 7.1.3), in then The but the be decrease a written bids is decrease as: of one in the increment the unit. type-6 The optimal 4.2 Three or more players In the discriminatory auction, whenbut we increase the the type-6 bidders player by bidThe one, the all type-6 same bidders as players in raiseto a their more two-player game bids fierce (see on competition. section both 7.1.2). units by one increment unit due where �x� Thus, when the numberdecrease of but players the increases, expectedticipants the revenue in bids always the increases auction. canis For with either 2. a the increase game number or with ofwas three par- 2 players, the expected revenue Thus, we see twoincreasing counteracting effects bids on reflect equilibriumthe the bidding competition. competitive behavior. The The effect; decreasingcurse the bids effect; reveal, more the on bidders, more thealso the bidders, other readily hand, visible the greater the that greaterand winner’s the the the high-value-bidders low-value-bidders potential belong to winner’s to theto curse. latter the have category. It former And more category is forwinner’s the bidders curse seller than effect. it is a better few, asFor more the bidders, competitive the calculations effectfrom become the dominates model more that complicated, the increasing butthe the one low-value-bidders number can to of see bidders drop evenpayoff. out more High-value-bidders of would would, induce of the course, auction, anticipate due this to and a bid negative according expected did not. And, since the bid does not affect the price, p will be the same in both In both the uniform and the Vickrey auctions, we see that bidders have strong payoff functions above. Hence, πi∗ πi�. incentives for demand reduction; if they bid the indicated value for the first ≥ unit and zero for the second, the seller gets zero revenue. Hence, as long as Conjecture 7 (Many players) The more bidders in the auction, the higher there are only two bidders, these can secure one unit each at the price of the bids. This is true for both the first-unit bid and the second-unit bid. zero. In a small-scale experiment, conducted in 2007 with master’s students at Orebro¨ University, we encountered this behavior in some sessions. A subject Proof 7 Given any realization of the two dice, we see from equation (7) that for further research is to test the model in a more thorough laboratory setting the conditional expected value weakly increases with the number of players. and see whether the winner’s curse conjecture stated in the introduction is And, as can be seen from conjectures 5 and 6, since both the first-unit and the valid. second-unit bids are dependent on that value, we have that both bids increase with the number of players.

In the Vickrey auction, when there are more than two bidders, all pure equi- 6 Acknowledgements libria disappear. This is an example of a non-core outcome; that is, there is always a coalition that can get a greater payoﬀ by blocking all proposed out- We would like to thank Lars Hultkrantz, Jan-Eric nilsson and Svante Mandell comes. This complication emanates from the discrete setting of the formats, for valuable comments on the paper. This study has been conducted within or, to be more precise, due to the restriction to only bid in integer values. the Centre for Transport Studies (CTS). The author is responsible for any remaining errors.

5 Conclusion References

Analytically identifying equilibrium strategies in a common value auction Alvares,´ F. and C. Mazón: 2010, ‘Comparing the Spanish and the discrimina- when more than one unit is demanded is hard due to its complex nature. tory auction formats: A discrete model with private information’. Economic Here, we build a model where strategies can be found and analyzed, at least Theory Online First. for a few bidders. We run into the problem of a non-core outcome in the Ausubel, L. M. and P. C. Crampton: 2002, ‘Demand Reduction and Ineffi- Vickrey auction when we have more than two players. ciency in Multi-Unit Auctions’. Mimeographed, Department of Economics, University of Maryland. For two-player games, the discriminatory auction delivers the greatest ex- Ausubel, L. M. and P. Milgrom: 2006, ‘The Lovely but Lonely Vickrey Auc- pected revenue. Due to their zero-revenue equilibria, both the uniform and tion’. Combinatorial Auction, chapter 1. the Vickrey auctions are far from the discriminatory auction in revenue rank- Back, K. and J. P. Zender: 1993, ‘Auctions of Divisible Goods: On the Ratio- ing. The Vickrey auction is a little better at surrendering revenue as compared nale for the Treasury Experiment’. The Review of Financial Studies 6(4), to the uniform auction, since its zero-revenue equilibrium is weaker and be- 733–764. cause of its second, demand-revealing, equilibrium. But the Vickrey auction Engelbrecht-Wiggans, R. and C. M. Kahn: 1998a, ‘Multi-Unit Auctions with is rarely implemented in real life auctions because of its complicated nature; Uniform Prices’. Economic Theory 12, 227–258. see, for example, Ausubel and Milgrom (2006). Engelbrecht-Wiggans, R. and C. M. Kahn: 1998b, ‘Multi-Unit Pay-Your-Bid Auction with Variable Awards’. Games and Economic Behavior 23(1), 25– For the discriminatory auction, we are able to find strategies for a larger 42. number of bidders. We see, for example, that both high signal carriers and Levin, D.: 2005, ‘Demand Reduction in Multi-Unit Auctions: Evidence from a low signal carriers change their strategies when we increase the number of Sportscard Field Experiment: Comment’. The American Economic Review bidders in the auction; bidders with a low signal reduce their bids and high 95(1), 467–471. signal bidders raise theirs. Since the probability of winning is low for the low Lindén, J., A. Lunander, and J.-E. Nilsson: 1996, ‘Revenues in Multi-Unit signal holders, we see an increase in revenue when increasing the number of Common Value Auctions - An Experimental Study of three Sealed-Bid bidders. Mechanisms’. Mimeographed, Dalarna University.

20 21 ESSAY I 6(4), (1), 25– Economic 23 . The American Economic Review . , 227–258. The Review of Financial Studies 12 21 Mimeographed, Department of Economics, Games and Economic Behavior . . on: 2010, ‘Comparing the Spanish and the discrimina- Economic Theory Mimeographed, Dalarna University Online First Combinatorial Auction, chapter 1 (1), 467–471. en, J., A. Lunander, and J.-E. Nilsson: 1996, ‘Revenues in Multi-Unit ¨ Orebro University, we encountered this behavior in some sessions. A subject 733–764. Uniform Prices’. 42. Sportscard Field Experiment: Comment’. 95 Common Value AuctionsMechanisms’. - An Experimental Study of three Sealed-Bid Auction with Variable Awards’. nale for the Treasury Experiment’. tory auction formats: A discreteTheory model with private information’. ciency in Multi-Unit Auctions’. University of Maryland tion’. ´ Engelbrecht-Wiggans, R. and C. M. Kahn: 1998a, ‘Multi-Unit Auctions with Levin, D.: 2005, ‘Demand Reduction in Multi-Unit Auctions: Evidence from a Lind´ Engelbrecht-Wiggans, R. and C. M. Kahn: 1998b, ‘Multi-Unit Pay-Your-Bid References Alvares, F. and C. Maz´ 6 Acknowledgements We would like to thankfor Lars Hultkrantz, valuable Jan-Eric comments nilsson on andthe the Svante Centre Mandell paper. for Thisremaining Transport study errors. Studies has (CTS). been The conducted author within is responsible for any Back, K. and J. P. Zender: 1993, ‘Auctions of Divisible Goods: On the Ratio- In both the uniform andincentives the for Vickrey auctions, demand we reduction; seeunit if that and bidders they zero have bid strong forthere the the are indicated second, value only the forzero. two seller the In bidders, gets first a these zero small-scaleat revenue. can experiment, Hence, secure conducted as in onefor long 2007 further unit as research with is each master’s toand at test students see the the model whether price invalid. the a of more winner’s thorough curse laboratory conjecture setting stated in the introduction is Ausubel, L. M. and P. C. Crampton: 2002, ‘DemandAusubel, Reduction L. and M. Ineffi- and P. Milgrom: 2006, ‘The Lovely but Lonely Vickrey Auc- will be the same in both p . � i 20 π The more bidders in the auction, the higher ≥ ∗ i π Given any realization of the two dice, we see from equation (7) that did not. And, since the bid does not affect the price, payoff functions above. Hence, Conjecture 7 (Many players) the bids. This is true for bothProof the first-unit 7 bid andthe the conditional second-unit bid. expected valueAnd, as weakly can increases be seensecond-unit with from bids the conjectures are 5 number dependent andwith on 6, of the since that players. both number value, the of we first-unit players. have and that the bothIn bids the increase Vickrey auction,libria when disappear. there are This more isalways than a an two coalition example bidders, that of allcomes. can a pure get This equi- non-core a complication greater outcome; emanatesor, payoff that from to by is, the be blocking there discrete more all is precise, proposed setting due out- of to the the formats, restriction to only bid in integer values. 5 Conclusion Analytically identifying equilibriumwhen strategies more in thanHere, a one we unit common build a is valuefor model demanded auction a where is strategies few hardVickrey can bidders. auction be due when We found to we run and have its analyzed, into more complex at than the least two nature. For problem players. two-player of games, apected the non-core revenue. discriminatory outcome Due auction inthe to delivers Vickrey the their auctions the zero-revenue are far greatesting. equilibria, The from both ex- Vickrey the auction discriminatory the isto auction a uniform in little the and better revenue uniform rank- at surrendering auction,cause revenue since as of compared its its zero-revenue second,is equilibrium demand-revealing, rarely is equilibrium. implemented weaker in Butsee, and real the for be- life Vickrey example, auctions Ausubel auction because and of Milgrom its (2006). complicatedFor nature; the discriminatorynumber auction, of we bidders. are Welow see, able signal for to carriers example,bidders change find that in their strategies the both strategies for auction; highsignal when bidders bidders a signal we raise with carriers larger theirs. increase asignal and Since low the holders, the signal number we probability reduce of see of bidders. their winning an bids is increase low and in for high revenue the when low increasing the number of Milgrom, P. and R. Weber: 1982, ‘A Theory of Auctions and Competitive 7 Appendix Bidding’. Econometrica 50(5), 1089–1122. Myerson, R.: 1981, ‘Optimal auction design’. Mathematics of Operations Re- search 6(1), 58–73. 7.1 Objection functions, equilibria and payoffs in the discriminatory auction Noussair, C.: 1995, ‘Equilibria in a Multi-Unit Uniform Price Sealed Bid Auc- tion with Multi-Unit Demands’. Economic Theory 5, 337–351. Riksgälden: 2007, ‘Will we benefit from changing auction format?’. Technical 2 We only allow integer-value-bids, (bi,1,bi,2) Z+, and let t = 6. Since the value Report 2007:2, Central Government Borrowing - Forecast and Analysis. function is symmetrical and we have a symmetrical∈ joint distribution, only Riley, J. G. and W. F. Samuelson: 1981, ‘Optimal Auctions’. The American types will be of importance when bidding; thus, we will look for a symmetrical Economic Review 71(3), 381–393. equilibrium. Vickrey, W.: 1961, ‘Counterspeculation, Auctions, and Competitive Sealed Tenders’. The Journal of Finance 16(1), 8–37. Wang, J. J. D. and J. P. Zender: 2002, ‘Auctioning Divisible Goods’. Economic Theory 19, 673–705. Wilson, R.: 1977, ‘A Bidding Model of Perect Competition’. The Review of 7.1.1 2 players Economic Studies 44(3), 511–518. Wilson, R.: 1979, ‘Auctions of Shares’. Quarterly Journal of Economics 93, 675–689. l 7 3 7 Wilson, R. B.: 1969, ‘Competitive Bidding with Disparate Information’. Man- π (bi(ti)) = ti + ki(bi(ti),bj(ti)) pi(bi(ti),bj(ti)) 12 2 4 − agement Science 15(7), 446–448. t 1 + [(ti + j)ki(bi(ti),bj(j)) pi(bi(ti),bj(j))]. j=1,j =t12 − � i

where b (t )=b (t ) for all t T . j i i i i ∈ i The discriminatory auction’s pure Bayesian equilibrium strategy proﬁle with adherent payoﬀ from this game is:

35 b∗(1) = (2, 2) π (2, 2) = 0.73 ⇒ 1 48 ≈ 49 b∗(2) = (3, 3) π (3, 3) = 1.02 ⇒ 2 48 ≈ 71 b∗(3) = (4, 4) π (4, 4) = 1.48 (16) ⇒ 3 48 ≈ 49 b∗(4) = (6, 6) π (6, 6) = 1.02 ⇒ 4 48 ≈ 79 b∗(5) = (7, 7) π (7, 7) = 1.65 ⇒ 5 48 ≈ 39 b∗(6) = (8, 8) π (8, 8) = 2.44. ⇒ 6 16 ≈

Proof 8 (Proof) The equilibrium prescribes equal bids on both units and the payoﬀ is:

22 23 ESSAY I (16) ))]. j )) ( i j t ( ,b j = 6. Since the value ) i t ,b t ) ( i i t b ( ( i i b p ( i p − , and let 2 + − )) j Z ( )) 73 02 48 02 65 44. j i ...... ∈ t 0 1 1 1 1 2 ( ,b ) j ) 2 i ≈ ≈ ≈ ≈ ≈ ≈ i, t ,b ( ) i 23 ,b i b t 1 71 49 79 39 35 49 48 48 48 16 48 48 ( ( i, i i b . k i ( ) i T j k 4) = 6) = 7) = 8) = 2) = 3) = ∈ + i i 7 4 (4, (6, (7, (8, (2, (3, t 3 4 5 6 1 2 [(t π π π π π π + i 1 12 t i 3 2 ⇒ ⇒ ⇒ ⇒ ⇒ ⇒ �=t The equilibrium prescribes equal bids on both units and the t ) for all i t 7 =1,j ( 12 j i 3) 4) 6) 7) 8) 2) b , , , , , , + )) = )= i i t t ( ( i j b b ( (5) = (7 (6) = (8 (2) = (3 (3) = (4 (4) = (6 (1) = (2 l ∗ ∗ ∗ ∗ ∗ ∗ b b b b b b π Proof 8 (Proof) payoff is: The discriminatory auction’s pureadherent Bayesian payoff equilibrium from strategy this profile game with is: where 7 Appendix 7.1 Objection functions, equilibria and payoffs in the discriminatory auction We only allow integer-value-bids, (b 7.1.1 2 players function is symmetricaltypes and will we be of have importanceequilibrium. a when bidding; symmetrical thus, joint we will distribution, look only for a symmetrical , 93 Man- Economic The American The Review of 5, 337–351. Mathematics of Operations Re- Quarterly Journal of Economics (1), 8–37. 22 16 Economic Theory (5), 1089–1122. 50 (3), 511–518. (3), 381–393. (7), 446–448. 44 71 15 , 673–705. Econometrica The Journal of Finance 19 6(1), 58–73. alden: 2007, ‘Will we benefit from changing auction format?’. Technical tion with Multi-Unit Demands’. search Economic Review Report 2007:2, Central Government Borrowing - Forecast and Analysis. Theory Economic Studies 675–689. agement Science Bidding’. Tenders’. Myerson, R.: 1981, ‘Optimal auction design’. Noussair, C.: 1995, ‘Equilibria in a Multi-Unit Uniform Price Sealed Bid Auc- Riksg¨ Riley, J. G. and W. F. Samuelson: 1981, ‘Optimal Auctions’. Wang, J. J. D. and J. P. Zender: 2002, ‘Auctioning DivisibleWilson, Goods’. R.: 1977, ‘A Bidding Model ofWilson, Perect R.: Competition’. 1979, ‘Auctions of Shares’. Wilson, R. B.: 1969, ‘Competitive Bidding with Disparate Information’. Milgrom, P. and R. Weber: 1982, ‘A Theory of Auctions and Competitive Vickrey, W.: 1961, ‘Counterspeculation, Auctions, and Competitive Sealed t 1 1 i− t 7 3 7 t 1 41 11 ti ti i i π(b∗) π(b�) + t 9 5t +7+ π(b∗)= 2 (ti + j) ti + + ti + ti + − ≥ 12 4 2 i − � 3 �− i � 3 � 12 j=1 − � 3 � 12 2 4 − � 3 � 1 69 1 ti ti 1 = + ti 8 > 0, ti Ti. 1 − ti 1 49 7 ti = 2 j + + t 7 12 4 2 − � 3 � ∀ ∈ 12 −�3 � 12 4 2 i − � 3 � j =1 ti 2 Since the difference is positive, the bidder will not make any profit by bidding 1 − ti 1 41 11 ti = 2 j + + ti 9 lower than b . 12 −�3 � 12 4 2 − � 3 � ∗ j =1 On the other hand, the player may deviate by raising the bid as compared to We will make use of the last equation when we scrutinize what happens if the the equilibrium bid; thus, b�(ti)=b(ti + l), where l Z+, and thereby mimic a ∈ bidder deviates by bidding lower, which we do first, while we make use of the type-(ti + l) player. Then, she will always win both units when playing against middle equation if she deviates by bidding higher. another type-ti bidder and, depending on which type she mimics, both will win one unit each when playing against each other. This shows that the best she Suppose that the bidder deviates and bids b�(ti)=b(ti l), where l Z+, can do is to mimic a type-(ti + 1) player, but it is not good enough since she i.e, she mimics a type-(t l) player. Then, she will always− lose both∈ units will then only get the following payoff: i − when playing against another type-ti bidder and, depending on which type she mimics, both will win one unit each when playing against each other. This means that the best she can do is to mimic a type-(ti 1) player and thereby ti 1 − 1 − ti +1 get the following payoff: π(b�)= 2 (t + j) t +1+ 12 i − i � 3 � j=1 7 3 7 t +1 + 2 t + t +1+ i 12 2 i 4 − i � 3 � ti 2 1 − ti 1 1 t +1 � i π(b )= 2 (ti + j) ti 1+ − + (ti + ti + 1) ti +1+ 12 j=1 − − � 3 � 12 3 − � � ti 1 1 ti 1 1 − ti +1 14 1 3 ti +1 + (ti + ti 1) ti 1+ − = 2 j 1 + t + 12 − − − � 3 � i 12 j=1 − −� 3 � 12 2 4 −� 3 � t 2 i− 1 ti 1 1 ti 1 1 ti +1 = 2 j +1 − + ti − + ti 12 j=1 −� 3 � 12 −� 3 � 12 3 −� � t 1 1 i− t +1 1 = 2 j i 2(t 1) 12 −� 3 � − 12 i − ti ti 1 j=1 using the relation 3 1 −3 , we get � �− ≤� � 14 1 3 ti +1 1 ti +1 + ti + + ti 12 2 4 −� 3 � 12 −� 3 �

ti 2 t t +1 1 − ti 1 1 ti using the relation i i , and merging the last three terms, we get π(b�) 2 j +2 + 4(ti 2) + ti +1 � 3 �≤� 3 � ≤ 12 −�3 � 12 − 12 −�3 � j =1 ti 2 1 − ti 1 ti = 2 j + 4ti 8+ti +1 ti 1 12 j=1 −�3 � 12 − −�3 � 1 − ti 1 21 ti ti π(b�) 2 j + 2ti +2+7ti + 14 + ti ti 2 ≤ 12 −�3 � 12 − 2 − � 3 � −�3 � 1 − ti 1 ti j=1 = 2 j + 5ti 7 . ti 1 12 j=1 −�3 � 12 − −�3 � 1 − ti 1 25 ti = 2 j + +6ti 15 12 −�3 � 12 2 − � 3 � j=1 Taking the difference between the payoff when a player places the equilibrium bid, and the payoff if she deviates downwards, we have Then, we have that

24 25 ESSAY I � i 3 t −� i t + � i 3 t � � 14 3 +1 . i � − i t i 3 t T , and thereby mimic a 2 21 + � � ∈ −� i Z i 3 t + t 3 4 i ∈ 3 ∀ +1 t i + l 15� +7+� t � i � � i t t 1) − 0, 1 2 5 i −� 3 +1 t − 3 3 +1 +1 i > i i t t +2+7 i i t t 14 12 �− i , where +6 i t ) � 2(t 3 t l i 2 1 + 3 12 t 2 1 25 12 − + 25 � 9� � i + 8 t player, but it is not good enough since she − +1+� − 1 1 +1+� +1+� − i i b( � 12 12 t t i i 3 +1 i � t t t i + + 2 t 11 1 2 �, and merging the last three terms, we get 3 +1 + 1) )= − i 3 i +1 i � � − − t t + + 3 +1 ) t i i � i ( i −� 3 3 t t � t j t b 1 4 4 7 4 41 69 + −� 3 +1 + 1) i −� −� − −� i + 3 4 t i t j j j j ( t i �≤� 1 1 t i 12 12 3 + t + 3 2 2 2 2 2 2 i � −� i t = ≥ 1 1 t i ) 1 2 − − −1 −1 −1 ( t =1 =1 =1 =1 =1 � i i i i i bidder and, depending on which type she mimics, both will win j j j j j t t t t t b 2 i ( . 1 1 1 1 1 1 7 1 π ∗ player. Then, she will always win both units when playing against 12 12 12 12 12 12 12 12 12 14 b ) − l = + = + = + + ≤ ) ) )= � � ∗ + b b b i ( ( ( t π π π anothert type- ( type- using the relation Then, we have that will then only get the following payoff: On the other hand,the the equilibrium bid; player thus, may deviate by raising the bid as compared to one unit each whencan do playing is against to each mimic other. a This( type- shows that the best she Since the difference islower positive, the than bidder will not make any profit by bidding , + Z � i 3 t ∈ � l + � i i t 3 t − , where ) l player and thereby 7 4 � +1−� i � 3 − t i + 1) t 1 i i � t t i − � 3 3 − t 3 2 i i . 1 3 i t t 12 t � 9� � 7 i +1−� 3 t � − i )=b( 7 −� � 12 i − i 1 2) + i t t 1 i t ( t � 3 −� + − 2 − 3 b − 7 2 11 i 8+t 7 i t i 1 t 12 + + − − � 4(t i i i 3 + t 4 4 t t 1 49 41 � 12 4 5 1+� 24 bidder and, depending on which type she 1+� � i + + 1 − 1 1 �, we get i 1 1 − 12 12 i 12 12 t 1 t i 3 − � t player. Then, she will always lose both units 3 − i i + + i + + 3 t t t ) − l − � � � � − ) i i ) i i 3 3 t t j 3 3 ≤� − t t j 1) i 1 + t + i −� −� − i +2−� −� −� +1−� t t i ( j j j j j ( j t �− i 3 t 2 2 2 + 2 2 2 2 2 � i 2 2 2 t −1 −1 −2 =1 =1 =1 − − − −2 −2 ( =1 =1 =1 =1 =1 i i i i i i j j j i i t t t j j j j j t t t t t 1 1 1 1 1 1 1 1 1 12 12 12 12 12 12 12 12 12 = = = = + = ≤ )= )= ) � � ∗ b b b ( ( ( π π π Taking the difference between thebid, and payoff when the a payoff if player she places deviates the downwards, equilibrium we have when playing against anothert type- using the relation We will make usebidder of deviates the by last biddingmiddle equation lower, when equation which we if we scrutinize she do what deviates first, happens by if while bidding the we higher. Suppose make that use the of bidder the deviates and bids mimics, both willmeans win that one the best unit she each can when do playing is to against mimic each a other.( type- This i.e, she mimics a( type- get the following payoff: 1 49 7 ti 25 ti And, if we assume equal probability for each type, the expected payoff to a π(b∗) π(b�) + ti 7 6ti + 15 − ≥ 12 4 2 − � 3 �− 2 − � 3 � bidder is: 1 1 5 t 1 6 13 = t +8 i > 0, t T . EπD = π = 0.54. 12 −4 − 2 i � 3 � ∀ i ∈ i i 6 i 24 ≈ i=1 Thus, no one has an incentive to deviate from the proposed equilibrium strategy. 7.1.3 4 players Then, if we assume there is equal probability of getting either signal, a bidder’s expected payoff for a game with two players would be: 13 15 7 π(b(t )) = t + k (b (t ),b (t ),b(t ),b (t )) p (b (t ),b (t ),b(t ),b (t )) , i 48 8 i 16 i i i j i l i m i − i i i j i l i m i 1 6 25 EπD = π = 1.39. 6 1 i i + [(t + j)k (b (t ),b (j),b(t ),b (t )) p (b (t ),b (j),b(t ),b (t )] 6 i=1 18 ≈ i i i i j l i m i i i i j l i m i j=1,j =t16 − � i 6 1 7.1.2 3 players + [(ti + j)ki(bi(ti),bj(j),bl(j),bm(ti)) pi(bi(ti),bj(j),bl(j),bm(ti)] j=1,j =t16 − � i 6 1 3 5 7 + [(ti + j)ki(bi(ti),bj(j),bl(j),bm(j)) pi(bi(ti),bj(j),bl(j),bm(j)] π(b(t )) = t + k (b (t ),b (t ),b(t )) p (b (t ),b (t ),b(t )) , 48 − i i i i i j i l i i i i j i l i j=1 ,j =it 8 4 8 − � 6 1 + [(ti + j)ki(bi(ti),bj(j),bl(ti)) pi(bi(ti),bj(j),bl(ti)] where bj(ti)=bi(ti) for all ti Ti. j=1,j =t12 − ∈ � i 6 1 The discriminatory auction’s unique pure strategy Bayesian equilibrium profile + [(t + j)k (b (t ),b (j),b(j)) p (b (t ),b (j),b(j))] , i i i i j l i i i j l (for 4 players) with the payoff: j=1,j =t24 − � i where b (t )=b (t ) for all t T . j i i i i ∈ i 91 b∗(1) = (1, 1) π1(1, 1) = 0.18 The discriminatory auction’s unique pure strategy Bayesian equilibrium profile ⇒ 512 ≈ 247 (for 3 players) with its payoff is: b∗(2) = (3, 3) π (3, 3) = 0.16 ⇒ 2 1536 ≈ 653 b∗(3) = (4, 4) π (4, 4) = 0.43 5 ⇒ 3 1536 ≈ b∗(1) = (2, 2) π (2, 2) = 0.16 ⇒ 1 32 ≈ 403 b∗(4) = (6, 6) π (6, 6) = 0.26 11 ⇒ 4 1536 ≈ b∗(2) = (3, 3) π (3, 3) = 0.34 ⇒ 2 32 ≈ 1033 b∗(5) = (7, 7) π (7, 7) = 0.67 67 ⇒ 5 1536 ≈ b∗(3) = (4, 4) π (4, 4) = 0.70 ⇒ 3 96 ≈ 559 b∗(6) = (9, 9) π (9, 9) = 0.36. 15 ⇒ 6 1536 ≈ b∗(4) = (6, 6) π (6, 6) = 0.47 ⇒ 4 32 ≈ 95 Proof 10 It follows the same lines as the one for two players, but is quite b∗(5) = (7, 7) π5(7, 7) = 0.99 messy. Therefore, we omit it here but it can be obtained from the author upon ⇒ 96 ≈ 19 request. b∗(6) = (9, 9) π (9, 9) = 0.59. ⇒ 6 32 ≈ And the expected payoff to a bidder is: Proof 9 It follows the same lines as the one for two players, but it is quite 6 messy. Therefore, we will omit it here but it can be obtained from the author D 1 11 Eπ = πi = 0.34. upon request. i 6 32 ≈ i=1

26 27 ESSAY I , )) )] i i )] t i t )] ( t ( j ( ( m m m m ,b ,b ) ,b ) ,b i i ) ) t t j ( j ( ( l l ( l l ,b ,b ,b ) ,b ) ) i ) j t j j ( ( ( ( j j j j ,b ,b ,b ,b ) ) ) i ) i i i t t t t ( ( ( i ( i i i b b b b ( ( ( i ( i i i p p p p − − − − )) )) )) i )) i i t t j t ( ( ( ( m m m m ,b ,b ,b ,b ) ) ) ) i 34. 54. i j t j . . t ( ( ( 0 0 ( l l l l ,b ,b ,b ≈ ≈ 16 43 26 67 36. ,b ) ) ) ) 18 i j j j 0. 0. 0. 0. 0. . t ( ( ( 0 32 24 11 13 ( j j j j ≈ ≈ ≈ ≈ ≈ ≈ ,b ,b ,b = = ,b ) ) ) i i i i i ) i t t t π π t 27 ( ( ( 559 91 247 653 403 i i i ( 1033 1536 512 1536 1536 1536 1536 i b b b 6 6 =1 =1 b ( ( ( i i i i i ( . i i k k k 6 6 1 1 k ) ) ) T 9) = 1) = 3) = 4) = 6) = 7) = j j j = = ∈ + + + 7 i (9, (1, (3, (4, (6, (7, D D 16 i i t i i i 6 1 2 3 4 5 π π π π π π + [(t [(t [(t Eπ Eπ i t 1 1 1 16 16 48 8 i i i 15 ⇒ ⇒ ⇒ ⇒ ⇒ ⇒ ) for all �=t �=t �=t i 6 6 6 t ( i =1,j =1,j =1,j 1) 3) 4) 6) 7) 9) b 48 13 j j j , , , , , , + + + It follows the same lines as the one for two players, but is quite )= )) = i i t t ( j b( b (4) = (6 (5) = (7 (6) = (9 (1) = (1 (2) = (3 (3) = (4 ( ∗ ∗ ∗ ∗ ∗ ∗ b b b b π b b 7.1.3 4 players Proof 10 messy. Therefore, we omitrequest. it here but it can be obtained fromAnd the the author expected upon payoff to a bidder is: The discriminatory auction’s unique pure strategy(for Bayesian equilibrium 4 profile players) with the payoff: And, if we assumebidder equal is: probability for each type, the expected payoff to a where , )] , i ))] t j ( ( l )) l i t ,b ,b ( ) ) l j j ( ( ,b j j ) i t ,b ,b ( ) ) i j i t t . � ( ( ,b i i i i ) 3 b t T i b ( t ( i i ( ∈ p i p b i t ( − i − ∀ + 15� p i )) )) i 39. − j t . 6t ( ( 1 l l )) − i ,b ,b ≈ t 0, ) ) ( 2 l j j 25 34 70 47 99 59. 16 > ( ( ...... 25 18 ,b j j 0 0 0 0 0 0 ) i � ,b ,b = t �− i ) ) ≈ ≈ ≈ ≈ ≈ ≈ ( 3 i t i i i 3 j t t t � π 26 ( ( 5 i i ,b 11 32 67 96 15 32 95 96 19 32 32 7� b b ) 6 =1 i ( ( i +8 i i t . − i ( i k k 1 6 i i t ) ) T t b 2) = 3) = 4) = 6) = 7) = 9) = j j 5 2 ( 7 2 = i ∈ k + + i − (2, (3, (4, (6, (7, (9, D i + t i i 1 2 3 4 5 6 1 4 π π π π π π 7 8 4 [(t [(t Eπ 49 − + 1 1 12 24 i t 1 1 i i 12 12 ⇒ ⇒ ⇒ ⇒ ⇒ ⇒ 5 4 ) for all �=t �=t i = ≥ 6 6 t ( ) i � =1,j =1,j 3) 4) 6) 7) 9) 2) b 3 8 b j j , , , , , ( π + + )= )) = i − It follows the same lines as the one for two players, but it is quite i t ) t ( ∗ j b b( b (2) = (3 (3) = (4 (4) = (6 (5) = (7 (6) = (9 (1) = (2, ( ( ∗ ∗ ∗ ∗ ∗ ∗ b b b b b π b π The discriminatory auction’s unique pure strategy(for Bayesian equilibrium 3 profile players) with its payoff is: 7.1.2 3 players Proof 9 messy. Therefore, we willupon omit request. it here but it can be obtained from the author where Thus, no one hasegy. an incentive to deviate from the proposedThen, equilibrium if strat- we assume there isexpected equal payoff probability for of getting a either game signal, with a bidder’s two players would be:

ESSAY II Multi-unit common value auctions: A laboratory experiment with three sealed-bid mechansims

Joakim Ahlberg

VTI - Swedish National Road and Transport Research Institute, P.O. Box 55685, SE-102 15 Stockholm, Sweden Tel: +46 8 555 770 23. E-mail address: [email protected]

Abstract

This study addresses a discrete common value environment with independent (one-dimensional) private signals, where the seller offers two identical units and the buyers have (flat) demand for both. Each session is conducted with 2, 3 or 4 buyers. Three auction formats are used: the discriminatory, uniform and Vickrey auctions which are all subjected to a variation in the number of bidders and to repeating bid rounds on each subject. The main findings are that there are no significant differences between the uniform and the discriminatory auction in collecting revenue, while the Vickrey auction comes out as inferior. More bidders in the auction result in a greater revenue and level out the performance across the mechanisms. Demand reduction is visible in the experiment, but it is not as prominent as anticipated. Moreover, subjects come closer to equilibrium play over time. Finally, the winner’s curse is less severe than what is reported for inexperienced bidders in other studies.

Keywords: Laboratory Experiment; Multi-Unit Auction; Common Value Auc- tion

JEL codes: C91; C72; D44

1 Introduction

Common value (CV) auctions with single unit demand have been studied for quite some time, both theoretically and in the laboratory. The main focus of the experimental research on CV auctions has been on the winner’s curse problem, that is, the adverse selection eﬀect produced by a win if not accounted for. But research on multi-unit demand is scarce. The winner’s curse problem has not been addressed in this literature; the emphasis in both theoretical and experimental research, when the items for sale are substitutes,

1 ESSAY II [email protected] : 1 E-mail address Joakim Ahlberg 55685, SE-102 15 Stockholm, Sweden sealed-bid mechansims C91; C72; D44 Laboratory Experiment; Multi-Unit Auction; Common Value Auc- Tel: +46 8 555 770 23. laboratory experiment with three VTI - Swedish National Road and Transport Research Institute, P.O. Box Multi-unit common value auctions: A 1 Introduction Common value (CV) auctionsquite with single some unit time, demandof both have the been theoretically experimental studied and for researchproblem, on in that CV the is, auctions laboratory.counted has the for. The been adverse But main on research selection the focusproblem on effect multi-unit winner’s has demand curse not produced is been byoretical scarce. addressed The a and in winner’s experimental win this curse research, literature; if the when not emphasis the ac- in items both for the- sale are substitutes, Abstract This study addresses a(one-dimensional) discrete common private value signals, environment with whereand independent the the buyers seller have2, offers (flat) 3 two demand or identical for 4 units and buyers. both. Vickrey Three Each auctions auction session which formatsbidders is are are conducted and all used: with to subjected the discriminatory, repeating tothat uniform a there bid are variation rounds in no oninatory significant the each differences auction number subject. between in of the collecting The uniforminferior. revenue, main and More while findings the bidders the discrim- are in Vickreythe the auction performance auction comes result out across in as theexperiment, a but mechanisms. greater it is Demand revenue not and reductioncloser as level prominent is to out as equilibrium visible anticipated. Moreover, play in subjectsthan over come the what time. is Finally, reported the for winner’s inexperienced curse bidders is in lessKeywords: other severe studies. tion JEL codes: has been on demand reduction. 1 A phenomenon in some auction formats is question is that the more bidders in an auction, the larger is the revenue for that bidders have an incentive to reduce the demand for units other than the the seller. Third, we see demand reduction, but we do not see any extreme first, since these bids may become the market clearing price. It is found that demand reduction at all, that is, zero bidding on the second unit. Fourth, we demand reduction leads to substantial revenue losses for the sellers. There is find that subjects do learn to play equilibrium strategies in the course of the also a literature concerning mechanism design issues, complementariness and game, at least in the discriminatory auction. Moreover, they continue to learn synergies between items, and the role of package bidding (see, for example, until the final rounds. Kagel and Levin (2011) for a start in these areas). For the last question, we find that the winner’s curse (WC) is highly present; The prevalent static multi-unit auction formats in the literature are the dis- mostly in the uniform and discriminatory auctions, but also in the Vickrey criminatory, the uniform, and the Vickrey auction. The first two formats are auction. We distinguish between bidding above the conditional expected value those used in the field, whereas the last is never used due to its (allegedly) (of winning) up to the naive expected value and above the naive expected complicated nature, even though it has nice demand-revealing properties; see, value. It is twice as common to bid in the first interval, which (partly) indicates for example, Rothkopf et al. (1990). When we add the common value environ- that subjects have difficulties in understanding the winner’s curse. ment, the ranking of these auction formats in term of revenues becomes an open question. There is also an ongoing discussion in the market for treasury The theoretical model in this article emanates from Ahlberg (2009) where it is bonds, as well as in the markets for CO2 allowances, on which of the first presented more thoroughly. There is little earlier theoretical and experimental two formats above should be used. (Back and Zender (1993) summarize this work on multi-unit demand common value auctions against which to directly debate in the independent private value (IPV) case.) compare our results, except for the theoretical article from Alvares´ and Mazón (2010). They have a theoretical model similar to ours in a continuous setting. This study features a discrete auction, in the sense that the values of the Much of the theory that exists focuses on independent private value (IPV) unit and bidding are only allowed in integer numbers with independent (one- settings, or, to some extent, interdependent value settings. The contribution dimensional) private signals, where the seller offers two identical units and of the present study to the experimental literature is a different common value the buyers have demand for both. The three auction formats (discussed in generation, which is made somewhat simpler for ease of understanding. There the above paragraph) are tried and subjected to a variation in the number of is reason to believe that subjects do not understand the concept of the winner’s bidders and to repeating bid rounds (15 - 20 rounds) on each subject. Five curse, and overbid as a result. Second, we want to contribute to the ongoing main questions are scrutinized. (i) which auction format gives the greatest debate on which of the static auction formats one should use in practice, when revenue?; (ii) how does the number of bidders affect revenue?; (iii) is there the value of the object(s) is common to all bidders. demand reduction in the uniform and Vickrey auctions?; (iv) what are the implications of repeating the auction several rounds on the subjects, that is The rest of the paper is organized as follows: Section 2 focuses on earlier do we see any learning effects?; and (v), is there a winner’s curse, that is research in relation to the stated questions, section 3 presents the theory do bidders ignore the informational content inherent in winning, and bid too and the hypotheses, and section 4 outlines the experimental design. Section high? 5 contains the results, section 6 discusses them and section 7 concludes the study. All computations are found in the Appendix. Starting with revenue, we find that the Vickrey auction always gives the least overall revenue, especially in small group sizes. The uniform and the discriminatory auctions run a close race and cannot be separated. This was quite 2 Earlier Research unexpected due to the non-expected result in 2-player groups. (The hypothesis for the uniform auction is that, in 2-player groups, the subjects play more according to the extreme demand reduction prediction. But, in general, they In previous analytical research, Ausubel and Crampton (2002) have shown do not.) For large group sizes, the difference in revenue between the Vickrey that, in the interdependent value case where the item for sale is infinitely and the other two formats disappears completely. The answer to the second divisible, in many cases, the discriminatory auction outperforms the uniform price auction but, in general, the revenue ranking between the two is ambigu- 1 Even though Vickrey (1961) was the first to point out the inefficiency of multi-unit ous. In an IPV setting, Engelmann and Grimm (2009) investigate the three auctions in general, Ausubel (2004) and Ausubel and Crampton (2002) emphasized, auction mechanisms described above and two open counterparts; the open in common value settings, that the inefficiencies are due to demand reduction. uniform auction and the Ausubel auction, which is a dynamic Vickrey auc-

2 3 ESSAY II on ´ Alvares and Maz´ 3 2 Earlier Research In previous analyticalthat, research, in Ausubel the anddivisible, interdependent Crampton in value (2002) many case cases, haveprice the where shown auction discriminatory but, the auction in item outperforms general,ous. the the for In revenue uniform an sale ranking between IPVauction is the setting, mechanisms two infinitely Engelmann is described ambigu- anduniform above Grimm auction and (2009) and investigate two the the open Ausubel three counterparts; auction, the which is open a dynamic Vickrey auc- (2010). They have a theoreticalMuch model of similar the tosettings, ours theory in or, that a to exists continuous some setting. of focuses extent, the on present interdependent study independent value togeneration, settings. the private which experimental The value is literature contribution made (IPV) is somewhat ais different simpler reason common to for believe value ease that of subjectscurse, understanding. do and There not understand overbid the as conceptdebate a of on the result. which winner’s of Second, the wethe static want value auction to of formats one the contribute should object(s) to use the is in common practice, ongoing when The to all rest bidders. ofresearch the in paper relation isand to the organized the hypotheses, as and stated5 follows: section contains questions, Section 4 the section 2 outlines results,study. All 3 focuses the section computations experimental 6 presents on are design. discusses the found earlier Section them in theory the and Appendix. section 7 concludes the question is that thethe more seller. bidders Third, in wedemand an reduction see auction, at the demand all, larger reduction,find that is that but is, the subjects zero we revenue do bidding do for game, on learn at not the to least see second in play any the unit. equilibriumuntil discriminatory Fourth, extreme the strategies we auction. final in Moreover, rounds. the they course continue to of learn the For the last question, wemostly find in that the theauction. winner’s uniform We curse and distinguish (WC) between discriminatory is bidding(of above highly auctions, the present; winning) but conditional expected also up value value. in to It is the the twice as Vickrey naive commonthat to expected subjects bid in value have the difficulties and first interval, in above which understanding (partly) the indicates theThe naive winner’s theoretical curse. expected model in thispresented article more emanates thoroughly. from There Ahlberg iswork (2009) little on where earlier it multi-unit theoretical is demand and experimental compare common our value results, auctions except against for which the to theoretical article directly from allowances, on which of the first 2 CO 2 A phenomenon in some auction formats is 1 Even though Vickrey (1961) was the first to point out the inefficiency of multi-unit two formats above shoulddebate be in used. the (Back independent and private Zender value (IPV) (1993) case.) summarizeThis this study featuresunit a and discrete bidding are auction,dimensional) only in allowed private in the signals, integerthe sense where numbers with buyers the that independent have seller the (one- the demand offers values above for two of paragraph) both. identical are the bidders tried The units and and three and subjected to auction to repeatingmain a formats bid questions variation (discussed in rounds are in the (15revenue?; scrutinized. number (ii) - of (i) how 20 which does rounds)demand auction on the reduction format each number in gives subject. ofimplications the the of bidders Five uniform greatest repeating affect and the revenue?;do Vickrey auction (iii) we auctions?; several is see (iv) rounds there do any what on bidders are learning the ignore the subjects, the effects?;high? informational that and content is (v), inherent in is winning, there and bid aStarting too winner’s with revenue, curse, we find thatoverall that revenue, is the especially Vickrey in auction smallinatory always group gives auctions sizes. the run The least unexpected uniform a due and close to the race the discrim- sis non-expected and for result cannot the in uniform be 2-playeraccording auction groups. separated. to is (The the that, This hypothe- in extreme wasdo 2-player demand not.) groups, quite reduction For the prediction. large subjects But,and group play in the sizes, more general, other the they two difference formats in disappears revenue between completely. the The Vickrey answer to the second 1 has been on demand reduction. auctions in general, Ausubel (2004)in and common Ausubel value and settings, Crampton (2002) that emphasized, the inefficiencies are due to demand reduction. that bidders have anfirst, incentive to since reduce these the bidsdemand demand may for reduction become units leads the other to marketalso than substantial clearing a the revenue price. literature losses It concerning forsynergies is mechanism the between design found sellers. issues, items, that There complementariness andKagel is and and the Levin role (2011) of for package a bidding start in (see,The these prevalent for areas). static example, multi-unitcriminatory, auction the formats uniform, in andthose the the literature used Vickrey are auction. in The the thecomplicated first dis- nature, field, two even whereas though formats it the are for has example, last nice Rothkopf is demand-revealing et properties; al. neverment, see, (1990). used the When due we ranking add to ofopen the its question. common these value There (allegedly) auction environ- is formatsbonds, also in an as term ongoing well discussion of as in revenues the in becomes market the an for markets treasury for tion. They find that the revenues are greater if a sealed-bid format is used as various types of auctions and is not eliminated, only somewhat mitigated, compared to an open auction; the revenues depend less on which pricing rule by experience or even by using expert bidders. But experimental studies on is employed. common value, multi-unit auctions are scarce.

Kagel and Levin (2001) theoretically predict that as the number of bidders But one, notably, is Ausubel et al. (2009), which experimentally tests alterna- increases, demand reduction will diminish. They confirm this behavior asym- tive auction designs suitable for pricing and removing troubled assets. They metrically in a laboratory experiment; that is, in which subjects behave ac- make use of the same static and dynamic uniform auction as this study and cording to theory only if their rivals decrease in number, not if they increase. Engelmann and Grimm (2009) above, except that their dynamic format is an Katzman (1995) also provides a theory indicating that the prevalence of de- Ausubel descending clock auction. The units for sale are not identical, and mand reduction decreases with the number of participants even though some they sell the units individually or as pooled units. And, for some sessions, demand reduction will always be present. Engelbrecht-Wiggans et al. (2006) bidders also know their liquidity needs. They find that the static and dynamic also establish that there is no difference in the first-unit-bid between the uni- auctions resulted in similar prices. However, the dynamic auctions resulted form auction and the Vickrey auction when the number of bidders increases in substantially higher bidders’ payoffs, which enabled the bidders to better from two to three (or five), even though the second-unit-bid is always greater manage their liquidity needs. The dynamic auction was also better in terms in the Vickrey auction. of price discovery, as well as for reducing the bidder error.

Concerning demand reduction, Noussair (1995) showed in a seminal paper that Another study is Manelli et al. (2006) which experimentally compares the the bid on the second unit was always lower than its value, in contrast to the static Vickrey auction with the Ausubel auction, also known as the dynamic first bid, which was always demand-revealing. The degree of under-revelation Vickrey auction, in both an IPV setting and an interdependent value (IV) depends on whether the bid sets the price or not. Ausubel and Crampton setting, in which the values are affiliated. They conclude that due to overbid- (2002) provide a formal proof of demand reduction in the uniform auction. ding in both types of auctions, but slightly more in the Vickrey auction, the Katzman (1995) and Engelbrecht-Wiggans and Kahn (1998) also analyze auc- revenue from the Vickrey auction is greater, while the efficiency is lower in the tions that involve demand reduction. Ausubel auction. But in the IV setting, they observe less overbidding and a trade-off between efficiency and revenue; the Vickrey auction is more efficient Demand reduction has been confirmed in experimental research to various while the revenue is higher in the Ausubel auction. degrees, for example in a field experiment by List and Lucking-Reiley (2000) in which two players with two-unit demand bid for two units through the uniform price, the English, or the Vickrey auction, also replicated in a laboratory experiment by Porter and Vragov (2006). Another laboratory experiment 3 Experimental Design where demand reduction is confirmed is by Kagel and Levin (2001); they let one bidder with two-unit demand compete against a robot bidder with unit demand and playing the dominant strategy. The experiment used students from KTH (the Royal Institute of Technology) as experimental subjects. They were from different Master’s programs in En- With respect to learning, we have the evolutionary paradigm, or what Nelson gineering, and the experiment took place in May 2009. In total, 152 unique and Winter (2002) call the ”competence puzzle”, which roughly means that subjects participated in the experiment. individuals typically do not have the vast computational and cognitive powers that are imputed to them by the optimization-based theories (such as that in The subjects were recruited for computer sessions consisting of a series of this article). But, since learning, guided by clear short-term feedback, can be auction periods. Each subject participated in one of three possible auction remarkably powerful even in addressing complex challenges, the evolutionary formats; hence, the design is between subjects. In each period, two identical response to the competence puzzle focuses on the role of learning and practice. units of a commodity were sold to the two highest bids, and these two bids could come from the same bidder or two different bidders. The units had no The research on the winner’s curse is vast, starting with Capen et al. (1971) meaning for the participants apart from the money they could bring forth. who claimed that oil companies suffered from low returns. A comprehensive Only the subject(s) who won the units earned profit(s), calculated as the survey of theory and experiments in single-unit, common value auctions is induced value of the item minus the price paid for it. When there were ties, offered by Kagel and Levin (2002). They show that the WC is pervasive across the winning bids were randomly selected.

4 5 ESSAY II 5 3 Experimental Design The experiment used students fromas KTH experimental (the subjects. Royal They Institutegineering, were of and from Technology) different the Master’s experimentsubjects programs took participated in in place En- the in experiment. May 2009. InThe total, subjects 152 were unique auction recruited periods. for Each computerformats; subject hence, sessions participated the consisting in designunits of is one of a between of a subjects. series three commoditycould In of were come possible each sold from auction period, to themeaning two same the identical for bidder two the or highestOnly participants two bids, different apart the and bidders. from these subject(s) Theinduced two the units who value bids money had of won they no the thethe could item winning units bring minus bids the forth. earned were randomly price profit(s), selected. paid calculated for it. as When the there were ties, various types ofby auctions experience and or is evencommon not by value, multi-unit using eliminated, auctions expert only are bidders. scarce. somewhat But mitigated, experimentalBut studies one, notably, on is Ausubeltive et auction al. designs (2009), which suitablemake experimentally for use tests pricing alterna- of and theEngelmann removing same and troubled static Grimm assets. and (2009)Ausubel above, dynamic They except uniform descending that auction clock their asthey dynamic auction. this format sell The is study the units an and bidders for units also sale know individually their are or liquidityauctions needs. not as They resulted identical, find pooled in and that units.in the similar static substantially And, prices. and higher dynamic for However, bidders’manage some the payoffs, their dynamic sessions, which liquidity auctions enabled needs.of the resulted The price bidders dynamic discovery, as to auction well better was as also for better reducingAnother the in bidder study terms error. isstatic Manelli Vickrey et auction with al.Vickrey the (2006) auction, Ausubel which in auction, alsosetting, experimentally both in known compares an which as the the IPV theding values dynamic setting are in affiliated. and both They an typesrevenue conclude from of interdependent that the auctions, due value Vickrey but to auction (IV) Ausubel is slightly overbid- greater, more auction. while in But the the in efficiencytrade-off is Vickrey between the lower auction, efficiency IV in and the the setting,while revenue; they the the Vickrey observe revenue auction is less is higher overbidding more in and efficient the a Ausubel auction. 4 tion. They find thatcompared the to revenues an are open greateris auction; if employed. the a revenues sealed-bid depend format less is on used which as pricingKagel rule and Levin (2001)increases, demand theoretically reduction predict will thatmetrically diminish. in as They confirm a the this laboratory numbercording behavior to of experiment; asym- theory that bidders only is,Katzman if in (1995) their which rivals also decrease subjects providesmand in a behave reduction number, theory decreases ac- not indicating with ifdemand the that they reduction number increase. the of will prevalence participants always ofalso even be de- establish though present. that some Engelbrecht-Wiggans thereform et is auction al. no difference and (2006) in thefrom the Vickrey two first-unit-bid to auction between three when the (orin the uni- five), the number even Vickrey of though auction. bidders the second-unit-bid increases is always greater Concerning demand reduction, Noussair (1995) showedthe in bid a seminal on paper the that first second bid, unit which was was alwaysdepends lower always than demand-revealing. on The its degree value, whether in(2002) of the contrast under-revelation provide to bid a the sets formalKatzman (1995) the proof and price Engelbrecht-Wiggans of and ortions demand Kahn that (1998) not. reduction also involve in Ausubel demand analyze auc- reduction. the and uniform Crampton auction. Demand reduction hasdegrees, been for confirmed example in inin a experimental which field research experiment two byuniform to players List price, with various the and English, two-unit Lucking-Reiley ortory (2000) demand the experiment Vickrey by bid auction, Porter for and alsowhere Vragov replicated (2006). two demand in Another units reduction a laboratory labora- is experiment throughone confirmed the bidder is with by two-unitdemand Kagel demand and and compete playing Levin the against (2001); dominant they a strategy. let robot bidderWith with respect unit to learning,and we have Winter the (2002) evolutionary call paradigm,individuals or typically the do what ”competence not Nelson puzzle”, havethat the which are vast computational roughly imputed and to means cognitivethis them that powers article). by the But, optimization-based sinceremarkably theories learning, powerful (such guided even as by in that clearresponse addressing in short-term to complex feedback, the challenges, competence can the puzzle be evolutionary focuses on the role ofThe learning research and on practice. thewho winner’s claimed curse that is oil vast,survey starting companies of with suffered theory Capen fromoffered and et low by Kagel al. experiments returns. and (1971) A in Levin (2002). comprehensive single-unit, They show common that the value WC auctions is pervasive is across The procedure for generating the common value was as follows: The value of One of the justifications for the starting balance is that, even if participants the units for sale was generated by two random integers which were added play the risk-neutral Nash equilibrium, losses may occur. A starting balance together. Both units had the same value for the potential buyers, i.e. bidders also imposes opportunity costs for overly aggressive bidding, and is enough had flat demand curves. To construct a tight market, we let the integers be for errors made during bidding and a reasonable return for participating if chosen from the set 1, 2, ..., 6 . Thus, the possible values for each item aggressive bidders shut them out of the auction. were 2, 3, . . . , 12 .{ The bidders} were not fully informed about this value, though.{ Rather, each} bidder was independently and randomly shown one of the two integers as a private signal. For expository reasons, we displayed these signals as dice. Subjects were told that two dice were rolled and added up as 4 Theory and Hypotheses the common value, but each of them was only allowed to see one of the dice, independently of each other. Thus, the distribution of both the value and the The theoretical model in this article originates from Ahlberg (2009) where it signal was common knowledge. When the die was displayed on the screen, the is presented more thoroughly. An excerpt from it is available in Appendix A. subjects bid on both units. Only integer-value-bids between 0 and 12 were allowed. Subjects were also informed that the order of the bids was irrelevant. Starting with the revenue question, and beginning with 2-player groups, equa- Bidder instructions are find in Appendix C. tion 9 in the Appendix shows there to be a unique equilibrium strategy in the discriminatory auction; it prescribes the player to bid equal amounts on both Hence, a common value environment is created where private signals may be units. For the uniform and Vickrey auctions, there is no unique strategy; the used to create unbiased estimates of the value of the items. If ti is the signal, uniform auction has a multitude of equilibria, whereas the Vickrey auction the common value will lie in V = ti + 1, 2,...,6 and an unbiased estimator of has dualistic equilibria. One thing in common for both, however, is that they 7 { } the value (ex ante) is thenv î = ti+ 2 . Thus, the signals are positively correlated have extreme demand reduction equilibria, i.e. equilibria that prescribe a zero (affiliated) with the value. The underlying distribution of the private signals bid on the second unit and thereby transform the players into single-unit de- in the experiment was common knowledge; that is, everyone was told that she manders. Equations 11 and 13 in the Appendix also show these to be the would see one of the two dice and her competitors might, but not necessarily, payoff-dominating equilibria. The equations show that the equilibrium bid for see the same die. the first unit is to bid the conditional expected value, which makes the equilibrium risk dominate the other equilibria. (Bidders could also bid above this The group size was limited to two, three and four participants, respectively. value for the first unit, but with a higher risk.) Two approaches to allocating subjects to groups were used. In the first, each Thus, using the unique strategy in the discriminatory auction and the payoff- participant always competed against the same number of opponents, but not dominated equilibria in the two other auctions, the following expected revenues necessarily against the same opponents. Before each new round, all partici- for a two-player game emerges (see eq. 14 and 15): pants within each group size were re-randomized against each other. In the second approach, all participants were re-randomized against each other before each new round, irrespective of the group size. The reason for re-randomizing E[RD(2 Players)] = 11.22 each new round was to counteract subject-specific effects and tacit collusion. E[RU,V (2 Players)] = 0 Moreover, in advance of every new round, the common value of the last round was displayed on the screen alongside the two winning bids and the price paid where U stands for the uniform, D for the discriminatory and V for the Vickrey for the two units. Moreover, to ensure that comparisons among auction for- pricing rule. mats were not driven by particular configurations of value, the two integers constituting the value were randomly generated for each new auction. Since the ex ante expected value for the two units for sale is 14, we see that, in the discriminatory auction, the seller captures the major part of the value at Each subject got SEK 100 as a participation fee, or show-up fee, and a starting stake, but zero in the two other formats. Thus, the discriminatory auction gets balance of SEK 50 to cover losses. Profit and losses were added to this balance. the highest ranking. As regards the two other formats, the payoff-dominating If a participant’s balance went negative, he or she was suspended from the equilibrium for the uniform auction is somewhat more robust (in the unique- auction and had to leave (with the participation fee). The others were paid in ness of the bidder’s best response) than the payoff-dominating equilibrium of cash at the end of the experiment. the Vickrey auction (see section 9.5.3), which thus indicates that the Vickrey

6 7 ESSAY II for the Vickrey V 7 for the discriminatory and D 22 . (2 Players)] = 11 (2 Players)] = 0 D U,V stands for the uniform, R [ R [ U E E where 4 Theory and Hypotheses The theoretical model inis this presented article more originates thoroughly. from An excerpt Ahlberg from (2009) it whereStarting is with it the available in revenue question, Appendixtion and A. 9 beginning in with the 2-player groups, Appendixdiscriminatory equa- shows auction; there it to prescribes beunits. the a For player unique the to equilibrium uniform bid strategyuniform and equal in amounts Vickrey auction the on auctions, has both therehas a is dualistic no multitude equilibria. unique of One strategy;have thing equilibria, extreme the in whereas demand common reduction the for equilibria,bid Vickrey both, i.e. on equilibria however, auction the that is second prescribe thatmanders. unit a they zero and Equations thereby 11 transformpayoff-dominating and the equilibria. players The 13 into equations in single-unit showthe that de- the first the unit Appendix equilibrium bid is alsolibrium for to risk show bid dominate the these the conditionalvalue to other for expected equilibria. be the value, (Bidders first which the could unit, makes also but the bid with equi- above aThus, this using higher the risk.) unique strategydominated equilibria in in the the two discriminatory other auctionfor auctions, the and a following the two-player expected game payoff- revenues emerges (see eq. 14 and 15): One of the justificationsplay for the the risk-neutral starting Nashalso balance equilibrium, imposes is losses that, opportunity may evenfor costs occur. if errors A for participants starting made overly balance aggressive aggressive during bidders bidding, bidding shut and and them is out a of enough reasonable the return auction. for participating if pricing rule. Since the ex ante expectedthe value discriminatory for the auction, two the unitsstake, seller for but sale captures zero is in the 14, the majorthe we two other see part highest formats. that, of ranking. Thus, in As the theequilibrium discriminatory regards value auction for at the gets the two other uniformness formats, auction of the is the payoff-dominating somewhat bidder’s morethe best robust Vickrey response) (in auction than the (see the unique- section payoff-dominating 9.5.3), equilibrium which of thus indicates that the Vickrey is the signal, i t and an unbiased estimator of 6} 2,..., , 6 {1 . Thus, the signals are positively correlated + 7 2 }. Thus, the possible values for each item i t + i t = were re-randomized against each other. In the = V i v {1, 2, ..., 6 participants were re-randomized against each other before }. The bidders were not fully informed about this value, all within each group size {2, 3, . . . , 12 the value (ex ante) is then ˆ (affiliated) with the value.in The the underlying experiment was distribution commonwould of knowledge; see the that one is, private of everyone signals see the was told two the that dice same she and die. her competitors might, but not necessarily, The group size wasTwo limited approaches to to allocating two,participant subjects three always and to competed groups four against werenecessarily participants, the used. respectively. same against In number the the ofpants same first, opponents, each opponents. but not Beforesecond each approach, new round, all partici- the common value will lie in were though. Rather, each bidderthe two was integers independently as a andsignals private randomly signal. as For shown dice. expository one Subjects reasons,the of we were common displayed told these value, that but twoindependently each dice of of were each them rolled other. was andsignal Thus, only was the added allowed common distribution up to knowledge. of as When seesubjects both the one the die bid of value was the on and displayedallowed. on dice, the both Subjects the were screen, units. also the Only informedBidder that integer-value-bids instructions the between are order find 0 of in the and Appendix bids 12 was C. irrelevant. were Hence, a common valueused environment to is create created unbiased where estimates private of signals the may value be of the items. If The procedure for generatingthe the units common for value wastogether. sale as Both was follows: units generated The had valuehad by the of flat same two value demand random for curves.chosen integers the To which from potential construct were buyers, the a i.e. added set tight bidders market, we let the integers be each new round, irrespective ofeach the new group round size. wasMoreover, The to in reason counteract advance for subject-specific of re-randomizing every effectswas new displayed and round, on tacit the the common collusion. screenfor value alongside of the the the two two last winning round units.mats bids Moreover, and were the to not price ensure driven paid constituting that by the comparisons particular value among were configurations randomly auction of generated for- value, for the each new two auction. Each integers subject got SEK 100balance as of a SEK participation 50 fee, to orIf cover show-up losses. fee, a Profit and and a participant’s losses starting auction balance were added and went to had negative, this to balance. he leavecash (with at or the the she participation end fee). was of The suspended the others experiment. were from paid the in auction should deliver weakly more revenue than the uniform auction. The same is also true for the uniform auction, because of diminishing incentives for demand reduction the more players there are. From conjectures 7 and 8, When there are more than two players in the auctions, the extreme demand- we have that both the bids on the first and the second units (weakly) increase reduction strategy of bidding zero on the second unit disappears. (Since it is with the number of participants. Thus, in terms of expected revenue, we have always an equilibrium to bid the conditional expected value on the first unit, notwithstanding the group-size, the price-setting bid will never be zero; thus, U U U a zero-bid on the second unit does not gain anything.) However, the bidders E[R (2 Players)] >E[R (3 Players)] >E[R (4 Players)]. must now be cautious about the first unit bid, as it can be the price-setting bid. Following the comparison of conjectures 7 and 8 in Appendix B for the Going from two to three players, we have a clear revenue ranking following uniform auction with the strategies for the discriminatory auction in section the disappearance of the extreme demand reduction. The next step is less 9.5.1, we have that the uniform auction always gives a greater revenue than pronounced, but at least there is a distinct difference in revenue. the discriminatory auction when there are 3 or 4 players. Hence, Thereby, we have the following hypothesis:

U D Hypothesis 2 (The number of bidders) The revenue will increase with E[R (3 Players)] >E[R (3 Players)] the number of bidders. Moreover, we expect to see a greater diﬀerence between U D E[R (4 Players)] >E[R (4 Players)]. two and three-player groups as compared to three and four-player groups. This should be true for both the discriminatory and the uniform auctions.

In the Vickrey auction, all pure equilibria disappear due to non-core outcomes A third hypothesis concerns the uniform and the Vickrey auctions. We have since there is always a coalition for which the total payoff becomes higher; seen (above) that, when there are only two bidders, there could be zero bids notwithstanding the individual strategy. Thus, we can formulate the following on the second unit. This, in turn, gives zero revenue to the seller. Hence, hypothesis: Hypothesis 3 (Demand reduction) When there are two bidders, bids on Hypothesis 1 (Revenue Comparison) If there are only two bidders in the the second unit will be (much) lower than the expected value (i.e. under- auction, the discriminatory auction will outperform the two other mechanisms, revealing) in the uniform and Vickrey auctions. and the Vickrey auction will have a marginally higher rank than the uniform auction. With more bidders, the uniform auction is likely to give more revenue The fourth hypothesis concerns profit maximization and learning, i.e., evolu- than the discriminatory auction. tionary aspects of the bidding process. Each settled round gives feedback on the performance which, correctly interpreted, gives an indication of how to The second topic is how the number of bidders influences the bidding and bid in the next round. So even if subjects do not understand how to compute hence, the revenue. The expected revenue is calculated using the unique strate- an equilibrium strategy, they may roughly learn a rule of thumb. gies for the three group sizes of the discriminatory auction (see eq. 14 in the Appendix): Thus, an iterative process may be needed to approach equilibrium play.

Hypothesis 4 (Learning) Subjects are likely to use strategies closer to (the- E[RD(2 Players)] = 11.22 (1) oretical) equilibrium play over time. D E[R (3 Players)] = 12.38 (2) Last, the winner’s curse (WC) is scrutinized. The first to recognize the WC E[RD(4 Players)] = 12.63. (3) was Capen et al. (1971) who argued that the low rates of return among oil companies in the 1960s and 1970s on OCS lease sales, year after year, resulted If we start with a two-player game and increase the number of bidders by one, from bidders’ ignorance about the informational consequences of winning. the revenue increases by almost 10 percent. If we go from the three to the four-player game, the revenue increase is just 2 percent. Since going to five Hence, we define the WC as the adverse selection effect of bidders neglecting players only increases the revenue by 1.5 percent, we also see that using four the information a win will produce. That is, that the announcement of winning players actually captures the idea of ”many” bidders. Thus, in this setting, leads to a decrease in the estimated value of the objects, if not accounted for the expected revenue seems to increase concavely with the number of bidders. when bidding (given a symmetric game and that the high signal holder(s)

8 9 ESSAY II (4. Players)] U R [ The revenue will increase with >E When there are two bidders, bids on 9 (3 Players)] Subjects are likely to use strategies closer to (the- U R [ >E (2 Players)] U R [ E The same is also true forfor the uniform demand auction, reduction because of thewe diminishing incentives more have that players both there thewith are. bids the From on number conjectures the of first 7 participants. and Thus, and the in 8, second terms units of (weakly) expected increase revenue, we have the second unitrevealing) in will the be uniform and (much) Vickrey lower auctions. The than fourth hypothesis the concerns expectedtionary profit aspects value maximization of and (i.e. the learning,the under- bidding i.e., performance process. evolu- which, Eachbid correctly settled in round interpreted, the gives gives next feedbackan an round. on equilibrium So indication strategy, even of they if may subjects how roughly do to learn not aThus, understand rule an how of iterative to process thumb. compute may be neededHypothesis to approach 4 equilibrium (Learning) play. oretical) equilibrium play over time. Last, the winner’s cursewas (WC) Capen is et scrutinized. al.companies The in (1971) the first who 1960s to and arguedfrom recognize 1970s bidders’ that on the ignorance OCS the WC about lease low sales, the rates year informational after of consequences year,Hence, return of resulted we winning. among define oil thethe WC information as a the win will adverseleads produce. selection That to effect is, a that of decrease the bidderswhen announcement in neglecting of the bidding winning estimated (given value a of the symmetric objects, game if and not accounted that for the high signal holder(s) Going from two tothe three disappearance players, of wepronounced, the have but a extreme at least clear demand there revenue reduction. is ranking a The following Thereby, distinct next we difference have step in the revenue. is following hypothesis: less Hypothesis 2 (The numberthe of number of bidders) bidders.two Moreover, we and expect three-player to groups as seeshould a compared to be greater three true difference and between for four-player both groups. the This discriminatoryA and third the hypothesis uniform concerns auctions. seen the (above) uniform that, and whenon the there the Vickrey are second auctions. only unit. We two This, have bidders, in there turn, couldHypothesis gives be zero 3 revenue zero (Demand to bids reduction) the seller. Hence, (1) (2) (3) If there are only two bidders in the 5 percent, we also see that using four . 8 (4. Players)] (3 Players)] D D 22 38 63. R R . . . [ [ >E >E (2 Players)] = 11 (3 Players)] = 12 (4 Players)] = 12 (4 Players)] (3 Players)] D D D U U R R R R R [ [ [ [ [ E E E E E In the Vickrey auction, allsince pure equilibria there disappear is duenotwithstanding to always the non-core a individual outcomes strategy. coalition Thus,hypothesis: we for can which formulate the the following total payoff becomesHypothesis higher; 1 (Revenue Comparison) auction, the discriminatory auction willand outperform the the two Vickrey other auction mechanisms, auction. will With more have bidders, a thethan marginally uniform the auction higher discriminatory is rank auction. likely than to the give uniform more revenue The second topichence, is the revenue. how The expected the revenuegies is number for calculated of using the the bidders three uniqueAppendix): strate- group influences sizes the of bidding the and discriminatory auction (see eq. 14 in the auction should deliver weakly more revenue than theWhen uniform there auction. are morereduction than strategy two of players in biddingalways the zero an auctions, on equilibrium the the to extreme secondnotwithstanding bid demand- the unit the group-size, disappears. conditional the expected (Sincea price-setting value it zero-bid bid on is will on the never the firstmust be unit, second now zero; unit be thus, does cautiousbid. not about Following gain the the anything.) first comparison However,uniform unit of the auction bid, conjectures bidders with as 7 the9.5.1, it and strategies we can 8 for have be in the that the Appendixthe discriminatory the price-setting discriminatory B auction uniform auction for in auction when the section there always gives are 3 a or greater 4 revenue players. than Hence, If we start with athe two-player game revenue and increases increase the byfour-player number game, almost of the bidders 10 by revenue percent.players one, only increase If increases is we the justplayers go revenue 2 by actually from 1 percent. captures the Since thethe three expected going idea to revenue to of seems the five to ”many” increase bidders. concavely with Thus, the in number this of bidders. setting, win(s) the objects). The underlying cause in this study is that, even though Moreover, our way of generating the signal from the CV makes the CV up- 7 the signal plus the expected value (EV) of the other integer is an unbiased wardly biased from the signal. That is to say, even thoughv î = ti + 2 is unbi- estimator of the value, the max operator of all ti +7/2 is not; it is a convex ased, ti underestimates the CV. In, for example, Kagel et al. (1987), where the function and thus overestimates the value. signal is drawn from a set consisting of � of the CV, there is a certain region where the signal by itself is an unbiased± estimator of the CV. We believe that In the present design, the lower bound estimate of the value is (ti + 1) and the latter method will produce more WC due to the fact that in fifty percent the upper bound is (ti + 6). The strategy of bidding the risk-free lower bound of the draws, the signal is below the CV. strategy never yields a negative payoff, whereas bidding above the upper bound would ensure a negative payoff. 2 The unbiased, though naive, EV of the items There is vast experimental evidence of the WC for both inexperienced players is E(v t )=(t +7/2). It is naive in the sense that it is the EV, independent of and professionals in single-unit auctions, see Kagel (1995). Due to the inherent | i i winning the item(s). Define E(v ti >t i) as the EV, for player i, conditional demand reduction equilibria in two of the auction formats, the WC should be | − on having the highest signal, ti. Since the informational content of winning lower on the second unit for sale. Thus, we conclude that: leads to a decrease in the estimated value, and cannot be lower than the lower Hypothesis 5 (Winner’s curse) The winner’s curse will be apparent, but bound, it must lie in the interval ti +1,ti +7/2 . For ti 1, 2,...,6 it is { } ∈{ } not so much as reported in other experiments since the common value in the E(v ti >t i)=3ti/2. | − present experiment is biased upwards. And the WC should be considerably A bidder who does not take this fully into account and uses the naive EV smaller for the second unit, as compared to the first. instead of the appropriate EV conditional of winning when placing her bids could, upon winning, pay more than the estimated value of the object(s). The systematic failure to account for this is referred to as the winner’s curse. 5 Experimental Results

The difference E(v ti) E(v ti >t i) decreases with the signal ti; meaning | − | − that the greater the signal, the less the bidder has to shade the bid to account We conducted 15 or 20 rounds of bidding for each subject; the number was for the winner’s curse. Or, stated differently, it is worse to find out that one stochastically determined, not known in advance by the subjects (they did won with a low signal rather than a high. not know how many rounds they were going to play). The data description is in Table 1, which shows, for each format, the number of subjects, how many In the present analysis, we discriminate between bidding in the WC interval, rounds there were, the number of unique observations, and the average profit. which s then above E(v ti >t i) up to the naive E(v ti), and bidding above Each format consisted of groups of two, three and four bidders. | − | the latter. And since it is risk-free to bid (ti + 1), the interval in question, i.e. the WC interval, becomes t +2,t +3 . 3 The reason for the division of the No. of subjects No. of rounds Unique observationsπ ¯ { i i } intervals is that, theoretically, bidding above the naive EV has nothing to do Discriminatory 62 345 941 0.37 with the WC. Bidding above the naive EV will, on average, produce a negative profit. But since this discrimination is not made in other experiments, we will Uniform 44 256 745 0.36 also pool the result. Vickrey 46 286 777 1.04 Table 1 2 That is to say, bidding in the discriminatory auction. For the uniform and Vickrey Data summary auctions, where players do not pay what they bid, the word bidding should be interpreted as paying. 3 There were two experimental designs; one configuration where subjects re- Bidding above the unbiased estimate ti +7/2 is not rational, ex ante. But, there is also what Holt and Sherman (1994) call a loser’s curse. The expected value of mained in the same group size in all rounds, i.e. the number of competitors the item, conditional on not winning, is greater than the naive expected value. In was always constant for them, but the competitors changed; and another where 3ti+7 subjects were randomized without any constraints in all rounds, i.e. both the this model it is, for bidder i, E(v ti ti +7/2=E(v ti). The | − 2 | difference E(v ti

10 11 ESSAY II π is unbi- 7 2 + i t = i v 941745777 0.37 0.36 1.04 of the CV, there is a certain region � 345 256 286 ± 11 The winner’s curse will be apparent, but 62 44 46 No. of subjects No. of rounds Unique observations ¯ underestimates the CV. In, for example, Kagel et al. (1987), where the i t Discriminatory Uniform Vickrey 5 Experimental Results We conducted 15 orstochastically 20 determined, rounds not ofnot known bidding know in for how many advance each roundsin by subject; they Table the the were 1, going subjects which number torounds shows, was (they play). there for The were, did each the data format, number descriptionEach the of format is unique number consisted observations, of of and subjects, groups the how of average profit. many two, three and four bidders. where the signal by itselfthe is latter an method unbiased will estimatorof produce of the more the draws, WC CV. the due We signal believe to is that the below factThere the that is CV. in vast fifty experimental percent evidenceand of professionals the in single-unit WC auctions, fordemand see both reduction Kagel inexperienced (1995). equilibria players Due in to twolower the of on inherent the the auction second formats, unit the for WC sale. shouldHypothesis Thus, be we 5 conclude (Winner’s that: curse) not so much aspresent reported experiment in other issmaller experiments biased since for upwards. the the common And second unit, value the as in WC compared the to should the first. be considerably signal is drawn from a set consisting of Table 1 Data summary There were twomained experimental in designs; the one samewas configuration always group constant for where size them, but subjects insubjects the competitors all re- were changed; rounds, randomized and another without i.e. where number any the constraints of number in individual of allusing competitors competitors rounds, the i.e. and both highest the the least (or competitors squares lowest) changed. regression, bid But,influence. these as when Nor when different the the designs dependent design interacts do variable with not in rounds, auction have an format any or ordinary group significant Moreover, our way ofwardly generating biased from the the signal signal. from That the is to CV say, makes even the though CV ˆ up- ased, it is } ). The i 6 t | + 1) and v ; meaning ( i i t E 2,..., i, conditional 1, /2= , meaning that the i ∈{ t i ), and bidding above +7 t i i t 2 is not; it is a convex | / v ( >t E +7 }. For i +7 t 2 i /2 3t + 1), the interval in question, i.e. The reason for the division of the +7 i i /2 is not rational, ex ante. But, there t 3 )= i ,t ) as the EV, for player }. − +7 −i i ) decreases with the signal t +1 10 +3 i t i i {t i t ,t The unbiased, though naive, EV of the items t | ) up to the naive | >t . Since the informational content of winning v ) increases with the signal i 2 v ( i i t −i ( t t +2 | | E i E v v ( ( >t i, {t E i E t | − v − + 6). The strategy of bidding the risk-free lower bound ( ) i ) i E −i t | /2. /2). It is naive in the sense that it is the EV, independent of v i ( t t | i v t ( | v E That is to say, bidding in the discriminatory auction. For the uniform and Vickrey Bidding above the unbiased estimate ( function and thus overestimates the value. In the present design, the lower bound estimate of the value is (t E the upper bound is (t the latter. And since it is risk-free to bid ( intervals is that, theoretically, biddingwith the above WC. the Bidding naive above the EVprofit. naive But has EV since nothing will, this on to discriminationalso average, do produce is pool a not the negative made result. in other experiments, we will that the greater the signal,for the the less winner’s the bidder curse.won has Or, with to stated a shade low differently, the it signal bid to rather is account than worse a toIn high. the find present out analysis, thatwhich we one s discriminate then between above bidding in the WC interval, greater the signal, thecurse. By more inspection the of biddercurse is the might non-existent. data, have bid we conclude to that account in for this the experiment, loser’s the loser’s 2 3 win(s) the objects). Thethe underlying signal cause plus inestimator the this of expected study the value is value, (EV) that, the of even max though the operator other of integer all is an unbiased auctions, where playersinterpreted do as not paying. pay what they bid, the word bidding should be A bidder whoinstead does of not the take appropriatecould, this upon EV fully winning, conditional pay into of moresystematic account than winning failure the and when to estimated account placing uses value for of her the this the bids naive object(s). is The EV referredThe to difference as the winner’s curse. difference is leads to a decrease inbound, the it estimated must value, and lie cannot in be the lower interval than the lower strategy never yields a negative payoff, whereaswould bidding ensure above a the negative upper payoff. bound is also what Holtthe item, and conditional Sherman on (1994)this not call model winning, a it is loser’s greater is, curse. than for The the bidder expected naive expected value of value. In the WC interval, becomes on having the highest signal, winning the item(s). Define size is there a significant effect on the highest (or lowest) bid. Therefore, the Result 1 (Revenue Comparison) In 2 player groups, the discriminatory data from the two experimental designs is pooled. together with the uniform auction collects− significantly more revenue than the Vickrey auction. This is contrary to the hypothesis that the discriminatory Weuseafirst unit bid and a high bid interchangeably, meaning the (weakly) auction should outperform the uniform auction. When there were more bid- highest bid of the two bids that each subject submits. Likewise, a second ders, the hypothesis was that the uniform auction should give a weakly higher unit bid and a low bid refer to the (weakly) lower bid of the two. The non- revenue than the discriminatory auction; which did not happen either. parametric Wilcoxon(-Mann-Whitney) rank sum test is used for comparing data between treatments, if not stated differently. There seems to be no prob- Hypothesis 2: lem with dependencies within subjects, nor within groups. We have made tests with OLS and Panel data (random effects) models with revenue (price) as the Table 2 also indicates that larger auction groups give more revenue. All p- dependent variable. Revenue is explained by format, group-size, round and values except one are below 0.01; the one above is 0.1156 and concerns the design. We have also made interactions between group-size and format on the discriminatory auction between 3- and 4-groups. above. Moreover, we have used both the difference in bids and equilibrium bids as dependent variables, explained by the same covariates as the former, and Result 2 (The number of bidders) The result for both the uniform and interactions between group-size and format. The below presented results only the Vickrey auctions supports the hypothesis that the revenue increases with the changed marginally and, thus, the conclusions still hold. (The OLS regression number of bidders. For the discriminatory auction, the result is not as strong on revenue can be found in Appendix B.) because of the high significance level (15-percent level) between two group sizes; but the result, therefore, verifies that the revenue increase is concave in that Last, in the discriminatory auction, roughly 11 percent of the subjects went format. bankrupt. For the uniform auction, that portion was only about 4 percent, whereas the Vickrey auction had zero bankruptcies. Hypothesis 3:

Table 2 displays the average revenue for different auction formats (rows) and Demand reduction, or bid shading, means that bidders do not apply the group sizes (columns). The numbers inside the brackets are the revenues in demand-revealing strategies. In the discriminatory auction, the unique sym- Bayesian equilibrium, to the extent that it is found. (See eq 1 for the discrim- metric strategy for bidders is to bid equally on both units. Hence, there should inatory auction. The uniform and Vickrey auctions are the payoff dominating not be any demand reduction in that format. However, in both the uniform Bayesian equilibrium, that is, the extreme demand reduction equilibrium.) and the Vickrey auctions, there are optimal strategies that are both demand revealing and not. In the latter strategies, the bidders will always bid below the Group Size 2 3 4 pooled expected value for the second unit, possibly zero. We will also report demand reduction for the discriminatory auction. Discriminatory 11.70 (11.22) 13.41 (12.38) 14.12 (12.63) 12.81 (11.90) Uniform 11.70 (0) 12.89 14.69 12.97 For all three formats, first, Table 3 shows the frequency of the bid-spread and, second, the value of the bid-spread, third, given that the bids are not equal, Vickrey 9.49 (0) 11.65 14.59 11.22 what is the frequency for Bid 1 to be above the EV and, finally, given that Table 2 the bids are not equal, with what frequency Bid 2 is below the EV. In other Average revenue, with predicted revenue inside the parenthesis. words, the next to last column shows if the subjects engaged in a bid-spread overbid or not for the first unit, and, in the last column, if they underbid or Hypothesis 1: not for the second unit.

Overall, there are no significant differences between the discriminatory and Bid-spread Bid-spread value Bid 1 > EV Bid 2 < EV the uniform auction as concerns concerning revenue. The Vickrey format is Discriminatory 0.58 0.88 0.15 0.90 inferior, especially for small group sizes. Interestingly enough, the larger the Uniform 0.78 1.85 0.55 0.70 group size, the closer the revenues are between the auctions. Looking at 4- groups alone, the formats are not significantly different from each other. In Vickrey 0.73 1.41 0.50 0.69 the other group-sizes, and when groups are pooled, the Vickery auction collects Table 3 significantly less revenue than both other formats (p values < 0.01). Frequency and value of bid-spread. −

12 13 ESSAY II EV < EV Bid 2 > 1156 and concerns the 0.150.550.50 0.90 0.70 0.69 player groups, the discriminatory − 2 The result for both the uniform and In 0.88 1.85 1.41 13 01; the one above is 0. . 0.58 0.78 0.73 Bid-spread Bid-spread value Bid 1 Uniform Vickrey Discriminatory Table 3 Frequency and value of bid-spread. together with the uniformVickrey auction auction. collects significantly This moreauction is revenue should than contrary the outperform toders, the the the uniform hypothesis hypothesis auction. was thatrevenue When that than the there the the uniform discriminatory were discriminatory auction more auction; should bid- which give didHypothesis a not 2: weakly happen higher either. Table 2 also indicatesvalues except that one larger arediscriminatory auction auction below groups between 0 give 3- more and 4-groups. revenue. AllResult p- 2 (The number of bidders) Result 1 (Revenue Comparison) the Vickrey auctions supports the hypothesisnumber that of the bidders. revenue increases For withbecause the the of discriminatory the high auction, significance the levelbut result (15-percent level) the is between result, two not therefore, groupformat. as sizes; verifies strong that the revenue increase isHypothesis concave 3: in that Demand reduction, ordemand-revealing bid strategies. In shading, themetric means strategy discriminatory for that auction, bidders is the biddersnot to unique bid be do equally sym- any on not both demandand units. apply reduction the Hence, in Vickrey there the should auctions, thatrevealing there and format. not. are However, In optimal in the strategies latterexpected both strategies, that value the the for are bidders uniform will the both always secondreduction bid demand unit, below for the possibly the zero. discriminatory We auction. will also reportFor all demand three formats, first,second, Table 3 the shows value the of frequencywhat of the is the bid-spread, bid-spread the third, and, frequencythe given for bids that are Bid the not bids 1words, equal, are to the with not next be what equal, to aboveoverbid frequency last the or Bid column EV not 2 shows and, for is ifnot the finally, below the for given first the subjects the that unit, EV. second engaged and, In unit. in in other a the bid-spread last column, if they underbid or second pooled 01). 0. < values 4 p− 3 interchangeably, meaning the (weakly) 12 high bid 2 refer to the (weakly) lower bid of the two. The non- and a 9.49 (0) 11.65 14.59 11.22 11.70 (0) 12.89 14.69 12.97 11.70 (11.22) 13.41 (12.38) 14.12 (12.63) 12.81 (11.90) low bid and a Group Size Discriminatory Uniform Vickrey highest bid ofunit the bid two bidsparametric that Wilcoxon(-Mann-Whitney) rank eachdata sum subject between test treatments, submits. if is Likewise, notlem used stated with a for differently. dependencies There within comparing seems subjects,with to nor OLS be within and groups. no We Panel prob- have datadependent made (random tests variable. effects) models Revenue with isdesign. revenue We explained (price) have as also by the madeabove. format, interactions Moreover, we between group-size, have group-size used round and bothas format and the difference on dependent in the variables, bids explained andinteractions equilibrium by between bids the group-size and same format.changed covariates The marginally as below and, the presented thus, former, results theon conclusions only and revenue still can hold. be (The found OLS in regression AppendixLast, B.) in the discriminatorybankrupt. auction, For roughly the 11 uniformwhereas percent the auction, of Vickrey that the auction portion had subjects zero was went bankruptcies. onlyTable about 2 displays 4 the percent, averagegroup revenue sizes for (columns). different auction TheBayesian equilibrium, formats numbers to (rows) inside the and extent theinatory that auction. brackets it The are is uniform the found. andBayesian (See revenues equilibrium, Vickrey eq auctions that in 1 are is, for the the the payoff extreme discrim- dominating demand reduction equilibrium.) Hypothesis 1: Overall, there arethe no uniform significant auction differencesinferior, as between especially concerns the for concerning discriminatory smallgroup revenue. and group The size, sizes. Vickrey the Interestingly formatgroups closer enough, is alone, the the the revenues larger formats arethe the other are between group-sizes, and the not when groups significantly auctions.significantly are less different pooled, Looking revenue the from at than Vickery auction each both 4- collects other. other formats In ( size is there adata significant from effect the on twofirst the experimental highest unit designs (or bid is lowest) pooled.Weusea bid. Therefore, the Table 2 Average revenue, with predicted revenue inside the parenthesis. The uniform and the Vickrey auction have overlapping strategies, given the unit. All formats differ significantly from each other on both the frequency and same signal, even though the price rule differs between them. The discrimina- the value of the bid-spread. And since the uniform and Vickrey auctions do tory auction has both different strategies, given the same signal, and a different not differ in the under revealing of the second unit bid frequency, we conclude pricing rule. This explains the similar frequencies in the last two formats as that the uniform auction produces more demand reduction than the Vickrey well as the great discrepancy between them and those of the discriminatory auction. As for the discriminatory auction, where the symmetric equilibrium auction. does not predict demand reduction, the format has significantly lower values and frequency compared with the other two. Hypothesis 3 suggests that there should be, if not complete, at least a great under-revelation for the second bid in both uniform and Vickrey auctions. Hypothesis 4: First, not seen in the table, we conclude that there is very little extreme demand reduction behavior, that is, zero bids on the second unit, even though For the discriminatory auction, we measure learning as the share of first and this is the payoff-dominating strategy in games with two players for the uni- second unit bids, consistent with the theoretical, extended-equilibrium strategy where the extended-equilibrium strategy is defined as: b ,b (b∗ 1,b∗ + form and Vickrey auctions (only 5 percent of the bids in the uniform auction, 1 2 ∈ − and 3 percent in the two other formats). There is no significant difference in 1). Does the share of bids in this interval increase with the number of rounds zero-bids between the formats. played?

If we use the bid-spread as a measure for demand reduction, we see that the Equipped with this definition, a learning effect in the discriminatory auction can be seen in Table 4. Moreover, this effect is concave. Between rounds 4 6 uniform and the Vickrey auction have quite the same frequency of demand − reduction; whereas the discriminatory auction has a lower frequency. Never- and the middle rounds, it is significant at the one-percent level, and between theless, all three formats are significantly different from each other (p-values the middle rounds and the last three rounds, it is significant at the five-percent < 0.022). Comparing the values, there is a larger spread between the for- level. mats; which is a reflection of the significant difference between them (p-values Round 4 6 11 13 18 20 < 0.001). − − − Discriminatory 61 % 75 % 85 % The next-to-last column tells us that, for the uniform and Vickrey auction, Table 4 half of the subjects who engage in demand reduction also bid above the value Frequencies of (extended) optimal bids on the first unit bid. Thus, we must look at the second unit bid to understand if demand reduction is present. (It could be the case that both bids are above Regarding the uniform and the Vickrey auctions, we do not have any equilib- value, which would then not really be demand reduction.) This cannot be rium strategy prediction for more than two players. Thus, we concentrate on compared to the much lower frequency in the discriminatory auction, where 2-player groups, and measure learning as finding the payoff-dominating equi- the winning bids become the price. In the last column, we see the frequency of librium strategy. Hence, do subjects increase the number of zero-bids on the under-revealing bids. All bids should be under revealing in the discriminatory second unit along with the rounds played, or, at least lower the second unit auction, which is almost the case. More interesting, both the uniform and bid as the session continues? the Vickery auction have about seventy percent second unit bids below the expected value. 4 The two formats do not differ significantly from each other There was no such effect in the Vickrey auction; whereas there was a tendency in this respect. to it in the uniform auction. That is to say, the p-value was 0.1125 when comparing the second unit of the early rounds with the last three rounds. Result 3 (Demand reduction) We find evidence of demand reduction on the second unit but, in contrast to hypothesis 3, very few zero bids on the second Result 4 (Learning) In the discriminatory auction, the subjects moved to- wards the optimal strategy over time, consistent with hypothesis 4. The sub- 4 The 70% share of bid 2 below EV almost coincides with the Porter and Vragov jects also continued to learn in later periods, but to a lesser extent. That is, (2006) result. They had a 68% share in an IPV experiment with two bidders and the learning effect is concave (at least between the measuring points). In the two units for sale. But they just got a 30% share for the Vickrey auction. (One must uniform auction, the learning was barely significant, but the subjects seemed be cautious when making a comparison with their results since they have a different to weakly understand the demand-reduction equilibria over time (rounds). value system to the one in this study.) Moreover, we are now using the EV, and not the EV conditional of winning, since we are looking at all subjects. Comment:

14 15 ESSAY II 6 + ∗ − 1,b − ∗ b 1125 when ( 4. The sub- ∈ 2 ,b 1 b 20 − 13 18 − 6 11 15 − In the discriminatory auction, the subjects moved to- Discriminatory 61 % 75 % 85 % Round 4 : 1). Does the shareplayed? of bids in this interval increase withEquipped the number with of this rounds definition,can a be learning seen effect inand in Table 4. the the Moreover, middle discriminatory this rounds, auction effectthe it is middle is concave. rounds Between and significant rounds the atlevel. last 4 the three rounds, one-percent it level, is and significant at between the five-percent comparing the second unit of the earlyResult rounds 4 with the (Learning) lastwards three the rounds. optimal strategyjects over also time, continued to consistentthe with learn learning in hypothesis effect later isuniform periods, concave auction, but (at the to leastto learning a between weakly was the lesser understand barely measuring extent. the significant, points). That demand-reduction but In equilibria is, the over the Comment time subjects (rounds). seemed Regarding the uniform andrium the Vickrey strategy auctions, prediction we for do2-player more groups, not than and have two any measure players. equilib- librium learning Thus, strategy. as we Hence, finding concentrate do the on second subjects payoff-dominating unit equi- increase along the withbid number the as of rounds the zero-bids session played, on continues? or, the at leastThere lower was no the such second effectto in unit the it Vickrey auction; in whereas there the was uniform a tendency auction. That is to say, the p-value was 0. Table 4 Frequencies of (extended) optimal bids unit. All formats differthe significantly from value each other of onnot the both differ the bid-spread. in frequency And and thethat since under the the revealing of uniform uniform theauction. and auction second unit As Vickrey produces bid for more auctions frequency, demand thedoes do we discriminatory not reduction conclude than auction, predict demand where theand the reduction, Vickrey frequency the symmetric compared with equilibrium format the has other two. significantly lowerHypothesis 4: values For the discriminatory auction,second we unit measure bids, learning consistent asegy with the where the share the theoretical, extended-equilibrium of strategy extended-equilibrium first is and strat- defined as: , very few zero bids on the second 3 14 We find evidence of demand reduction on The two formats do not differ significantly from each other 4 022). Comparing the values, there is a larger spread between the for- 001). . . 0 0 The 70% share of bid 2 below EV almost coincides with the Porter and Vragov 4 The uniform and thesame signal, Vickrey even auction though have thetory overlapping auction price has strategies, rule both differs given different between strategies,pricing the given them. rule. the The same This discrimina- signal, and explainswell a the different as similar the frequencies greatauction. in discrepancy the between last them two and formats those as ofHypothesis the 3 discriminatory suggests thatunder-revelation there for should the be,First, second if not not bid seen complete, in inmand at the both reduction least table, behavior, uniform a we that great andthis conclude is, is that Vickrey zero the there auctions. bids payoff-dominating isform on strategy very and the in little Vickrey second games extreme auctions unit, with (only de- and 5 even two 3 percent players though percent of for in thezero-bids the the bids between uni- two in the other the formats. uniform formats). auction, There is noIf we significant use difference the in uniform bid-spread and as the a measurereduction; Vickrey for whereas auction demand the have reduction, discriminatory quitetheless, we auction the all see has three same that a formats the frequency< are lower of frequency. significantly demand Never- differentmats; from which each is other a (p-values reflection< of the significant difference between them (p-values The next-to-last column tellshalf of us the that, subjects foron who the engage the first in uniform unit demand bid. andif reduction Thus, demand also Vickrey we reduction bid must auction, is above look present.value, the at (It value the which could second would be unit thecompared bid then case to to not understand that the both really muchthe bids be lower winning are frequency bids above demand become in reduction.) theunder-revealing the price. bids. This discriminatory In All the cannot auction, bids last should where auction, be column, be we which under see revealing the is in frequencythe almost of the Vickery discriminatory the auction case. haveexpected More about value. seventy interesting, percent both second the unit uniform bids and below the (2006) result. They hadtwo units a for 68% sale. But sharebe they in cautious just an when got making a IPV a 30%value experiment comparison share system with for with to their the the two results Vickrey one bidders auction.the since in (One they EV and this must have conditional study.) a Moreover, of different we winning, are since now we using are the EV, looking and at not all subjects. in this respect. Result 3 (Demand reduction) the second unit but, in contrast to hypothesis There is especially one odd result here when compared to theory, which spurs EV b > EV Winner’s curse Total both anomalies in hypotheses 1 and 3. The subjects’ bids in 2-player groups n ≥ i c were expected to be (much) lower in the uniform and the Vickrey auctions. b1 b2 b1 b2 b1 b2 Even if the revenue does not go to zero as predicted by theory, it should at least be much lower (than the discriminatory auction). Maybe the competitive Discriminatory 0.42 0.35 0.42 0.57 0.18 0.20 element, or the joy of winning, 5 overtook any rationale in these groups. Even though some subjects understood having to play zero on the second unit and Uniform 0.37 0.32 0.47 0.42 0.17 0.13 high on the first, their opponents seldom did. Another prominent feature in Vickrey 0.36 0.30 0.34 0.33 0.12 0.10 the experiment is the low revenue outcome in the Vickrey auction for two- Table 5 and three-player games. Supposedly because of its complicated nature, the The frequency of winner’s curse bids and actually experienced winner’s curse. subjects did not seem to understand this. At a first glance, there is no significant difference between the first and second Hypothesis 5: unit prices within each auction type (the p-values starting from 0.1229 and rising.). But looking more closely reveals a pattern due to the lower frequency As described above, the WC interval is defined as bids above the EV con- of second unit bids/prices vs. first unit bid/prices in all three formats. (There ditional on winning, EVc = E(v ti >ti)=3ti/2, up to the naive EV, is an interval ranging from 0.05 to 0.07 between the two bids.) This is also | − EVn = E(v ti)=(ti +7/2). Since it may not be intuitive to grasp the under- confirmed with a p-value of 0.0545, when testing all formats together for dif- | lying cause of the winner’s curse, i.e. the convexity of the max function, bids ferences between the two bids. Hence, we have a distinct difference between in this interval could be rationalized on the basis of the fact that they are bids for subjects when pooling all auction formats. (at least) below the naive expected value. However, bids above E(v ti) are, on average, never individual-rational since they produce a negative profit| on Switching to a comparison between auction formats, and once more testing for average. 6 Moreover, to separate the random component from the actual bid, differences between bids in the WC-interval, we only find a difference between we distinguish between bidding in the WC interval and actually experiencing the discriminatory and Vickrey auctions; the p-value is 0.0928 when comparing a negative profit, i.e. suffering from the winner’s curse, in Tables 4 to 6 below. first unit prices.

In the uniform and Vickrey auctions, the bid is just a proxy for the price, Since the difference between the first and second unit bids is weak in this since subjects do (often) not pay what they bid. The price-setting bid could case, Table 8 shows the results when pooling first and second unit bids. It can come from any bidder in the uniform auction but, in the Vickrey auction, it is always another player’s bid that becomes the price-setting bid. Thus, in these EVn bi > EVc Winner’s curse Total two formats, we are measuring more like a generalized WC; a WC within each ≥ group. All bid frequencies, or prices, in table 5 are conditional on both winning b1 & b2 b1 & b2 b1 & b2 and having the high signal. That is, as stated in the above paragraph, even though we are using bid 1 and bid 2 in the table, it is the price that these bids Discriminatory 0.40 0.46 0.18 generated that we are measuring for the last two formats. Uniform 0.36 0.46 0.17

The table is to be interpreted as follows: In the discriminatory auction, 35 Vickrey 0.34 0.34 0.12 percent of the second-unit bids were in the WC interval. Of these, 57 percent Table 6 de facto gave a negative proﬁt. Thus, a total of 20 percent second-unit bids The frequency of winner’s curse bids and experienced winner’s curse, when bids 1 gave negative proﬁts. & 2 are pooled.

5 Cox et al. (1992) tried to explain overbidding in IPV first price auctions with ’joy be seen that the uniform and the discriminatory auctions are almost identi- of winning’ and Cooper and Hanming (2008) partly support a modified version of cal when analyzing the total WC. The Vickrey auction has almost the same the ’joy of winning’ hypothesis in an experimental study of the IPV second-price frequency of bids in the WC interval, but experienced WC is lower; hence, auction. it has a roughly 30 percent lower total WC when compared with the other 6 It could be individual-rational for 2-player groups in the uniform and Vickrey two. As above, the only difference in bids between the auctions is between auctions. But that hinges on how the other player bids, and it is still quite risky. the discriminatory and Vickrey auctions; now the p-value becomes somewhat

16 17 ESSAY II 1229 and . 2 2 b b & 1 1 b b 0928 when comparing 2 b 2 b & 0.34 0.12 0.46 0.17 0.46 0.18 1 b 1 Winner’s curse Total b Winner’s curse Total c c 07 between the two bids.) This is also 17 2 b > EV 2 b i > EV b & i 0.34 0.36 b 1 ≥ b 0545, when testing all formats together for dif- 05 to 0. ≥ . n . n 1 EV b EV Vickrey DiscriminatoryUniform 0.40 Discriminatory 0.42Uniform 0.35Vickrey 0.42 0.37 0.57 0.36 0.32 0.18 0.30 0.47 0.20 0.42 0.34 0.17 0.33 0.13 0.12 0.10 ferences between the twobids bids. for Hence, subjects we when have pooling a all distinct auctionSwitching formats. difference to a between comparison betweendifferences auction between formats, bids and once in more thethe testing WC-interval, discriminatory we for and only Vickrey auctions; find the afirst p-value difference is unit between 0. prices. Since the differencecase, between Table 8 the shows first the results and when second pooling first unit and bids second unit is bids. weak It can in this confirmed with a p-value of 0 be seen that thecal uniform when and analyzing the thefrequency total discriminatory of WC. auctions bids The areit in Vickrey almost has auction the identi- has a WC almosttwo. roughly interval, the As but 30 same above, percent experienced thethe lower WC discriminatory total only is and WC difference lower; Vickrey when hence, auctions; in now compared bids the with between p-value the becomes the somewhat other auctions is between Table 6 The frequency of winner’s& curse 2 bids are and pooled. experienced winner’s curse, when bids 1 rising.). But looking more closelyof reveals second a unit pattern bids/prices due vs.is to first an the unit lower interval bid/prices frequency ranging in all from three 0 formats. (There Table 5 The frequency of winner’s curse bids andAt actually a experienced first winner’s glance, there curse. unit is prices no significant within difference each between the auction first type and second (the p-values starting from 0 ) are, i t | v ( E /2, up to the naive EV, i t )=3 i − >t 16 i t | v overtook any rationale in these groups. Even ( 5 E = c /2). Since it may not be intuitive to grasp the under- EV +7 i t )=( i t | Moreover, to separate the random component from the actual bid, v ( 6 E = n It could be individual-rational for 2-player groups in the uniform and Vickrey Cox et al. (1992) tried to explain overbidding in IPV first price auctions with ’joy lying cause of thein winner’s this curse, i.e. interval the could(at convexity be least) of below the rationalized max the on function, naive the bids expected basis value. of However, the bids fact above that they are we distinguish between biddinga in negative the profit, WC i.e. interval suffering and from actually the experiencing winner’sIn curse, in the Tables 4 uniform tosince and 6 subjects below. Vickrey do auctions, (often)come the from not bid any pay bidder is what inalways they just the another uniform bid. player’s a auction bid The proxy but, thattwo price-setting in becomes formats, for bid the the we the Vickrey could are price-setting auction, measuring bid. price, group. it more Thus, All is like in bid a these frequencies, generalized or WC;and prices, a in having WC table the within 5 are each highthough conditional we signal. on are both That using winning is, bidgenerated 1 as that and stated we bid are in 2 measuring in the the for above table, the paragraph, itThe last is even two table the formats. price is thatpercent to these of bids be the interpreted second-unitde bids as were facto follows: in gave In the agave WC the negative negative interval. discriminatory profits. profit. Of these, Thus, auction, 57 a 35 percent total of 20 percent second-unit bids though some subjects understoodhigh having on to the play zero first,the on their experiment the opponents is second seldom unitand the did. and three-player low Another games. revenue prominent outcomesubjects Supposedly feature did in because in not the of seem Vickrey to its auction understand complicated this. forHypothesis nature, two- 5: the As described above,ditional the on WC interval winning, is defined as bids above the EV con- 6 5 There is especially oneboth odd result anomalies here in when hypotheseswere compared 1 to expected and theory, to which 3. spurs beEven The if (much) subjects’ lower the bids in revenueleast in the does be 2-player much uniform not groups lower (than and goelement, the the to or discriminatory Vickrey auction). zero the Maybe joy auctions. as the of competitive predicted winning, by theory, it should at auctions. But that hinges on how the other player bids, and it is still quite risky. of winning’ and Cooperthe and ’joy Hanming of (2008) winning’auction. partly hypothesis support in a an modified experimental version study of of the IPV second-price EV on average, never individual-rationalaverage. since they produce a negative profit on lower, namely 0.0692, when the first and second unit prices are pooled. Still, Result 5 (Winner’s curse) The winner’s curse is highly visible, but does the significance levels are quite weak. not have as large an effect on outcomes as in results from earlier experiments, c.f. Kagel and Levin (2002), for inexperienced bidders in single unit auctions. When bids/prices above the expected value are scrutinized which, as men- The cases of WC are also robust across the sample population and not just for tioned above, is not really a WC problem but shown here for reference, the a couple of bidders. Across group sizes, there is no difference between the sizes following table emerges (Table 7). Only the bid for the second unit in the dis- in bidding in the WC interval.

bi > EVn Negative proﬁt Total Comment:

b1 b2 b1&b2 b1 b2 b1&b2 b1&b2 As stated at the beginning, for players in 2-player groups, it could be an equilibrium strategy to bid on or above any of the expected values. But this Discriminatory 0.12 0.08 0.11 0.80 0.77 0.79 0.09 is only the case if both bidders bid zero on the second unit, which never Uniform 0.13 0.19 0.14 0.71 0.64 0.69 0.10 happened. Thus, bids from 2-player groups are included in the above results.

Vickrey 0.11 0.11 0.11 0.67 0.89 0.72 0.08 The impact-differences of the winner’s curse across distinct set-ups, i.e. other Table 7 experiments, could be explained from the construction of the common value Frequency of bids above the (naive) expected value, and the negative payoff. interval and the private signal generation. Here, if a player got a 6 (1) as a signal, he/she knew that the signal was the highest (lowest) possible. This is, of criminatory auction differs significantly from the others when analyzing bids course, valuable information. One purpose of this set-up was to make the idea above EVn. Thus, the pooled results (b1&b2) are also shown in the table, as is of common value clear and uncomplicated to understand and, thus, the win- the total WC. (To be comparable with the analysis above, all bid frequencies, ner’s curse would be mitigated. This was the case in the present experiment. or prices, in the table are conditional on both winning and having the high Moreover, due to demand reduction, the WC would be lower in multi-unit signal.) settings than in single-unit settings. As for the discriminatory auction, subjects are more cautious when bidding on the second unit, and the lion’s share of the second unit bids give a negative payoff, but the analysis of the second unit bids is to be taken with caution due to lack of data. The lack of data on the second-unit bids is shown in both 6 Discussion columns of pooled bids, since the pooled results highly resemble the first unit- bids. But, in total, the formats are not significantly different from each other in both bidding above expected value and making a negative profit. First, we notice that as the number of players increases, the pricing rules converge in collecting revenue. When there were only two bidders in the auction, all formats were significantly different in revenue raising, but when there were The result of pooling all bids above EVc is shown in Table 8. The formats are four bidders, the difference became insignificant. Thus, attracting bidders, or ensuring competition, could be much more important than selecting the auc- Bids giving negative profit tion form.

b1 & b2 There was one particularly odd result in the experiment, namely the high Discriminatory 0.28 revenue for 2-player groups in the uniform auction. This was rather unexpected because of the anticipated low revenue equilibria outcome of this group. One Uniform 0.31 possible explanation is the competitive element; subjects did not play the Vickrey 0.23 theoretical equilibrium at all; but wanted to win the object(s), no matter Table 8 what the costs. Holt and Sherman (1994) explain this as the joy of winning The percentage of bids giving negative profits, in total phenomenon in their study. In the present study, it was encountered not only in this particular group size, but was pretty common in all group sizes in all not significantly different from each other, so the ranking is ambiguous. auction formats.

18 19 ESSAY II 19 The winner’s curse is highly visible, but does : 6 Discussion First, we notice that asverge the in number collecting of revenue. players Whenall increases, there formats the were were pricing significantly only rules different twofour con- in bidders bidders, revenue in the raising, the but difference auction, ensuring when became there competition, insignificant. were could Thus, be attractingtion much bidders, form. more or important than selecting the auc- There was onerevenue particularly for 2-player groups odd in the resultbecause uniform auction. in of This the was the rather anticipated unexpected possible experiment, low revenue namely explanation equilibria the is outcometheoretical high of the equilibrium this competitive group. at element; One what all; the subjects but costs. did Holt wantedphenomenon not and in to their play Sherman win study. (1994) the Inin the explain the this object(s), present this particular study, as group it noauction the size, was formats. matter encountered joy but not of was only pretty winning common in all group sizes in all not have as largec.f. an effect Kagel on and outcomes Levin asThe (2002), in cases for of results inexperienced from WC bidders earlier area in experiments, also couple single robust of unit across bidders. auctions. the Acrossin sample group bidding sizes, population in and there is not the no just WC difference for interval. between theComment sizes As stated atequilibrium the strategy beginning, to bid foris on players only or in above thehappened. any 2-player case Thus, of groups, bids if the from it expected both 2-player values. could groups bidders But are be this bid includedThe an in zero impact-differences the of on above the results. theexperiments, winner’s could curse second across be unit, distinct explainedinterval set-ups, and which from i.e. the the never private other signal constructionnal, generation. of Here, he/she the if knew a common that playercourse, value the got valuable a signal information. 6 was One (1) the purposeof as highest of a common (lowest) this sig- value possible. set-up clear This wasner’s to and is, curse make uncomplicated of the would to idea be understandMoreover, mitigated. and, due This thus, to was the thesettings demand win- case than reduction, in in the the single-unit present settings. WC experiment. would be lower in multi-unit Result 5 (Winner’s curse) 2 b & 1 b 2 b & 1 b 2 b 2 b & 1 Negative profit Total 0.31 0.23 0.28 b 1 b ) are also shown in the table, as is 2 b is shown in Table 8. The formats are 2 & b c 1 & 18 1 b EV n Bids giving negative profit 2 b > EV i b 1 b Discriminatory Uniform Vickrey 0692, when the first and second unit prices are pooled. Still, . . Thus, the pooled results (b n Discriminatory 0.12Uniform 0.08 0.11Vickrey 0.80 0.13 0.77 0.19 0.79 0.11 0.14 0.11 0.09 0.71 0.11 0.64 0.67 0.69 0.89 0.10 0.72 0.08 EV the total WC. (To beor comparable prices, with in the analysis thesignal.) above, table all are bid frequencies, conditional on bothAs winning for and the having discriminatory theon the auction, high second subjects unit, and arepayoff, the more but lion’s cautious share the of when analysisdue the bidding to of second lack unit the of bids second data. givecolumns a The of unit negative lack pooled bids of bids, data is sincebids. on the to But, the pooled in be second-unit results total, bids taken highlyin the is resemble with both formats shown the bidding caution in are first above both unit- not expected significantly value different and from makingThe each a result other of negative pooling profit. all bids above Table 7 Frequency of bids above the (naive) expected value, and the negative payoff. not significantly different from each other, so the ranking is ambiguous. the significance levels are quite weak. When bids/prices abovetioned the above, expected is value notfollowing are table really scrutinized emerges a (Table which, 7). WC Only as problem the but men- bid shown for here the second for unit reference, in the the discriminatory auction differs significantlyabove from the others when analyzing bids lower, namely 0 Table 8 The percentage of bids giving negative profits, in total As for the winner’s curse, we chose to solely isolate the WC interval. Many demand-reduction equilibrium. Very few, indeed, grasped the idea of bidding experiments do not distinguish between the intervals and, thus, treat all bids zero when there were just two bidders in the uniform and the Vickrey auctions. above the EV conditional on winning as potential winner’s curse bids, which, Moreover, in the discriminatory auction, subjects learned to play equilibrium per definition, they are not. But, of course, all bids above the EV conditional of strategy over time. winning are dangerous and could give rise to a negative profit. Thus, we present the results from the bids above the naive EV, and then the experiment is The WC is still a problem; overall, subjects did not seem to understand the comparable with the results from other experiments. Another issue concerning adverse selection effect that winning produces. Regardless of group size or the WC was that it entailed no learning effects; subjects continued to suffer auction type, the WC was always there for about 17 percent for the two most from the winner’s curse in later rounds, and not just in the early rounds. They common auctions, and around 12 percent for the Vickrey auction. The WC in never really grasped the idea. this study is defined as bids/prices between the expected value conditional on winning and the (usual, naive) expected value, not bids above this expected Contrary to the non-learning in the WC problem, there was another type value. Bids above the naive expected value were somewhat less common, and of learning that we chose to discuss here because of lack of evidence in the quite the same in all three formats, around 9 percent. data. Subjects learned in the course of play, i.e. they adapted to what the other player(s) did in the auction and bid according to that. In other words, they were trying to find a best-response function. Nonetheless, because of the common value structure, where the random component played a part of the 8 Acknowledgements profit earned, it is hard to see the evidence in the data.

All subjects were inexperienced players, and one must be careful in drawing We would like to thank Lars Hultkrantz, Jan-Eric nilsson and Svante Man- policy recommendations from the result. But other research, Kagel and Levin dell for valuable comments on the paper. Also, thanks to Jan-Erik Sw¨ardh (2002) for example, has shown that overbidding is a robust feature, not only for important help with econometrics. This study has been conducted within for bidders with no experience, but also for professionals. the Centre for Transport Studies (CTS). The author is responsible for any remaining errors.

7 Conclusion References The present paper has studied the two most common auction formats used in the field, the discriminatory and the uniform auctions, as well as the Vickrey Ahlberg, J.: 2009, ‘Analyzes of discrete multi-unit, common value auctions: A auction, a more theoretical format. All three formats make use of two treat- study of three sealed-bid mechanisms’. Portuguese Economic Journal 8(1), ments; first, varying the number of bidders and, second, repeating the auction 3–14. ´ several times inside each session. Alvares, F. and C. Mazón: 2010, ‘Comparing the Spanish and the discriminatory auction formats: A discrete model with private information’. Economic The main conclusion was that the auction format is less important for rev- Theory Online First. enue generation when the number of bidders is large; there were no significant Ausubel, L. M.: 1999, ‘A Generelized Vickrey Auction’. Mimeographed, Uni- differences in the revenue of the three formats when there were four competi- versity of Maryland. tors in the auction. Neither of the discriminatory and the uniform auctions Ausubel, L. M.: 2004, ‘An Efficient Ascending-Bid Auction for Multiple Ob- could be distinguished as better at revenue generation than the other; only jects’. The American Economic Review 94(5), 1452–1475. the Vickrey auction could be classified as inferior, compared to the others, Ausubel, L. M. and P. C. Crampton: 2002, ‘Demand Reduction and Ineffi- when there were few bidders. One possible explanation for this could be the ciency in Multi-Unit Auctions’. Mimeographed, Department of Economics, complicated nature of the Vickrey auction, which subjects had difficulties in University of Maryland. understanding. Ausubel, L. M., P. Cramton, E. Filiz-Ozbay, N. Higgins, E. Y. Ozbay, and A. J. Stocking: 2009, ‘Common-Value Auctions with Liquidity Needs: An Interestingly enough in the experiment, almost no one understood the extreme Experimental Test of a Troubled Assets Reverse Auction’. Papers of peter

20 21 ESSAY II ardh 8(1), Economic Mimeographed, Uni- (5), 1452–1475. 94 Portuguese Economic Journal 21 Mimeographed, Department of Economics, . . on: 2010, ‘Comparing the Spanish and the discrimina- . Online First The American Economic Review ciency in Multi-Unit Auctions’. University of Maryland A. J. Stocking: 2009,Experimental Test ‘Common-Value of Auctions a with Troubled Liquidity Assets Needs: Reverse Auction’. An Papers of peter study of three sealed-bid mechanisms’. 3–14. tory auction formats: A discreteTheory model with private information’. jects’. versity of Maryland ´ Ausubel, L. M. and P. C. Crampton: 2002, ‘Demand Reduction and Ineffi- Ausubel, L. M., P. Cramton, E. Filiz-Ozbay, N. Higgins, E. Y. Ozbay, and 8 Acknowledgements We would like todell thank for Lars valuable Hultkrantz, comments Jan-Eric on nilsson the and paper. Svante Man- Also, thanks to Jan-Erik Sw¨ demand-reduction equilibrium. Very few,zero indeed, when grasped there were the just idea twoMoreover, bidders of in in bidding the the uniform discriminatory andstrategy auction, the over subjects Vickrey time. auctions. learned to play equilibrium The WC is stilladverse a selection problem; effect overall,auction subjects that type, did winning the not WC produces. was seemcommon always auctions, Regardless to there and understand of for around about the 12 groupthis 17 percent study size percent for is for the defined or the Vickrey aswinning auction. two bids/prices and most The between the WC the in (usual, expectedvalue. value naive) Bids conditional above expected on the value, naivequite not expected the bids value same above were in this somewhat all less expected three common, formats, and around 9 percent. References Ahlberg, J.: 2009, ‘Analyzes of discrete multi-unit, common value auctions: A for important help withthe econometrics. Centre This study for hasremaining Transport been errors. Studies conducted (CTS). within The author is responsible for any Alvares, F. and C. Maz´ Ausubel, L. M.: 1999, ‘A Generelized Vickrey Auction’. Ausubel, L. M.: 2004, ‘An Efficient Ascending-Bid Auction for Multiple Ob- 20 As for the winner’sexperiments do curse, not we distinguish choseabove between to the the EV solely intervals conditional and, isolateper on thus, the definition, winning treat they WC as all are interval. not. potential bids winning But, Many winner’s are of curse dangerous course, and all bids, could bids which, give abovethe rise the to EV results a conditional negative from of profit. Thus,comparable the we with present the bids results above fromthe other the experiments. WC Another naive was issue that EV, concerning from it the and winner’s entailed then curse no innever the learning later really rounds, experiment effects; grasped and subjects is not the just continued idea. in to the suffer earlyContrary rounds. They to theof non-learning learning in that the wedata. WC chose Subjects to problem, learned discussother there in here player(s) was the did because another in course ofthey the type of were lack auction trying play, of and to i.e. evidence bid findcommon they a according in value best-response adapted to structure, the function. that. to whereprofit Nonetheless, In the what earned, because other random it of the words, component the is hard played to a see partAll the of evidence subjects the in were the inexperiencedpolicy data. players, recommendations from and the one(2002) result. must But for be other example, careful research, has Kagelfor in shown and bidders drawing that Levin with overbidding no is experience, a but robust also feature, for not professionals. only 7 Conclusion The present paper has studiedthe the field, two the most discriminatory commonauction, and auction a the formats more uniform used auctions, theoretical in ments; as format. first, well varying All the as three number theseveral of formats Vickrey times bidders make inside and, use second, each of repeating session. two the treat- auction The main conclusion wasenue generation that when the the number auctiondifferences of in format bidders the is is revenue large; lesstors of there the important were in no three for the significant formats rev- auction.could when there Neither be were of distinguished four thethe as competi- discriminatory Vickrey better and auction at thewhen could revenue there uniform generation be were auctions than classified few thecomplicated as bidders. nature other; inferior, One of only possible compared theunderstanding. explanation to Vickrey for auction, the this which others, could subjects be had the difficultiesInterestingly in enough in the experiment, almost no one understood the extreme cramton, University of Maryland, Department of Economics - Peter Cram- tions:Evidence from a sportscard field experiment’. The American Economic ton. Review 90(4), 961–972. Back, K. and J. P. Zender: 1993, ‘Auctions of Divisible Goods: On the Ratio- Manelli, A. M., M. Sefton, and B. S. Wilner: 2006, ‘Multi-Unit Auctions: nale for the Treasury Experiment’. The Review of Financial Studies 6(4), A Comparison of Static and Dynamic Mechanisms’. Journal of Economic 733–764. Behavior & Organization 61(2), 304–323. Capen, E., R. Clapp, and W. Campbell: 1971, ‘Competitive Bidding in High- Nelson, R. R. and S. G. Winter: 2002, ‘Evolutionary Theorizing in Economics’. Risk Situations’. Journal of Petroleum Technology 23(6), 641–653. The Journal of Economic Perspectives 16(2), 23–46. Cooper, D. J. and F. Hanming: 2008, ‘Understanding Overbidding In Second Noussair, C.: 1995, ‘Equilibria in a Multi-Unit Uniform Price Sealed Bid Auc- Price Auctions: An Experimental Study’. The Economic Journal 118(532), tion with Multi-Unit Demands’. Economic Theory 5, 337–351. 1572–1595. Porter, D. and R. Vragov: 2006, ‘An Experimental Examination of Demand Cox, J. C., V. L. Smith, and J. M. Walker: 1992, ‘Theory and Misbehavior of Reduction in Multi-Unit Versions of the Uniform-Price, Vickrey, and English First-Price Auctions: Comment’. The American Economic Review 82(5), Auctions’. Manageral and Decision Economics 27(6), 445–458. 1392–1412. Rothkopf, M. H., T. J. Tiesberg, and E. P. Kahn: 1990, ‘Why are Vickrey Engelbrecht-Wiggans, R. and C. M. Kahn: 1998, ‘Multi-Unit Auctions with Auctions Rare’. Journal of Political Economy. Uniform Prices’. Economic Theory 12, 227–258. Vickrey, W.: 1961, ‘Counterspeculation, Auctions, and Competitive Sealed Engelbrecht-Wiggans, R., J. A. List, and D. H. Reiley: 2006, ‘Demand reduc- Tenders’. The Journal of Finance 16(1), 8–37. tion in multi-unit auctions with varying numbers of bidders: Theory and Wilson, R.: 1977, ‘A Bidding Model of Perect Competition’. The Review of evidence from a field experiment’. International Economic Review 47(1), Economic Studies 44(3), 511–518. 203–231. Engelmann, D. and V. Grimm: 2009, ‘Bidding behaviour in multi-unit auctions - An experimental investigation’. Economic Journal 119(537), 855–882. Holt, C. A. and R. Sherman: 1994, ‘The Loser’s Curse’. The American Eco- nomic Review 84(3), 642–653. Kagel, J. H.: 1995, Auction: Survey of experimental research. In Kagel, J.H., Roth, A.E. (Eds.), The Handbook of Experimental Economics. Princton University Press. Kagel, J. H., R. M. Harstad, and D. Levin: 1987, ‘Information Impact and Allocation Rules in Auctions with Affiliated Private Values: A Laboratory Study’. Econometrica 55(6), 1275–1304. Kagel, J. H. and D. Levin: 2001, ‘Behavior in multi-unit demand auctions: Ex- periments with uniform price and dynamic Vickrey auctions’. Econometrica 69(2), 413–454. Kagel, J. H. and D. Levin: 2002, Common Value Auctions and the Winner’s Curse. Princton University Press. Kagel, J. H. and D. Levin: 2011, ‘Auctions: A Survey of Experimental Re- search, 1995 - 2010’. Working Paper. Katzman, B. E.: 1995, ‘Multi-Unit Auctions with Incomplete Information’. Working Paper, University of Miami. Lebrun, B. and M. . Tremblay: 2003, ‘Multiunit pay-your-bid auction with one-dimensional multiunit demands’. International Economic Review 44(3), 1135–1172. Cited By (since 1996): 1. Levin, D.: 2005, ‘Demand Reduction in Multi-Unit Auctions: Evidence from a Sportscard Field Experiment: Comment’. The American Economic Review 95(1), 467–471. List, J. A. and D. Lucking-Reiley: 2000, ‘Demand reduction in multiunit auc-

22 23 ESSAY II The Review of Journal of Economic , 337–351. 5 The American Economic (6), 445–458. 27 (2), 23–46. 16 (1), 8–37. 23 16 Economic Theory (2), 304–323. 61 (3), 511–518. 44 Journal of Political. Economy Manageral and Decision Economics (4), 961–972. The Journal of Finance 90 Tenders’. tion with Multi-Unit Demands’. Reduction in Multi-Unit Versions of theAuctions’. Uniform-Price, Vickrey, and English Auctions Rare’. Economic Studies A Comparison of Static and Dynamic Mechanisms’. The Journal of Economic Perspectives tions:Evidence from a sportscard field experiment’. Review Behavior & Organization Vickrey, W.: 1961, ‘Counterspeculation, Auctions, and Competitive Sealed Noussair, C.: 1995, ‘Equilibria in a Multi-Unit Uniform PricePorter, Sealed D. Bid Auc- and R. Vragov: 2006, ‘An Experimental Examination ofRothkopf, Demand M. H., T. J. Tiesberg, and E. P. Kahn: 1990, ‘Why are Vickrey Manelli, A. M., M. Sefton, and B. S.Nelson, Wilner: R. R. 2006, and S. ‘Multi-Unit G. Winter: Auctions: 2002, ‘Evolutionary Theorizing in Economics’. Wilson, R.: 1977, ‘A Bidding Model of Perect Competition’. (3), (5), (1), 6(4), (532), 44 82 47 Princton 118 Econometrica (537), 855–882. The American Eco- (6), 641–653. 119 23 The American Economic Review The Economic Journal , 227–258. International Economic Review . The Review of Financial Studies International Economic Review 12 22 The American Economic Review Economic Journal Common Value Auctions and the Winner’s (6), 1275–1304. 55 Working Paper Journal of Petroleum Technology Economic Theory (3), 642–653. Auction: Survey of experimental research. In Kagel, J.H., 84 Econometrica (2), 413–454. (1), 467–471. Price Auctions: An Experimental Study’. 1572–1595. First-Price Auctions: Comment’. periments with uniform price and69 dynamic Vickrey auctions’. Curse. Princton University Press. search, 1995 - 2010’. Working Paper, University of Miami. one-dimensional multiunit demands’. cramton, University of Maryland, Departmentton. of Economics - Peter Cram- nale for the Treasury733–764. Experiment’. Risk Situations’. 1135–1172. Cited By (since 1996): 1. Sportscard Field Experiment: Comment’. 95 1392–1412. Uniform Prices’. tion in multi-unitevidence auctions from with a varying field numbers203–231. experiment’. of bidders: Theory and - An experimental investigation’. nomic Review Roth, A.E. (Eds.),University The Press. Handbook of Experimental Economics. Allocation Rules in AuctionsStudy’. with Affiliated Private Values: A Laboratory Cooper, D. J. and F. Hanming: 2008, ‘Understanding Overbidding In Second Cox, J. C., V. L. Smith, and J. M. Walker: 1992, ‘Theory and Misbehavior of Kagel, J. H. and D. Levin: 2001, ‘Behavior in multi-unit demand auctions: Ex- Kagel, J. H. and D. Levin: 2002, Kagel, J. H. and D. Levin:Katzman, 2011, B. ‘Auctions: E.: A 1995, Survey of ‘Multi-Unit Experimental AuctionsLebrun, Re- with B. Incomplete and Information’. M. . Tremblay: 2003, ‘Multiunit pay-your-bid auction with Back, K. and J. P. Zender: 1993, ‘Auctions of Divisible Goods: OnCapen, the E., Ratio- R. Clapp, and W. Campbell: 1971, ‘Competitive Bidding in High- Levin, D.: 2005, ‘Demand Reduction in Multi-Unit Auctions: Evidence from a List, J. A. and D. Lucking-Reiley: 2000, ‘Demand reduction in multiunit auc- Engelbrecht-Wiggans, R. and C. M. Kahn: 1998,Engelbrecht-Wiggans, ‘Multi-Unit R., J. Auctions A. with List, and D. H. Reiley: 2006, ‘Demand reduc- Engelmann, D. and V. Grimm: 2009, ‘Bidding behaviour in multi-unit auctions Holt, C. A. and R. Sherman: 1994,Kagel, ‘The J. Loser’s H.: Curse’. 1995, Kagel, J. H., R. M. Harstad, and D. Levin: 1987, ‘Information Impact and 9 Appendix A Let D1,D2 be the random variables describing the outcome of the two integers, or dice, respectively. Then, the value for each bidder is the realization of the two variables, hence v = d1 + d2. Define ki(b) as the number of items won by player i if The theoretical model is found in Ahlberg (2009), but an excerpt is presented here. strategy profile b is employed.

Now, let b =(b1,...,bn) be a strategy profile and let v = d1 + d2 be a random variable. Then, player i’s payoff function πl : b R is defined as 9.1 The Model i → πl(b, v)=k (b)v p (b). (4) i i − i In a Bayesian game, players update their prior beliefs by Bayes’ rule, as well as the Thus, the payoff is the number of items won multiplied by the realized value of these opponents’ payoff functions, once they learn their types. items, minus the price the winner has to pay for them.

9.2 Posterior beliefs 9.4 Expected value functions

Let the type-vector of all players except player i be denoted by t i =(ti,...,ti 1, − − Since players only get to see one integer, i. e. their signal, they have to use the ti+1,...,tn). Then, given a player’s own type ti, denote μ(t i ti) as player i’s con- − | expected value when calculating their value, which is v(ti)=ti +7/2, that is, the ditional probability, or posterior beliefs, about her opponents’ types. value of her signal plus the expected value of the other die. But in a Bayesian game they also need to calculate their competitors’ value, given their own signal. This

n 1 conditional expected value for the other players is dependent on how many players 1 2 − 1 1 n 1 1+ n −1 6 all types in t i are equal to ti, there are in the game/auction. 2 − · 2 − · − ⎧ n 1 1 1 μ(t i ti)= − n 1 At least one type in t 1 is different from ti. − | ⎪ x · 2 − · 6 − The fact that induces this is that they can all see the same integer or different ⎨⎪ 1 1 �n 1 � all types in t i are different from ti. integers. The more signals (players), the more accurate becomes the conditional 2 − · 6 − ⎪ expected value. That is, with many bidders, we approach the true value. This is an ⎪ where x is defined⎩ as the number of players who are of the same type as player application of information aggregation, studied by Wilson (1977). The conditional i. The first equation works as follows: If all other players, except player i, see the expected value is defined as: same value as ti, two things can happen. Either they all see the same die as player 1 i, which happens with probability 2n 1 , or at least one of the others sees a different − n 1 2 − 1 1 v (t t )=v(t t ) die with the same value as ti, which happens with probability n −1 . (The first i i i i i 2 − 6 − | − | · n 1 term is the probability that at least one sees a different die and the second term is 1 2 − 1 n 1 (ti + t)+ n −1 2ti all tj = ti, the probability that the die has the same value as t .) = 2 − 2 − · (5) i ⎧ ti + tj (where tj t i) some tj = ti, ⎨ ∈ − � The second equation says that if one of the non-i players sees a different value than 2n 1 1 n 1 1 1 ⎩ n−1 ti + n 1 t all tj = ti, player i, the belief for player i becomes − n 1 . In the last row, all non-i 2 − 2 − x 2 − 6 = (6) · · ⎧ players see a different value than player i, and, since we only have two dice, they ti + tj (where tj t i) some tj = ti, � � 1 1 ⎨ ∈ − � all see the same die. This happens with probability n 1 . 2 − · 6 The first row in⎩ equation (5) says that if the non-i players are of the same type, ti, as player i, two things can happen. Either they see the same die as player i, which 1 9.3 Strategy and payoff occurs with probability n 1 , or at least one of them sees a different die. The value 2 − for the former becomes ti +7/2 for player i, while the value for the latter becomes ti + ti =2ti. The strategy for each player is a to assign two (integer-)bids, one for each item, from her signal. Formally, the strategy for player i is a mapping from her signal space In the second row of the same equation, we see the value if one, or both, is of a 2 Ti = 1, 2,...,6 to the two-dimensional space of integers, bi : Ti Z , where different type than player i. Then, since there are only two distinct integers, the { } → + bi(ti)=(bi,1,bi,2). value becomes the sum of the integer values. Equation (6) is just a simplification.

24 25 ESSAY II , if i t i (4) (5) (6) i, which be a random /2, that is, the 2 d +7 + i 1 d = , , )=t i i i t t t v ( , , i i v �= �= t t . j j is defined as t t = = players are of the same type, b) j j ( R i t t p → all all − b ) some ) some v : i l i i, while the value for the latter becomes b) −i −i π t t ( 2t i · 25 k ∈ ∈ b) as the number of items won by player 1 ( j j i 1 − t t k 1 )= n− − 2 n 2 b, v t ( 1 l i − 1 π /2 for player (where (where n , or at least one of them sees a different die. The value t)+ 2 . Define 1 i. Then, since there are only two distinct integers, the for the other players is dependent on how many players 2 + − +7 1 d + n i i ) be a strategy profile and let i 2 t t j j t ) + n ( t t i i’s payoff function 1 1 1 t | − −1 + + d i 1 n n− n i i − 2 2 2 t t t = is employed. ( ,...,b v 1 v ⎧ ⎨ ⎩ ⎧ ⎨ ⎩ b b = = )= be the random variables describing the outcome of the two integers, i . =( t i | 2 t i, two things can happen. Either they see the same die as player b −i t ,D ( =2 1 i i v D t + i Now, let strategy profile or dice, respectively. Then,variables, the hence value for each bidder is the realization of the two for the former becomes Thus, the payoff is the numberitems, of minus items the won multiplied price by the the winner realized value has of to these pay for them. 9.4 Expected value functions Since players onlyexpected get value to when see calculating one their integer, value, which i. is e. their signal, they have to use the Let variable. Then, player In the second rowdifferent of type the than same player equation, we see the value if one, or both, is of a value becomes the sum of the integer values. Equation (6) is just a simplification. as player occurs with probability value of her signal plusthey the also expected value need ofconditional to the expected other calculate value die. their But competitors’ in a value, Bayesian given game their own signal. This there are in the game/auction. The fact thatintegers. induces The this more isexpected signals that value. (players), That they is, the can withapplication more many all of bidders, accurate information see we becomes aggregation, approachexpected the studied the the value same true by is conditional value. Wilson integer defined This (1977). as: or is The an different conditional The first row in equation (5) says that if the non-i t . i , t −1 i i’s con- , where . i 2 + i, see the t ,...,t Z . (The first i t 1 6 · , → i =( t i 1 −1 T 1 −i n− − : t ) as player is different from 2 n i 2 i t | b −1 t −i . t 1 6 · μ( . In the last row, all non-i 1 are different from are equal to 1 6 1 · n− .) 2 −i −i 1 i t t t − 1 n players sees a different value than be denoted by 2 , denote i · i is a mapping from her signal space t � i x −1 n i, and, since we only have two dice, they � 24 At least one type in all types in all types in , or at least one of the others sees a different 1 1 6 − 1 · n 2 1 −1 1 6 1 − becomes n · i 2 n− 1 2 , which happens with probability i − 1 t n 2 1 6 1+ · · · � 1 1 x −1 1 1 n− n− n 2 2 to the two-dimensional space of integers, � ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ ). , two things can happen. Either they all see the same die as player 6} i 2 t i, )= ,b i 1 ). Then, given a player’s own type t | i, n i 2,..., is defined as the number of players who are of the same type as player b − t i, the belief for player x {1, μ( )=( ,...,t i = t ( +1 i i i . The first equation works as follows: If all other players, except player , which happens with probability The second equation says thatplayer if one of the non-i same value as where i The strategy for each player isher a signal. to assign Formally, the two (integer-)bids,T strategy one for for each player item, from Let the type-vector of all players except player 9.3 Strategy and payoff 9 Appendix A The theoretical model is found in Ahlberg (2009), but an excerpt is presented9.1 here. The Model In a Bayesian game, playersopponents’ update payoff their functions, prior once beliefs they by learn Bayes’ their rule, types. as well as the 9.2 Posterior beliefs players see a differentall value see than the player same die. This happens with probability i ditional probability, or posterior beliefs, about her opponents’ types. die with the same value as b t term is the probabilitythe that probability at that least the one die sees has a the different same die value and as the second term is The term expectation above means expectation over the possible outcomes of the From equation 8 in the Appendix A, we can derive the following unique pure integer values. From player i’s perspective, if we also take expectations over all Bayesian equilibrium strategy for two bidders: t i, we get the expected value for a competitor, given player i’s type. That is, we − must combine the posterior beliefs with the conditional expected value to get the ti ti expected value for any competitor. Hence, the expected value for a competitor to b∗(t )=(t + ,t + ), (9) i i � 3 � i � 3 � player i is defined as where x is a ceiling function which maps x to the smallest following integer, i.e. � � x = min n Z n x . The striking feature is that players bid the same amount � � { ∈8 | ≥ } Et i [v(t i ti)] = μ(t i ti)v(t i ti). (7) on both units. − − | − | − | t i − When we increase the bidders by one, all bidders but the type-6 player bid the same If we also take the expectation over all t , the terms will sum up to seven as they i as in a two-player game. The type-6 players raise their bids on both units by one should. But for any given t , the expected value for a competitor will not be t +7/2. i i increment unit. This is the case since we have to take into account that the competitor may get the signal from the same die as player i. If we look at the four-player game, we get the same increase in the bids for the type-6 players as in the three-player game, but a reduction in the bids for the type- 1 players. The reduction is one increment. The optimal strategy when there are four 9.5 Equilibrium players can then be written as:

A Bayesian equilibrium of this game with a ﬁnite number of types ti, for each ti ti b∗(ti)=(ti + ,ti + ), (10) player i, and a common prior distribution μ, and pure strategy spaces Ti is a Nash � 2 � � 2 � equilibrium of the ”expanded game” where each player i’s space of pure strategies 2 Ti 2 where x is a ﬂoor function which maps x to the largest previous integer, i.e. is the set (Z+) of maps from Ti to Z+. � � x = max m Z m x . � � { ∈ | ≤ }

2 Ti Given strategy profile b( ), and bi� ( ) (Z+) ,let(bi� ( ),b i( )) denote the profile · · ∈ · − · where player i plays b ( ) and the other players follow b( ), and let i� · · 9.5.2 Uniform auction (bi� (ti),b i(t i))=(b1(t1),...,bi 1(ti 1),bi� (ti),bi+1(ti+1),...,bn(tn)) − − − − Conjecture 6 (Two players) In a two-player game, no other equilibrium payoff denote the value of this profile at (ti,t i). Then, since all types have positive prob- dominates the following: − abilities, the bid/strategy bi(ti) is a (pure strategy) Bayesian equilibrium if player i maximizes her expected utility conditional on ti for each t i: − b∗(t )=(b∗,b∗)=( v(t t ) , 0), (11) i 1 2 � j| i � l where x is the nearest integer to x upwardly. bi(ti) arg max μ(t i ti)[ki(bi� ,b i)v(t i ti) pi(bi� ,b i)]. (8) � � ∈ 2 − | − − | − − bi� Z+ t i ∈ − Proof 1 Suppose that player j utilizes b∗(tj). Any attempt to win 2 units for player i would make her second unit bid set the price. And since the bid from player j is We only allow integer-value-bids 7 . Since the value function is symmetrical and b (t ) v(t 1) =4, player i must bid at least 5 to win. The payoff for using b (t ) 1∗ i ≥� j| � ∗ i we have a symmetrical joint distribution, only types will be of importance when is the expected value minus the price paid, which is zero, hence π∗ = ti +7/2, while bidding; thus, we look for a symmetrical equilibrium. the expected value for using the alternative strategy would be π 2(t +7/2 5). � ≤ i − Then, we have that π >π implies (at best) 2(t +7/2 5) > (t +7/2) t > 6, � ∗ i − i ⇒ i which is impossible. 9.5.1 The Discriminatory auction As a matter of fact, when there are two bidders, any bid above v(t t ) on the � j| i � In this auction, conditional on winning, for each item won, every bidder pays the first unit is an equilibrium bid. In an IPV setting, Levin (2005) has shown that any price of her bid on that item. 8 Lebrun and Tremblay (2003) give a more general proof of this result when values 7 A pure strategy Bayesian equilibrium in R does not exist. are private.

26 27 ESSAY II . ) i is 6, t 5) (9) ( j (11) (10) > ∗ − b i t on the /2 � /2, while ) ⇒ i t +7 | +7 j i i t t t ( /2) 2( �v = +7 ≤ ∗ i � π t ( π > 5) − /2 to win. The payoff for using +7 i 5 t 2( . Any attempt to win 2 units for player ) j t ( ∗ 27 b upwardly. , x 0) In a two-player game, no other equilibrium payoff , , �, utilizes ) �) �) must bid at least i i i j t 2 3 i t t | implies (at best) � � j t ∗ ( + + }. i i x �v }. The striking feature is that players bid the same amount x >π ≤ �,t �,t � , player i i ≥ 2 3 π t t )=( m � � | ∗ 2 n =4 | Z + + ,b Z � ∗ 1 i i ∈ 8 b t t 1) ∈ | j t is a floor function which maps x to the largest previous integer, i.e. {m is a ceiling function which maps x to the smallest following integer, i.e. is the nearest integer to ( Suppose that player v )=( )=( )=( � � i i i t t t �x ( ( ( �x �x� max min{n ≥� ∗ ∗ ∗ b b b ) = = i t Lebrun and Tremblay (2003) give a more general proof of this result when values ( would make her second unit bid set the price. And since the bid from player ∗ 1 on both units. is the expected value minus the price paid, which is zero, hence where Proof 1 i b which is impossible. As a matter of fact, when there are two bidders, any bid above 8 Conjecture 6 (Two players) dominates the following: 9.5.2 Uniform auction When we increase the biddersas by one, in all a bidders two-player butincrement game. the unit. type-6 The player type-6 bid the players same raise theirIf bids on we both look unitstype-6 at by players the as one in four-player the1 game, three-player players. game, we The but reduction get is aplayers one the reduction can increment. in same The then the optimal be increase bids strategy written for in when the as: there the type- are four bids for the From equation 8Bayesian in equilibrium strategy the for Appendix two bidders: A, we can derive the following unique pure are private. the expected value for using the alternative strategy would be �x� first unit is an equilibrium bid. In an IPV setting, Levin (2005) has shown that any Then, we have that where �x� where (7) (8) /2. +7 i t )) is a Nash , for each n i i t t ( T n ,...,b ) )]. i’s type. That is, we )) denote the profile : · ( +1 −i −i i t t −i ,b ( � i ), and let b ,b · i’s space of pure strategies ( ) +1 · i l i b( ( p � i ,b b ) − i t for each ) ( i � i i t t | ,b ,let( ) −i i does not exist. t T ( ) −1 R i v μ, and pure strategy spaces 2 + t ) , the terms will sum up to seven as they . ( i Z ) t ). Then, since all types have positive prob- −i i ( . 26 t −1 | i 2 + −i ,b ∈ � i Z −i b ,t ) t i ( . Since the value function is symmetrical and · i. ( i ( to 7 v � i ) b i ,...,b )[k i ) i T t ) is a (pure strategy) Bayesian equilibrium if player | 1 t i | t i t i’s perspective, if we also take expectations over all ( −i ( 1 − t i , the expected value for a competitor will not be t b b i t ), and μ( of this game with a finite number of types · μ( ) and the other players follow i · i b( − ( t − � i t ))=( b −i t 2 + ( of maps from )] = i i ∈Z t −i plays � i | T b ) i ,b −i arg max ) 2 + t i ( t ∈ v ( [ � i ) i is defined as i b , and a common prior distribution t ( − i i ( t i b E , we get the expected value for a competitor, given player Bayesian equilibrium A pure strategy Bayesian equilibrium in maximizes her expected utility conditional on −i abilities, the bid/strategy we have a symmetricalbidding; thus, joint we distribution, look only for types a symmetrical will equilibrium. be of importance when equilibrium of the ”expandedis game” the where set each (Z player i where player In this auction, conditionalprice on of her winning, bid for on each that item item. won, every bidder pays the 9.5.1 The Discriminatory auction 9.5 Equilibrium A 7 The term expectation aboveinteger means values. expectation From over player t the possible outcomes of the should. But for any given Given strategy profile player must combine the posteriorexpected beliefs value with for the anyplayer conditional competitor. expected Hence, value the to expected get value the for a competitor to If we also take the expectation over all denote the value of this profile at (t We only allow integer-value-bids This is the case sincesignal we from have the to same take die into account as that player the competitor may get the bid weakly above the upper endpoints of the distribution, if the reservation price other bids, which the zero bid did not. And, since the bid does not affect the price, is zero, is an equilibrium. This is indeed true also in this model, but the proposed p will be the same in both payoff functions above. Hence, π π . i∗ ≥ i� equilibrium risk dominates all other equilibria. Conjecture 9 (Many players) The more bidders in the auction, the higher the But there also exist other equilibria. If both bidders bid 1, 2 or 3 on the second bids. This is true for both the first-unit bid and the second-unit bid. unit, irrespective of ti, the bids also become equilibrium bids. But since it is highly unclear on which of these equilibria the subjects would coordinate, the zero-bid on Proof 4 Given any realization of the two dice, we see from equation (6) that the the second unit is focal as well as payoff-dominating in undominated strategies. conditional expected value weakly increases with the number of players. Besides, as can be seen from conjectures 7 and 8, since both the first-unit bid and the second-unit When there are more than two players in the game, two things happen. First, bid are dependent on that value, we have that both bids increase with the number of we have to correct downwards instead of upwards, as above, because now there is a players. chance that someone’s first-unit bid may become the price-setting bid. And, second, as a result of the first, the zero bid on the second unit is no longer an equilibrium. This is the case since there are now at least three bidders and two units, and all 9.5.3 The Vickrey auction three bidders have a weak incentive to bid the true (expected) value of the first unit. In the Vickrey auction, a player who wins ki units pays the ki highest losing bids of the other players - that is, the ki highest losing bids not including her own. Hence, Conjecture 7 (More than two players) When there are more than two bidders the winner is asked to pay an amount equal to the externality she exerts on other in the auction, it is an undominated strategy to bid the following on the first unit: competing bidders.

The Vickrey auction is known to have an ex post equilibrium,orano-regret equilib- b∗(t )= v(t t ) . (12) rium. That is, an ex post equilibrium is a Bayesian equilibrium with the additional 1 i � j| i � requirement that even if all players’ signals were known to a particular bidder, it where x is defined as the nearest integer of x downwardly. would still be optimal for her not to alter her strategy, that is, she would not suffer � � from any regret. 9 This Bayesian strategy is: Proof 2 First, note that to bid more than b1∗ will incur an expected loss if the bid is above both v(t t ) and the price. That is, suppose that player i bids b > v(t t ) . � j| i � 1� � j| i � Then if b >p> v(t t ) ,alossofp v(t t ) will be realized on that unit. ti ti 1� j i j i b∗(t )=(t + +2,t + + 1), � | � −� | � i i � 2 � i � 2 � Second, suppose that the bid is below the equilibrium bid b b� , then nothing would change if the player were to raise the bid to ≥� | � 1 Proof 5 If the type-ti bidder bids less, the number of units that she wins is at most v(tj ti) . Next, if the bid is below v(tj ti) and above the price, v(tj ti) >b� >p, � | � � | � � | � 1 what she would win by bidding b∗(ti). For any of the units won, the prices will be nothing would change here either if the bid was increased to v(t t ) . The last case � j| i � the same as before, but she will forgo some surplus for units that she did not win. is if the value is greater than the price and the price is weakly greater than the bid, v(t t ) >p b . Now, if the player raised the bid to v(t t ) , she would win a � j| i � ≥ 1� � j| i � If she instead bids b(ti) >b∗(ti), then she wins at least as many units as before. unit at a more profitable price. Thus, to bid the proposed equilibrium bid on the first The prices for the first kti units will remain the same as if she bid b∗(ti). For any unit is (weakly) dominant in expectation. additional units, however, the price paid will be too high, since for k>kti the price is greater than the value for the item(s). Now, by the last conjecture, when there are more than two players, the bid on the second unit will be weakly bounded from below by the first-unit bid from the low But, as in the uniform auction, we have an extreme demand reduction strategy. type player. Thus This equilibrium is the same as in the uniform auction, i.e.:

Conjecture 8 (More than two players) The second unit bid is weakly bounded by 4, i.e. b2(ti)∗ b1∗(1) = 4. b∗(t )=( v(t t ) , 0). (13) ≥ i � i| i � Proof 3 If player i, say, bids below 4, she will win at most one unit and get the 9 For this to be true in this setting, we must have that the value function satisfies payoff: π =(t +7/2 p)k , where k 1. If the player bids 4, the payoff will be: what Ausubel (1999) calls value monotonicity and value regularity. This, indeed, is i� i − i� � ≤ π =(t +7/2 p)k , where k k since the bid b (t ) now competes against the true for the value function in this paper. i∗ i − i∗ i∗ ≥ i� 2 i ∗

28 29 ESSAY II (13) the price . For any i ) t i t ( ∗ b k>k . This, indeed, is . � i π highest losing bids of ≥ i k ∗ i π value regularity and units pays the i k exno-regret post equilib- ,ora equilibrium , 29 . For any of the units won, the prices will be + 1) ) i � t The more bidders in the auction, the higher the i ( highest losing bids not including her own. Hence, 2 t , then she wins at least as many units as before. ∗ i ) � b i k t + ( ∗ i units will remain the same as if she bid value monotonicity ,t . i t >b 0) k +2 ) bidder bids less, the number of units that she wins is at most i �, � i t ) i i 2 t b( t | This Bayesian strategy is: � i t 9 ( + i �v t is defined as the nearest integer upwardly. This is indeed an equilibrium: If thet type- Given any realization of the two dice, we see from equation (6) that the )=( )=( i i t t ( ( �·� ∗ ∗ b b For this to be true in this setting, we must have that the value function satisfies will be the same in both payoff functions above. Hence, 9.5.3 The Vickrey auction In the Vickrey auction, a player who wins 9 other bids, which thep zero bid did not. And, since the bid does not affect the price, the same as before, but she willIf forgo some she surplus instead for bids units that she did not win. what Ausubel (1999) calls Proof 5 where what she would win by bidding The prices for the first additional units, however, the price paid will be too high, since for the other players - that is, the Conjecture 9 (Many players) bids. This is true for both theProof first-unit 4 bid and theconditional second-unit expected value bid. weakly increasescan with be seen the from conjectures number 7bid of and are 8, players. dependent since Besides, both on the as players. that first-unit value, bid and we the have second-unit that both bids increase with the number of the winner is askedcompeting to bidders. pay an amount equal to theThe externality Vickrey she auction is exerts knownrium. on to That other have is, an anrequirement ex post that equilibrium even is ifwould a all still Bayesian be equilibrium players’ optimal with signalsfrom for the were her additional any known not regret. to to a alter her particular strategy, bidder, that it is, she would not suffer is greater than the value for theBut, item(s). as inThis the equilibrium uniform is auction, the same we as have in an the extreme uniform auction, demand i.e.: reduction strategy. true for the value function in this paper. , �. ) i (12) t >p | j � 1 t ( >b �v � ) > i t � 1 | b j �. The last case t ) ( i t | �, she would win a . Then, three cases �v bids j ) ∗ 1 t i 4, the payoff will be: i ( t | j �v b t i � ( t � /2 ( ) >p> �v is defined as the nearest integer of 2 i =( If player First, note that to bid more than t >p b )= � 1 | � i i +7 b j �. Next, if the bid is below � t π t i ) ) ( �x� ( t i i ∗ 1 t t v b | | j j t t =( 4, i.e. ( ( ≥� ∗ i Then if is if the value�v is greater than the price and the price is weakly greater than the bid, above both appear; first, if the bidp is below the value which, in turn, is weakly below the price, i.e. bid weakly above theis upper zero, endpoints is an ofequilibrium equilibrium. the risk This distribution, dominates is all if indeed other the true equilibria. reservation also price inBut this there model, also but existunit, the irrespective other proposed of equilibria. If both bidders bid 1, 2 or 3 on the second nothing would change here either if the bid was increased to where �v unclear on which ofthe these second equilibria unit the is subjects focal would coordinate, as well the asWhen zero-bid payoff-dominating on there in undominated arewe strategies. more have to than correct downwards twochance instead that players of someone’s upwards, first-unit in as bidas may the above, become a because game, the result now price-setting there of two bid.This is the And, things a is second, first, happen. the the zero casethree First, bid since bidders on there have the are second aunit. now unit weak is at incentive no least to longer three bid an bidders equilibrium. the andConjecture true two 7 (expected) units, (More value and thanin of all two the the players) auction, first it is an undominated strategy to bid the following on the first unit: π unit at a more profitableunit price. is Thus, (weakly) to dominant bid in the expectation. proposed equilibrium bidNow, on by the the first lastsecond conjecture, unit when there will are betype more weakly player. than bounded Thus two from players, below the by bid the onConjecture first-unit the bid 8 from (More the thanby low two players) Proof 3 payoff: Second, suppose that the bid is below the equilibrium bid Proof 2 Proof 6 The proof is as in the uniform auction, hence it is omitted. 10 Appendix B

But this demand reduction strategy is a much weaker equilibrium strategy in the Vickrey auction than in the uniform auction, because, if player i bids the above Variable OLS Robust strategy in the uniform auction, player j’s best response is to bid the same. That standard error is not entirely true in the Vickrey auction since you never pay what you bid, but what the other bids. Hence, in the Vickrey auction, player j can bid any number 2-player groups Reference below her conditional expected value for the second unit and still be an equilibrium 3-player groups 1.85*** 0.31 strategy. And, by the same token, any bid below the conditional expected value is an equilibrium bid. For this equilibrium, as in the uniform auction, we have that 4-player groups 3.80*** 0.33 any bid on the ﬁrst unit above the conditional expected value is an equilibrium bid. Vickrey auction 1.53*** 0.33 Uniform auction 1.80 0.32 9.6 Expected revenue Design 1 0.02 0.26 Intercept 11.40*** 0.28 In a (pure) common value auction, revenue is strongly negatively correlated with No of observations 766 proﬁt. And seen above, both the uniform and the Vickrey auctions have equilibria 2 that give the entire surplus to the buyer, which is the same as the expected value R 0.181 of the two integers, i.e. 7. This translates into zero revenue to the seller. Notes: a; Dependent variable is revenue (price).

The discriminatory auction, on the other hand, has a unique equilibrium, and to b; ***, ** and * denote difference from zero at the one, find the expected revenue, we calculate the probability for each set of possible joint five and ten percent significance level respectively. signals between the players. Then, we make use of the strategies implicitly inherent Table 9 in the signals to compute the price paid for each possible set of joint signals. Then, Regression on revenue (price) we have the expected revenue as the product of the intersection of the signals times the realized price in that outcome. For two players, player i and player j, it becomes:

E[R]=P (t t )p(b ,b ), i ∩ j i j where p(bi,bj) is the price paid. If there are three players, we instead calculate P (t t t )p(b ,b ,b ), and so on. i ∩ j ∩ k i j k By doing this computation, we have for the discriminatory auction:

E[RD(2 Players)] = 11.22 E[RD(3 Players)] = 12.38 (14) E[RD(4 Players)] = 12.63.

As we already have stated, both the uniform and the Vickrey auction give zero revenue:

E[RU,V (2 Players)] = 0 (15)

30 31 ESSAY II 0.28 0.26 0.31 0.33 0.33 0.32 Robust standard error 31 0.02 OLS 0.181 11.40*** : a; Dependent variable is revenue (price). 2 Variable Intercept No of observationsR 766 Notes b; ***, ** and * denotefive difference and from ten zero percent at significance the level one, respectively. 2-player groups3-player groups Reference 4-player groups 1.85*** Vickrey auction 3.80*** Uniform auction 1.53*** Design 1 1.80 Table 9 Regression on revenue (price) 10 Appendix B (14) (15) , it becomes: j bids the above i can bid any number j and player i ’s best response is to bid the same. That j 30 , ) j 22 38 63. . . . ,b i b ), and so on. k p( ) ,b j t j ,b ∩ i i b t ( p( ) is the price paid. If there are three players, we instead calculate ) P j (2 Players)] = 0 k t (2 Players)] = 11 (3 Players)] = 12 (4 Players)] = 12 ,b i The proof is as in the uniform auction, hence it is omitted. ]= U,V D D D ∩ b j R R R R R p( t [ [ [ [ [ E E E E E ∩ i t ( below her conditional expected valuestrategy. for And, the by second the unitan same and still token, equilibrium be any bid. an bid Forany equilibrium below this bid the on equilibrium, the conditional as first expected in unit value above the is the uniform conditional expected auction, value we is have an that equilibrium bid. P In a (pure) commonprofit. And value seen auction, above, revenuethat both is give the strongly the uniform negatively entire andof correlated surplus the the to with Vickrey two the auctions integers, i.e. buyer, have equilibria 7. which This is translates theThe into same discriminatory zero as revenue auction, the to onfind expected the the the value seller. expected other revenue, hand, wesignals calculate between has the the a probability players. for Then, uniquein each we equilibrium, the set make signals and use of to of possible to we compute the joint have the strategies the price implicitly expected paid inherent revenuethe for as realized each the price possible product in set that of of outcome. the For joint intersection two signals. of players, player Then, the signals times Proof 6 But this demand reductionVickrey strategy auction is than astrategy in much in the weaker the equilibrium uniform uniform strategyis auction, auction, in not because, player the entirely if truewhat player in the the other Vickrey bids. auction Hence, since in you the never Vickrey pay auction, what player you bid, but 9.6 Expected revenue where As we alreadyrevenue: have stated, both the uniform and the Vickrey auction give zero By doing this computation, we have for the discriminatory auction: 11 Appendix C Opponents: Before the beginning of each round, the program will randomly choose how many players you will be matched with. You can have one, two or three opponents. Your group-size will be seen on your screen. Bidder instructions for the uniform, common value auction: Bids: After receiving your information, that is, after seeing your die, you should decide on what you want to bid for the units. You are permitted to place equal or different bids for the units. 11.1 Introduction

11.3 Instructions Hello and welcome. You will participate in an experiment on economic decision- making. The purpose is to study sales by bidding, i.e. through an auction. Buy: Those who have placed the highest bid, and the next-to-highest bid, purchase You have the opportunity to win money through participation. The show-up fee is the units. This may be the same person or two different people. If there are ties SEK 100 ( 10), and by learning the rules of the game you have the opportunity among the (winning) bids, the program will randomly choose the winner(s). to earn more than that. On the other hand, you could also lose in the process. Price: The winners will pay a price equal to the highest bid that does not win, To ensure that you walk away with at least SEK 100 in your pocket, we give you That is, the highest bid that is rejected. All winners pay the same price for the a starting balance of SEK 50. If you lose this money, you will be excluded from units. the experiment. Your winnings, and the show-up fee, will be paid in cash after the experiment. Example: 2 units are sold. Three people (A, B, C) have the three highest bids: 10 (A), 9 (B), 8 (C). A and B purchase the units, and both pay 8. A rule that applies at all times is that all communication between participants is prohibited. If you have any questions, raise your hand and I will come to you and Gain/Loss: The winners make a profit equal to the difference between the (re- you may ask your question in a whisper. If I believe the question must be answered, demption) value and the price. If the difference is negative, a loss is the result. I will repeat it to everyone and give the answer. Example of profit: You won one unit, and the price was 6. The value of the unit was 8. You made a profit of 2 (8 6 = 2). − Example of a loss: You won one unit, and the price was 10. The value of the unit was 8. You then made a loss of 2 (8 10 = 2). 11.2 Design − − Note If you do not have one of the highest bids, nothing happens. The profit is Rounds: The experiment consists of several rounds. In each round, 2 identical zero. objects, or units, are to be sold through an auction. (How many rounds to actually be played will be unknown to you.) 11.4 Practical execution The commodities: We will name the units as unit A and unit B. Each of you has a value associated with owning these units and would like to buy them. We call Bidding: You will come to a (web-)page where you see two dice, one of them this the redemption value, which is the same for both units. without dots. The one with dots is your signal. Below the dice, there will be 2 fields, one for unit A and one for unit B. You place your bids for the two units in The redemption value: Before the start of each round, the value of the units is these fields. Only integers between 0 and 12 are possible. (The units are identical, randomly determined through the roll of two dice. The redemption value will then and each bid is for one of the two units.) be the sum of the dice. The value can thus never be less than 2, and the maximum is 12. Therefore, the (value) v belongs to the set 2, 3, , 11, 12 . However, you { ··· } Money: You will see what your current balance is before every game starts on the will not know what this value is. Instead, you will get private information about screen. The starting balance is 10 experimental currency. These will be converted this value. to SEK 5/1 at the end of the experiment. If you lose your starting balance, the auction is over for you. Information: Your information will consist of one of the dice; the other die will be hidden. Thus, you have to make your bids with only partial information of the Lost starting balance: If someone (or some) loses her starting balance, she will value. The program randomizes which of the two dice you will see. Other players no longer participate in the auction. This means that there will be one (or more) may, but must not, see the same die as you do. person(s) less in the auction. If that happens, the auction continues as usual

32 33 ESSAY II 2). − 10 = − 6 = 2). − 33 If someone (or some) loses her starting balance, she will You won one unit, and the price was 6. The value of the You won one unit, and the price was 10. The value of the Before the beginning of each round, the program will randomly choose 2 units are sold. Three people (A, B, C) have the three highest bids: The winners make a profit equal to the difference between the (re- /1 at the end of the experiment. If you lose your starting balance, the You will come to a (web-)page where you see two dice, one of them You will see what your current balance is before every game starts on the The winners will pay a price equal to the highest bid that does not win, After receiving your information, that is, after seeing your die, you should If you do not have one of the highest bids, nothing happens. The profit is Those who have placed the highest bid, and the next-to-highest bid, purchase 10 (A), 9 (B), 8 (C). A and B purchase the units, and both pay 8. unit was 8. You made a profit ofunit 2 was (8 8. You then made a loss of 2 (8 the units. This mayamong be the the (winning) same bids, person the or program two will different randomly people. choose If the there winner(s). That are is, ties the highestunits. bid that is rejected.Example: All winners pay the same price for the how many players youopponents. Your will group-size be will be matched seen with. on You your can screen. havedecide one, on two what or youor three want different to bids bid for for the the units. units. You are permitted to place equal demption) value and the price. IfExample the of difference profit: is negative, a loss is the result. zero. Example of a loss: screen. The starting balance isto 10 SEK experimental 5 currency. Theseauction will be is converted over for you. without dots. The onefields, one with for dots unit is Athese your fields. and Only one signal. integers for Below between unitand 0 the B. each and dice, You bid 12 place are there is your possible. for will bids (The one for be units of the are 2 the two identical, units two in units.) no longer participate in theperson(s) auction. less This means in that the there will auction. be If one (or that more) happens, the auction continues as usual Bidding: 11.4 Practical execution 11.3 Instructions Buy: Price: Gain/Loss: Opponents: Bids: Note Lost starting balance: Money: : 12}. However, you 11, , ··· 3, {2, 32 Before the start of each round, the value of the units is We will name the units as unit A and unit B. Each of you has Your information will consist of one of the dice; the other die will 10), and by learning the rules of the game you have the opportunity The experiment consists of several rounds. In each round, 2 identical objects, or units, are tobe be sold played through will an be auction. (How unknown many to rounds you.) to actually a value associated withthis owning the these redemption units value, and which would is like the to same buy for them. both We units. call randomly determined through the rollbe of two the dice. sum The of redemption the valueis dice. will 12. The then value Therefore, can the thus (value) never be v less belongs than to 2, the and set the maximum will not know whatthis this value. value is. Instead, you will get private information about be hidden. Thus, you havevalue. to The make program your randomizes bids which withmay, of but only partial the must information two not, of dice see the you the will same see. die Other as players you do. 11.2 Design Rounds: The commodities: The redemption value: 11 Appendix C Bidder instructions for the uniform, common value auction 11.1 Introduction Hello and welcome.making. You The will purpose participate is in to an study sales experiment byYou on bidding, have the economic i.e. opportunity through decision- to anSEK win auction. 100 money ( throughto participation. The earn show-up more feeTo than is ensure that. that On youa walk the starting away other with balance hand, atthe of least experiment. you SEK SEK Your could 50. winnings, 100 alsoexperiment. and If in the lose you your show-up pocket, in lose fee, we will the this give be money, process. you paid you inA will cash rule be after that the excluded appliesprohibited. from at If all you have timesyou any is may questions, ask that raise your all question yourI in communication hand will a between and repeat whisper. participants I If it is I will to believe come everyone the and to question give you must the be and answer. answered, Information: without them but, since we need to have even groups, the program randomizes Bidder Instructions for the discriminatory and Vickrey auctions: which players are going to play in subsequent rounds. You may have to pass a round or two. You will be given notice about that on your screen. The item price in the Instructions above is changed for the two other auction formats; for the discriminatory auction it is: One round: After you enter your bids in the fields, press the button ”Add bids”. When everyone has pressed the button, bids are ranked. Those who have placed the highest bids purchase units at a price that is determined by the pricing rule Price: The winners pay a price equal to their own placed bid. for each auction. Example: 2 units are sold. Three people (A, B, C) have the three highest bids: If there are more winning bids than units for sale, the program randomizes the 10 (A), 9 (B), 8 (C). A and B purchase the units, and they pay 10 and 9, winners. The balance is recalculated and a new round starts. On the screen you respectively. will see the redemption value for the units, the price, the winning bids, own won units, and own profits/losses. And for the Vickrey auction, we have: The end: After a certain number of rounds, the experiment will end and you will come to a page showing what you have earned in the experiment. Price: The winner(s) pays a price equal to the highest bid that does not win, not including his own. That is, the highest bid that is rejected and comes from someone else. 11.5 Summary Example 1: 2 units are sold. Four people (A, B, C, D) have the four highest bids: 10 (A), 9 (B), 8 (C)and 7 (D). A and B purchase the units, and both pay You will play a certain number of rounds and, in each round, two identical units 8. • are for sale. Example 2: 2 units are sold. Three people (A, B, C) have the four highest bids: 11 (A), 10 (B ),9(B ), 8 (C). A and B purchase one unit each, A pays 9, and You will play against one, two or three opponents. On the screen you will see the 1 2 • B pays 8 (since 9 is his bid). number of opponents you have in the current round. Example 3: 2 units are sold. Three people (A, B, C) have the four highest bids: In each round, all players in an auction have the same redemption value for both 7(A1), 6 (A2), 5 (B), 4 (C). A purchases both units; for the first he pays 5 and • units. for the second he pays 4.

Each player only gets an informational signal about the true value. Subjects may • or may not see the same information as their opponents. Otherwise, the instructions for the formats are the same.

You place two bids, one for each unit. You are allowed to place equal or diﬀerent • bids on the units.

You start with 10 experimental currency. If you lose this, the experiment is ﬁn- • ished for you, and you are excluded from the experiment. But you can also earn more, depending how you and your opponents act.

34 35 ESSAY II : 35 ), 8 (C). A and B purchase one unit each, A pays 9, and 2 ), 4 (C). A purchases both units; for the first he pays 5 and B ),9( 1 ), 5 (B 2 2 units are sold. Four people (A, B, C, D)2 units have are the sold. four Three people highest (A, B, C) have the four highest bids: 2 units are sold. Three people (A, B, C) have the four highest bids: B 2 units are sold. Three people (A, B, C) have the three highest bids: ), 6 (A 1 The winners pay a price equal to their own placed bid. The winner(s) pays a price equal to the highest bid that does not win, A 10 (A), 9respectively. (B), 8 (C). A and B purchase the units, and they pay 10 and 9, bids: 10 (A), 9 (B),8. 8 (C)and 7 (D). A and B purchase11 the (A), units, 10 and ( both pay for the second he pays 4. B pays 8 (since 9 is his bid). 7( Example: not including his own.someone That else. is, the highestExample bid 1: that is rejected and comes from Example 2: Example 3: Bidder Instructions for the discriminatory and VickreyThe auctions item price inmats; the for Instructions the above discriminatory is auction changed it for is: the two otherPrice: auction for- And for the Vickrey auction, we have: Price: Otherwise, the instructions for the formats are the same. 34 After you enter your bids in the fields, press the button ”Add bids”. After a certain number of rounds, the experiment will end and you will You will play a certainare number for of sale. rounds and, in eachYou round, will two play identical against units one,number two of or opponents three you opponents. have On in the the screenIn you current each will round. round, see all the playersunits. in an auction have the same redemptionEach value player for only both gets anor informational may signal not about see the the true same value. Subjects informationYou may place as two their bids, opponents. onebids for on each the unit. units. You are allowed toYou place start equal with or 10 different ished experimental for currency. you, If and youmore, you lose depending are this, how excluded the you from and experiment the your is experiment. opponents fin- But act. you can also earn When everyone has pressedthe the highest button, bids bids purchase arefor units ranked. each Those at auction. who a have price placed thatIf is there determined are by more thewinners. winning pricing The bids rule balance than iswill units recalculated see and for the a sale, redemption newunits, the value round for and program the starts. own randomizes units, profits/losses. On the the the price, screen the you winning bids, own won come to a page showing what you have earned in the experiment. without them but, sincewhich we players need are to goinground have to or even play two. groups, You in the will subsequent be program rounds. given randomizes You notice may about have that to on your pass screen. a 11.5 Summary • • • • • • The end: One round:

ESSAY III Multi-unit common value auctions: An experimental comparison between the static and the dynamic uniform auction

Joakim Ahlberg

VTI - Swedish National Road and Transport Research Institute, P.O. Box 55685, SE-102 15 Stockholm, Sweden Tel: +46 8 555 770 23. E-mail address: [email protected]

Abstract

It is still an open question whether the dynamic or the static format should be used in multi-unit settings, in a uniform price auction. The present study conducts an economic experiment in a common value environment, where it is found that it is more a question of whether the auctioneer wants to facil- itate price discovery, and thereby lessen the otherwise pervasive overbidding, or if only the revenue is important. The experiment in the present paper provides evidence that the static format gives a signiﬁcantly greater revenue than the dynamic auction, in both small and large group sizes. But a higher revenue comes at a cost; half of the auctions in the static format yield negative proﬁts to the bidders, the winner’s curse is more severely widespread in the static auction, and only a minority of the bidders use the equilibrium bidding strategy.

Keywords: Laboratory Experiment; Multi-Unit Auction; Common Value Auc- tion

JEL codes: C91; C72; D44

1 Introduction

In many auctions, such as for CO2 allowances, electricity, bonds, etc, the auctioneer wants to sell many items at the same time, and bidders are usually not content with buying just one unit. All units for sale have the same value for bidders in these auctions. That is, the proﬁt is linear, or is a multiple of the number of units won. For some of them, as a ﬁrst approximation, the value is also equal across bidders because the value of the unit often depends on

1 ESSAY III [email protected] : allowances, electricity, bonds, etc, the 1 2 CO auction E-mail address Joakim Ahlberg bidders because the value of the unit often depends on 55685, SE-102 15 Stockholm, Sweden across C91; C72; D44 Laboratory Experiment; Multi-Unit Auction; Common Value Auc- static and the dynamic uniform Tel: +46 8 555 770 23. VTI - Swedish National Road and Transport Research Institute, P.O. Box experimental comparison between the Abstract It is still anbe open used question in whether multi-unitconducts settings, the an in dynamic economic a or experiment uniformis the in price found static a auction. that format common The it should itate present value is price study environment, more discovery, where and a it or thereby question if lessen of only the the whether otherwisevides revenue the pervasive evidence is overbidding, auctioneer that important. the wants The static tothe experiment format in facil- dynamic gives the a auction, significantly present in greaterenue paper revenue both comes pro- than at small a andprofits cost; large to half group the of sizes. bidders,static the But auction, the auctions a and winner’s in higher only curse thestrategy. a rev- static is minority format of more the yield severely bidders negative widespread use in the the equilibrium bidding Keywords: tion JEL codes: 1 Introduction In many auctions, such as for Multi-unit common value auctions: An auctioneer wants to sell manynot items content at with the buying samefor time, just bidders and one in bidders unit. these are auctions. Allnumber usually That units is, of for the units profit sale won. isis have linear, For the also or some same is equal of a value multiple them, of as the a first approximation, the value some outside parameter, common to all bidders. 1 Such auctions are referred The two allowance auctions above employ a static, sealed bid, uniform price to as common value auctions. Even though the valuation of the items across auction. Both theoretical and experimental economic research suggests that bidders is identical in a common value auction, it is unknown at the time of a dynamic auction format is preferable to the static auction in conducting bidding. Bidders’ information only consists of a (privately known) signal. allowance auctions. In the theoretical literature, Milgrom and Weber (1982) show, in single-unit, affiliated value auctions, that the informational content in When there are secondary markets, as in the emission permits market or the open auctions reduces the bidder’s uncertainty about the (affiliated) value and bond market, the price in the secondary market can be a good estimator of the thus, bidders are able to bid more aggressively in them. In the experimental price in the auction; that is, a common price signal for all bidders. But, private literature, Kagel et al. (1987) report a pervasive bidding above value in the signals also exist. For example, when there are market-dominant participants single-unit, IPV static auctions, which is, however, alleviated in the dynamic in the auction, they could, due to their (demand) size, be price drivers. This format. See also Kagel (1995). is especially true when there is a fixed quantity for sale, and big participants need/want a large share of the supply. Then, their own demand is one type In the common value (CV) environment, an essential advantage of dynamic, or of private signal. Signals can also contain information such as (under hand) open, bidding is that the bidding process reveals information about the other political information about change of rules, or technology changes which are bidders’ estimates of the value. Consequently, the winner’s curse is likely to not known to every participant in the auction. be mitigated in the open auction. The argument is that, by using tentative price information, bidders are better able to make more precise calculations Comparing the CO2 allowance auctions in the USA and the EU, that is the about the value; thus the open auction facilitates price discovery. Regional Greenhouse Gas Initiative (RGGI) and European Union Emission Trading Scheme (EU ETS), respectively, a striking difference is that the clearing price in the EU ETS is more than five times higher than in the RGGI. The seminal closed-form equilibrium analysis of the winner’s curse (WC) was This discrepancy has a couple of different explanations; the fixed quantity, made by Wilson (1969), and has since then been shown by Bazerman and Samuelson i.e. the shortage of units for sale, and the number of bidders, given the same (1983) in various experimental environments. In the present experiment, we supply. The two are correlated, and they are intertwined with the bidders’ de- discriminate between bidding above the conditional expected value (of win- mand. The quantity cap in the RGGI has been non-binding, the reserve price ning) and the more naive conventional expected value. The rationale is that has been met in the last six auctions, whereas, for the EU ETS, the clearing bidding above the naive expected value has nothing, strictly speaking, to do price fluctuates with the number of bidders; the more bidders, the higher the with the WC; it will transmit negative profit in the mean. Whereas bidding price. 2 in the WC interval, which is defined as bidding in between the two expected values and winning, could ensure a negative profit; it depends on how other In the present study, we try to replicate the two allowance auctions mentioned, players bid. but without varying the cap (i.e. supply). The experiment makes use of two group sizes, the first includes a large group of bidders that more or less has Both these arguments run in favor of open bidding, rather than sealed bid- the same relation between demand and supply as the EU ETS, and, second, ding. The open paradigm is also widely used by the Federal Communication a smaller group of bidders that has half the demand (and players) of the first Commission when selling radio frequencies in the USA. In IPV settings, some group. For each group size, there are also two different demand sizes. Even research, e.g. Klemperer (2002) and Engelmann and Grimm (2009), instead though it is theoretically not the same to cut the number of bidders’ demands calls for caution due to the facilitative facilitating effect of the open format on in half as to increase the supply twofold, there is experimental evidence that, collusion between bidders, since all bids, or quantities demanded, are visible contrary to the predictions of a Nash equilibrium, bidding does not decrease for all participants still in the auction. in response to an increased number of bidders. See Kagel (1995) and Ahlberg (2011). Multi-unit, common value experiments are rare, and the experiment in the 1 For the CO2 allowance auction case, the value is a proxy for the social abatement present study contributes to the ongoing debate on open or sealed bid auction cost; in the electricity auction, the value comes from the electricity price; whereas mechanisms inside the uniform price mechanism. One exception is the closely in the bond case, the value is driven by the interest rate. related experiment conducted by Ausubel et al. (2009), which is focused on 2 Data from RGGI can be found at http://www.rggi.org/market/market troubled assets and liquidity needs. They find that, even though the formats monitor and data from the EU ETS can be found at rendered similar prices, the open format gave substantially higher (bidder) http://ec.europa.eu/clima/policies/ets/auctioning/second/index en.htm. payoffs as well as reduced bid errors.

2 3 ESSAY III 3 ensure a negative profit; it depends on how other could The two allowance auctionsauction. above Both employ a theoretical static, anda sealed experimental dynamic bid, economic auction uniform researchallowance format price suggests auctions. is that In preferable theshow, to in theoretical single-unit, the literature, affiliated value static Milgrom auctions,open that and auction auctions the reduces Weber informational in the (1982) content bidder’s in conducting thus, uncertainty about bidders the are (affiliated) able valueliterature, and to Kagel bid et more al.single-unit, aggressively (1987) IPV in report static them. auctions, a Informat. which pervasive the See is, bidding experimental also however, Kagel above alleviated (1995). in value the in dynamic the In the common value (CV) environment,open, an bidding essential advantage is of that dynamic,bidders’ the or bidding estimates process of reveals thebe information value. about mitigated Consequently, the in the other price the winner’s information, open curse bidders auction. is areabout The likely better the to argument value; able is thus to that, the make by open more auction using precise facilitates tentative price calculations The discovery. seminal closed-form equilibriummade analysis by of Wilson (1969), the and winner’s has(1983) since curse then in (WC) been was shown various by experimental Bazermandiscriminate and environments. between Samuelson In bidding the abovening) present the and experiment, conditional the we more expectedbidding naive value above conventional the (of expected naive win- with value. expected the The value rationale WC; has it is nothing,in will that the strictly transmit WC speaking, negative interval, tovalues which profit do and is in winning, defined the asplayers mean. bidding bid. Whereas in bidding between the two expected Both these arguments runding. in The open favor of paradigmCommission is open when also selling bidding, radio widely rather frequenciesresearch, used than in e.g. by the sealed the Klemperer USA. bid- Federal Incalls (2002) Communication IPV for and settings, caution due Engelmann some to andcollusion the Grimm between facilitative bidders, (2009), facilitating effect since instead for of all all the bids, open participants format or still on in quantities the demanded, auction. are visible Multi-unit, common value experimentspresent study are contributes to rare, themechanisms and ongoing inside debate the the on uniform experiment open pricerelated or in mechanism. sealed experiment the One bid conducted exception auction is bytroubled the assets Ausubel closely and et liquidity al.rendered needs. (2009), similar They which find prices, is that,payoffs the focused as even open well though on as the format reduced formats gave bid substantially errors. higher (bidder) /market en.htm. /index Such auctions are referred 1 /second 2 /policies/ets/auctioning allowance auctions in the USA and the EU, that is the 2 /clima CO allowance auction case, the value is a proxy for the social abatement 2 CO 2 Data from RGGI can be found at//www.rggi.org/market http: For the monitor and data from the EU ETS can be found at to as common valuebidders auctions. is Even identical though in thebidding. a valuation Bidders’ common of information value the only auction, items consists it of across is aWhen unknown (privately known) there at signal. are the secondary timebond markets, of market, as the in price in theprice the emission in secondary the permits market auction; can market that besignals or is, a also a the good common exist. estimator price of For signal the example,in for when the all there bidders. auction, But, are they private is market-dominant could, participants especially due true to when theirneed/want there a (demand) is size, large a be share fixedof price of quantity private drivers. for the signal. This sale, supply. Signalspolitical and Then, can big information their also participants about own contain changenot demand information of known is such rules, to one as every or type participant (under technology in changes hand) the which auction. Comparing are the //ec.europa.eu http: 2 In the present study, we trybut to without replicate the varying two the allowancegroup auctions cap mentioned, sizes, (i.e. the supply). firstthe The includes same experiment a relation makes large between usea demand group smaller of and of group two supply of biddersgroup. as bidders that that the For more has each EU or half ETS, groupthough less the and, it size, demand has is second, there (and theoretically players) are notin of the also half the same as two first to to different cutcontrary increase the demand to the number sizes. the supply of bidders’ predictions Even twofold,in demands there of response is a to experimental Nash an evidence(2011). equilibrium, increased that, bidding number of does bidders. not See decrease Kagel1 (1995) and Ahlberg some outside parameter, common to all bidders. cost; in the electricityin auction, the the bond value case, comes the from value the is electricity driven price; by the whereas interest rate. Regional Greenhouse GasTrading Initiative Scheme (RGGI) (EU ETS), and respectively, aing European striking price Union difference is in Emission that theThis the clear- EU discrepancy ETS has isi.e. a the more couple shortage than of of fivesupply. units different The times two for explanations; are higher sale, correlated, the than andmand. and fixed The in the they quantity are number quantity, the cap intertwined of RGGI. with inhas bidders, the the been given bidders’ RGGI met the de- has in same beenprice the non-binding, fluctuates last the with six reserve the price price. auctions, number whereas, of bidders; for the the more EU bidders, ETS, the the higher clearing the This study compares two different uniform price auctions; the static and the 2 Earlier research on static vs. dynamic formats ascending clock auction, both in a common value environment. To address the above questions, both formats are used in two group sizes: 3- and 6-bidder groups. Letting the configuration of the larger groups (in own demand) be The research on multi-unit, common value auctions is still embryonic. Much exactly two times that of the smaller groups, and letting the supply be equal has been done in the independent private value (IPV) field, especially with in both groups, is effectively comparing a loose and a tight cap at the same time single unit demand. Vickrey (1961) was the first to show that, in theory, (if bidding does not adapt to the increasing number of bidders). The loose cap, the static second-price auction produced efficient outcomes in the IPV set- 1 represented by the 3-player groups, has the relation 2 of supply (numerator) ting with single unit demand. Vickrey was also the first to state the revenue and aggregated demand (denominator), whereas the tight cap, or 6-player equivalence theorem that under certain conditions, any allocation mechanism S 1 groups, has the relation = 4 . Moreover, the two group sizes always have will lead to the same revenue for the seller. Riley and Samuelson (1981) and 1 D the relation 2 between a large demander (numerator) and a small demander Myerson (1981) then generalized the theorem. In contrast to this, laboratory (denominator). The tight cap resembles the EU ETS auctions conducted in experiments have proved that the dynamic second-price auction, the English Great Britain (but which are open to participants throughout the EU). auction, performs roughly as predicted by theory, whereas the static second- price auction does not. One rationale for that is that the transparency of the dynamic mechanism guides subjects; see for example Kagel (1995). The main results from the experiments are; In the multi-unit case, the seminal (game theoretic) article is by Wilson (1979), who, in an auction of shares, found collusive equilibria with prices lower than if the unit was sold as an indivisible unit. Later, Ausubel and Crampton (2002) The seller revenue is significantly greater in the sealed-bid format. But it • showed that the efficiency of the second-price, multi-unit auction may break comes at the cost of a considerably more negative profit for buyers, and down due to demand reduction. Demand reduction, which is the phenomenon nearly half of the auctions ended with a negative profit for the subjects. of bidders reducing demand (on marginal units) in favor of a lower market- In line with this is the considerably smaller amount of WC in the open • clearing price, has been shown in a number of experiments since then. In an- format, both bidding in the WC interval and experiencing a negative profit. alyzing the difference between the static and the dynamic uniform auction 3 There is also a notable quantity of bids above the conventional, naive, ex- in a model with two bidders, with two-unit demand, Engelmann and Grimm pected value, especially in the static format. (2009) see a larger share of demand reduction, especially extreme demand The more bidders (the tighter the market), the greater the revenue. • reduction, in the dynamic format versus the static in an IPV setting. Consis- None of the formats seem to result in high bids that coincide with individual • tent with that, they also find that the static version outperforms the dynamic rationality. That is, there is overbidding; less than 1/5 of all subjects’ first version in terms of collecting revenues as well as efficiency. Alsemgeest et al. unit bid/dropout is at, or below, the expected value of the unit. (1998) also report lower revenues in the English clock auction as compared to The demand reduction, measured as the bid spread, is visible in both for- the static version, due to demand reduction. • mats, but it is significantly lower in the dynamic auction. Vickrey (1961) also described an efficient mechanism in multi-unit settings in the IPV environment, nowadays called the Vickrey auction. Ausubel (2004) We conclude that the dynamic auction seems to be a better choice in common then came up with an open format that implements the same outcome as the multi-unit Vickrey auction in an IPV setting, and continues to be efficient in value environments, especially if the players are without experience. It facili- 4 tates price discovery, and thereby alleviates the overly aggressive bidding. The an affiliated value (AV) environment which is not the static Vickrey auction. choice between an open or a closed format may be more important than the Manelli et al. (2006) experimentally compare the static Vickrey auction with choice of price mechanism, especially in common value settings. the Ausubel auction, also known as the dynamic Vickrey auction, in both an IPV setting and an interdependent value (IV) setting, where the values are

3 The uniform auction is a generalized second-price auction, meaning that the price The remainder of the paper is organized as follows. Section 2 provides an paid is the highest losing bid. overview of some earlier research, section 3 introduces the experimental model 4 Aﬃliated value comes from Milgrom and Weber (1982), and roughly means that and delivers the hypotheses. Section 4 presents the experimental results, while a high value of one bidder’s estimate makes high values of the others’ estimates section 5 discusses them. Section 6 concludes the paper. more likely.

4 5 ESSAY III 3 which is not the static Vickrey auction. 5 4 The uniform auction is a generalized second-price auction, meaning thatAffiliated the value price comes from Milgrom and Weber (1982), and roughly means that paid is the highest losing bid. a high valuemore of likely. one bidder’s estimate makes high values of the others’ estimates Manelli et al. (2006)the experimentally Ausubel compare auction, the also staticIPV known Vickrey setting as auction the and with dynamic an Vickrey interdependent auction, value in (IV) both setting, an where the values are 3 4 in a model with(2009) two see bidders, with a two-unitreduction, larger demand, in Engelmann share the and dynamic oftent Grimm format with demand versus that, the reduction, they static also especiallyversion in find in an extreme that terms IPV the demand of setting. static(1998) Consis- version collecting also outperforms revenues report the as lower dynamic revenuesthe well in static as the version, efficiency. English due Alsemgeest clock to et auction demand al. as reduction. comparedVickrey to (1961) also describedthe an efficient IPV mechanism environment, in nowadaysthen multi-unit called settings came in the up with Vickreymulti-unit an auction. Vickrey open Ausubel auction format in (2004) thatan an implements affiliated the IPV value same setting, (AV) environment outcome and as continues the to be efficient in 2 Earlier research on static vs. dynamic formats The research on multi-unit,has common been value done auctions issingle in still the unit embryonic. independent demand. Much the private Vickrey value static (1961) (IPV) second-price field, wasting auction especially with the produced single with first efficient unitequivalence outcomes demand. to theorem in Vickrey that show under was the that, certain alsowill IPV conditions, the lead in any set- first to allocation theory, to the mechanism Myerson state same (1981) the revenue then for revenue generalizedexperiments the the have seller. theorem. proved Riley In that and contrastauction, the Samuelson to performs dynamic (1981) this, roughly second-price and laboratory as auction,price predicted the auction English by does theory, not. whereasdynamic One the mechanism rationale guides static for subjects; second- that see is for that example theIn Kagel transparency the (1995). multi-unit of case, the the seminalwho, (game in theoretic) an article auction is of bythe shares, Wilson found (1979), unit collusive was equilibria sold with as pricesshowed lower an that than indivisible if the unit. efficiency Later,down of due Ausubel the to and demand second-price, Crampton reduction.of multi-unit (2002) Demand bidders auction reduction, may reducing which is break demandclearing the price, (on phenomenon has marginal been units)alyzing shown in the in favor difference a of between number a of the experiments static lower since market- and then. the In dynamic an- uniform auction 5 of all subjects’ first of supply (numerator) / 1 2 4 . Moreover, the two group sizes always have 1 4 = S D between a large demander (numerator) and a small demander 1 2 The seller revenue iscomes significantly at greater the innearly cost the half of sealed-bid of a the format.In auctions considerably But ended line more it with with negative aformat, this negative profit both profit is bidding for for in the buyers, the theThere WC subjects. considerably and is interval and smaller also experiencing a amountpected a notable negative value, of quantity profit. especially WC of inThe bids in the more above static the bidders the format. open (the conventional,None tighter naive, of the ex- the market), formats seem therationality. to greater That result the in is, revenue. high there bids is that coincide overbidding; with less individual than 1 unit bid/dropout is at,The or demand below, reduction, the measured expectedmats, as value but of the it the bid is unit. significantly spread, lower is in visible the in dynamic both auction. for- the relation This study compares twoascending different clock uniform price auction, auctions;the both the above questions, in static both and a formatsgroups. the are common Letting used value in the two environment.exactly group configuration To sizes: two of address 3- times and that the 6-bidder in of larger both the groups, groups is smaller effectively (in groups, comparing(if a and own bidding loose letting does and demand) not a the tight adapt be supplyrepresented cap to at be by the the increasing equal the same number time 3-player of groups, bidders). The has loose the cap, relation (denominator). The tight capGreat Britain resembles (but the which EU are ETS open auctions to conducted participants throughout in the EU). The main results from the experiments are; We conclude that the dynamicvalue auction environments, seems especially to be iftates a the price better players discovery, choice and are in thereby without common alleviateschoice experience. the between overly It an aggressive facili- bidding. openchoice The or of a price closed mechanism, format especially may in be common value more settings. important than the The remainder ofoverview the of some paper earlier research, isand section delivers 3 organized the introduces hypotheses. as the Section experimentalsection follows. 4 model 5 presents Section the discusses experimental 2 them. results, Section provides while 6 an concludes the paper. and aggregated demandgroups, (denominator), has whereas the the relation tight cap, or 6-player • • • • • affiliated. They conclude that due to overbidding in both types of auctions, together in a pooled-unit auction. This is a pure CV environment but with but slightly more in the Vickrey auction, the revenue from the Vickrey auction non identical units. is greater, while the efficiency is lower in the Ausubel auction. But in the IV setting, they observe less overbidding and a trade-off between efficiency and One important implication of our (Kagel’s) way of generating the common revenue; the Vickrey auction is more efficient while the revenue is higher in value is the three distinct signal regions to which it gives rise, with a different the Ausubel auction. informational content. The most interesting region, which encompasses the larger mass of bids, lies in 20, 80 . (It is called region 2.) In this region, the Concerning the (pure) common value environment, much of the focus is on signal is always an unbiased{ estimator} of the true value, ex ante. The other the winner’s curse and very few studies focus on the multi-unit case. One, two regions, regions 1 and 3, contain signals in the interval s 0, ..., 19 and i ∈{ } notably, is Ausubel et al. (2009), which experimentally tests alternative auc- si 81, ..., 100 . The information that the signal is in one of these regions tion designs suitable for pricing and removing troubled assets. They make use can∈{ be used to compute} a more exact expected value than signals from region of the same static and dynamic uniform auction as the present study and 2. That is, in region 1 (region 3), the signal is a downward (upward) biased Engelmann and Grimm (2009) above, except that their dynamic format is an estimator of the true value. And the lower (higher) the signal is in region 1 Ausubel descending clock auction. The units for sale are not identical, and (region 3), the more downward (upward) biased it is. Signals at the endpoints they sell the units individually or as pooled units. And, for some sessions, can be used to compute an exact value. bidders also know their liquidity needs. They find that the static and dynamic auctions resulted in similar prices. However, the dynamic auctions resulted in Given signal si, the estimated valuation will be contained in vi [ max si ∈ { { − substantially higher bidder payoffs, which made it possible for the bidders to 10, 10 , min si + 10, 90 . Bidders can place a risk free bid by bidding the } { }} better manage their liquidity needs. The dynamic auction was also better in lower end-point in this interval. terms of price discovery, as well as at reducing bidder error. Two group sizes are used; 3-player groups and 6-player groups. As hypothesized, the two treatment groups can be seen as either representing a loose and a tight cap, respectively, or just plainly as two different group sizes. Inside the 3 The Model and Hypotheses smaller group, one bidder demands 4 units and two bidders 2 units each. The larger group has the same relationship between small and large demanders, that is two 4-unit demanders and four 2-unit demanders. Aggregated demand The experiment will take place inside a multi-unit, common value (CV) auc- is thus 8 (16) in 3-player (6-player) groups. The supply in each auction is 4 tion model where bidders have independent (private) signals. Four units will 1 1 units. Thus, we have the relationship 2 ( 4 ) between supply and aggregated be sold in each round, and all bidders place the same value v on each unit in demand in small (large) groups. a given auction, i.e. subjects have flat demand curves. The common value v is an integer drawn from a uniform distribution on the interval V = 10, 90 , { } In the static, the players bid in prices, whereas the dynamic is a quantity and the signal si is uniformly distributed around this value, and lies in the auction. In this quantity auction, the price is raised by means of a price clock interval S = v 10,v+ 10 0, ..., 100 , for i 1, 2, 3 (3-player groups) and players respond with the quantities desired at the prevailing price. The or i 1, ..., 6{ (6-player− groups).}⊆{ } ∈{ } ∈{ } quantity is restricted by an activity rule requiring monotonicity in quantities demanded, i.e. a dropout is irrevocable. This method for generating values comes from Kagel et al. (1987), where it is used in a single unit CV auction and can be contrasted to that used in In the sealed bid auction, all bidders submit, once and for all, their bids, and Manelli et al. (2006). In the latter study, all bidders get different private in- then the auctioneer ranks the bids from high to low. The four highest bids formation about the value, and the CV is calculated as the weighted average are deemed to be winning bids, and the owners of these bids pay the fifth of all bidders’ information. Thus, it is not a pure CV environment but an highest bid (b5) for each unit won. (When there are ties, the winning bids are interdependent value environment. Ausubel et al. (2009) use different meth- randomly determined.) Thus, if ki is the number of units won in the auction l ods for generating values. In the first method, they let the CV for a security for bidder i, bi 0, ..., 100 is the vector of bids for bidder i (where l 2, 4 (or unit) be the average of eight iid random variables, uniformly distributed is the the demand),∈{ and B}is the downward ranked vector of all bids.∈{ Then,} between 0 and 100, where a bidder’s private information about the unit is the the profit for each bidder is πi = ki(v B(5)). realization of one of the random variables. In the second method, the high- − value (U[50, 100]) and the low value (U[0, 50]) random variables are grouped The dynamic ascending auction is a natural generalization of the English

6 7 ESSAY III − 4} i and 2, {s 19} ∈{ price clock {max [ 0, ..., ∈ i v ∈{ (where l i i s region 2.) In this region, the ) between supply and aggregated 1 4 (5)). ( B 1 2 − 7 v ( is the number of units won in the auction i i k k 80}. (It is called = i is the vector of bids for bidder π 20, is the downward ranked vector of all bids. Then, l { B 100} 90}}. Bidders can place a risk free bid by bidding the , ..., 0 }. The information that the signal is in one of these regions + 10, , the estimated valuation will be contained in ∈{ ) for each unit won. (When there are ties, the winning bids are i i 5 i s b 100 b i, , ..., min{s 81 }, 10 ∈{ , i The dynamic ascending auction is a natural generalization of the English can be used to2. compute a That more is, exact in expectedestimator region value of than 1 the signals (region true from(region 3), region 3), value. the the And more signal the downwardcan is (upward) lower be biased a (higher) used it downward the is. to (upward) Signals signal compute biased at an is the exact in endpoints Given value. region signal 1 10 lower end-point in this interval. Two group sizes aresized, used; the two 3-player treatment groups groupsa and can tight be 6-player cap, seen groups. respectively, or as Assmaller just either hypothe- group, plainly representing one as a bidder two loose differentlarger demands and group 4 group sizes. units Inside has and the thethat two is bidders same two 2 relationship 4-unit units demanders betweenis each. and The small thus four 8 and 2-unit demanders. (16) largeunits. Aggregated in demanders, Thus, demand 3-player we (6-player) have groups. the The relationship supply in each auction is 4 demand in small (large) groups. In the static,auction. the In players this quantity bid auction,and in the players price prices, respond is whereas withquantity raised is the by the restricted means quantities dynamic of by desireddemanded, is a an at i.e. activity a the a rule quantity dropout prevailing requiring price. is monotonicity in irrevocable. The quantities In the sealed bidthen auction, all the bidders auctioneer submit,are ranks once the deemed and for bids to all,highest from be their bid high bids, winning ( to and bids, low. and The the four owners highest of bids these bids pay the fifth together in a pooled-unitnon identical auction. units. This is a pureOne CV important environment implication butvalue of with is our the three (Kagel’s) distinctinformational way signal content. of regions The to generating whichlarger most the it mass gives interesting common of rise, region, bids, with which a lies different in encompasses the signal is always antwo regions, unbiased regions estimator 1 of and 3, the contain true signals in value, the ex interval ante. The other is the the demand), and the profit for each bidder is for bidder randomly determined.) Thus, if s , v 90} {10, = V on each unit in v (3-player groups) 3} , 2 1, ∈{ i 50]) random variables are grouped , [0 100}, for U 6 0, ..., + 10}⊆{ ,v 10 is uniformly distributed around this value, and lies in the i − (6-player groups). s v { 6} 100]) and the low value ( = [50, S 1, ..., U ∈{ i affiliated. They conclude thatbut slightly due more in to the overbiddingis Vickrey in auction, greater, the both while revenue from the typessetting, the efficiency of Vickrey they is auction auctions, observe lower less inrevenue; the overbidding the Ausubel and Vickrey auction. a auctionthe But trade-off is Ausubel in between more auction. the efficiency efficient IV and while the revenueConcerning is the higher (pure) in the common value winner’s environment, curse muchnotably, and of is very Ausubel the few et focustion al. studies is designs (2009), focus suitable on which for on experimentally pricingof and tests the the removing alternative multi-unit troubled same auc- assets. case.Engelmann static They One, and make and Grimm use (2009) dynamicAusubel above, uniform except descending that auction clock their asthey dynamic auction. format the sell The is the present units an bidders for study units also sale and know individually their are or liquidityauctions needs. not as resulted They identical, in find pooled similar and that units.substantially the prices. static higher However, And, the and bidder dynamic for dynamic payoffs, auctionsbetter which some resulted made manage sessions, in it their possible liquidityterms for needs. of the The price bidders dynamic discovery, as to auction well was as also at better reducing in bidder error. 3 The Model and Hypotheses The experiment will taketion place model inside where a bidders multi-unit,be have common sold independent value in (CV) (private) each signals. auc- a round, Four given and units auction, all will i.e. biddersis place subjects an the integer have same drawn flat value from demand a curves. uniform The distribution common on value the interval interval and the signal or This method foris generating values used comes in fromManelli a et Kagel single al. et unit al. (2006).formation CV (1987), In about the where auction the latter it value, andof study, and can all the all be bidders CV bidders’ is get contrastedinterdependent information. calculated different value to Thus, as private environment. that the in- it Ausubelods weighted used et is average for in al. not generating (2009) values. a(or use In pure unit) different the be CV meth- first the method,between environment 0 average they but and of let an 100, eight the whererealization iid CV a of random for bidder’s one variables, private a information uniformly ofvalue security about distributed ( the the random unit is variables. the In the second method, the high- auction when selling more than one unit. In this auction, the price is gradually signals is not. (The max function is convex and thus overestimates the value.) raised by means of an integer price-clock from zero to one hundred, and players start with full demand and yield units as the price rises. The auction ends Assume that values are uniformly distributed and the signals are uniformly when there are only four units demanded left in the auction, and all winners distributed around the values. Then, if the value, and hence the signal, came pay the price that prevailed when the fifth-to-last unit was surrendered. Thus, from a continuous distribution, the conditional expected value for player i, if P (5) is defined as the price that prevailed when the fifth-to-last unit in with realized signal si, would be: the auction was surrendered, the profit-function is similar to the one above n 1 πi = ki(v P (5)), but now the bid B(5) has been changed to the clock-price Ei(v si >s i)=si 10 − (1) P (5). − | − − n +1

where s i is defined as the realized signals from the other players and n is In an IPV auction, with only one unit for sale, B(2) and P (2) would have − the number of players. Thus, in 3-player (6-player) groups, the bidders must the same value, if the distribution of the values and signals were continuous, scale down their expected value by 5 (7) from the signal to avoid falling prey by the revenue equivalence theorem. But we have two extensions from this: to the WC. We will use this measure when testing whether bidders account First, this is a common value auction and, second, there are four units for sale. for this adverse selection effect. The hypothesis, partly from Ahlberg (2011) Regarding the first extension, Milgrom and Weber (1982) showed that the dy- where there was a fairly large amount of WC in the static uniform auction, namic auction is always at least as good for revenue as the static counterpart is that they do not; but, once more, to a lesser degree in the dynamic auc- in a CV auction. But in the multi-unit case, the ranking is less clear, espe- tion because of its inherent price discovery mechanism. (The survey, by Kagel cially with CVs. From Vickrey (1961), we have that all weakly non-dominated (1995), of experiments with single-unit auctions also shows the presence of equilibria have one thing in common; namely, that the bid/dropout on the WC, to various degrees, for the inexperienced as well as professionals under a first unit should be the expected valuation of the unit. (The first unit means variety of circumstances.) the unit with the weakly highest bid/dropout.) For the subsequent units, the theory is still vague. Hypothesis 2 The winner’s curse will be present in both auctions, but more so in the static form. Thus, even though the dynamic auction is to collect weakly more revenue in contrast to the static auction, the experimental literature has supported From the above, we had: the dynamic auction for a long period of time because of its price discovery and transparency qualities. This is important since there appears to be a Hypothesis 3 The equilibrium strategy to bid or dropout at the conditional competitive effect, what seems to be a myopic joy of winning, that works in the expected value of the first unit should be more likely in the dynamic format other direction. That is, many other experiments, starting with Kagel et al. due to information revelation, but the problem may be to bid/dropout at the (1987) in an affiliated private value setting, have shown a pervasive bidding conditional expected value (equation 1) and not at the naive EV (si in region above the value in static uniform auctions, whereas this is alleviated in the 2). dynamic auction. This also carries over to CV settings, and Ahlberg (2011) showed, in another multi-unit, CV setting, substantial bidding above value in From equation 1, we had that the conditional expected value decreases with the static uniform auction. This overbidding affects the profit for the bidders, the number of bidders. From the Nash equilibrium theory, when there is just and often produces negative earnings. Thus, one unit for sale, we also have that the bids will decrease with the number of bidders. But, in contrast to this, Kagel et al. (1995) show that bidders fail Hypothesis 1 The static auction will, at the expense of the bidder profit, to respond to the Nash predictions in a single-unit, second-price auction with deliver the highest revenue of the two formats. CV. Ahlberg (2011) also shows this in a multi-unit setting. We believe the experimental literature to have more bearing also in this case; thus, we have In a common value auction, there is also an adverse selection effect called that: the winner’s curse (WC). It arises when bidders neglect the information a win will produce, and overbid as a result. The core of the WC is that the Hypothesis 4 Subjects’ bids will not decrease in response to an increased announcement of winning the auction leads to a decrease in the estimated number of bidders. Thus, instead of halving the supply, increasing the number value, if not accounted for when bidding. That is, even though the signal in of bidders, to construct a tighter market, will have the same effect. Hence, the region 2 is ex ante an unbiased estimator of the value, the largest of all bidders’ tighter the market, or the more bidders, the larger the revenue.

8 9 ESSAY III i, is (1) n in region i s 1 +1 − n n 10 − i s 9 )= −i >s i s | v ( i function is convex and thus overestimates the value.) E , would be: i s max The winner’s curse will be present in both auctions, but more The equilibrium strategy to bid or dropout at the conditional Subjects’ bids will not decrease in response to an increased is defined as the realized signals from the other players and i − s signals is not. (The Assume that values aredistributed uniformly around the distributed values. andfrom Then, the if a signals the continuous value, arewith distribution, and uniformly realized the hence signal the conditional signal, expected came value for player the number of players.scale Thus, down in their 3-player expected (6-player)to value groups, by the the 5 WC. bidders (7) Wefor must from will this the use signal adverse this to selectionwhere measure avoid effect. there falling when The was prey testing hypothesis, ais whether partly fairly that bidders from large they account Ahlberg amount dotion (2011) of because not; WC of but, its in once inherent(1995), the price more, of static discovery to mechanism. experiments uniform (The aWC, auction, with survey, to by lesser single-unit Kagel various degree degrees, auctions in forvariety also of the the inexperienced circumstances.) shows dynamic as the auc- well presence as professionals of underHypothesis a 2 so in the static form. From the above, we had: Hypothesis 3 expected value of thedue first to unit informationconditional revelation, should expected but be value the more (equation problem 1) likely may and in not be the at to dynamic the bid/dropout format naive at EV the ( 2). From equation 1, wethe had number that of the bidders. conditionalone From the expected unit Nash value for equilibrium decreases sale,of theory, with bidders. when we But, there also in is haveto contrast just that respond to to the this, the Kagel bidsCV. Nash et will predictions Ahlberg al. in decrease (1995) (2011) a with showexperimental single-unit, also that the literature second-price shows bidders auction number to this fail with havethat: in more bearing a also multi-unit in setting. this We case; believe thus,Hypothesis the we 4 have number of bidders. Thus,of instead bidders, of to halving construct thetighter a supply, tighter the increasing market, market, the will number or have the the more same bidders, effect. the Hence, larger the the revenue. where (2) would have P (2) and B (5) has been changed to the clock-price 8 B The static auction will, at the expense of the bidder profit, (5)), but now the bid P − v ( i k (5) is defined as the price that prevailed when the fifth-to-last unit in = P (5). i P In an IPVthe auction, same with value, only ifby one the the distribution unit revenue of for equivalence theFirst, sale, theorem. this values is and But a signals we commonRegarding were value have the auction continuous, two first and, extension, second, extensions Milgrom therenamic from are and auction four this: Weber is units (1982) always for showed that at sale. in the least a dy- as CV good auction.cially for with But revenue CVs. as in From the Vickrey theequilibria (1961), static we multi-unit counterpart have have case, one that all the thingfirst weakly ranking unit non-dominated in should is common; be lessthe namely, the clear, unit that expected with espe- the valuation the oftheory bid/dropout weakly the is highest on unit. still bid/dropout.) the (The vague. For the first subsequent unit units, means the Thus, even thoughin the contrast dynamic to auctionthe the is dynamic static to auction auction, for collectand the a weakly experimental transparency long more literature qualities. periodcompetitive has revenue This effect, of what supported time is seems to because important beother of a direction. since myopic its joy That of there price winning, is,(1987) discovery that appears many in works in to other an the experiments, affiliated beabove starting private the a value with value setting, Kagel indynamic have et static auction. shown al. uniform This a auctions, alsoshowed, pervasive in whereas bidding carries another this over multi-unit, to is CVthe setting, CV alleviated static substantial uniform settings, in bidding auction. and the above Thisand value Ahlberg overbidding often in affects (2011) produces the negative profit earnings. for Thus, the bidders, Hypothesis 1 deliver the highest revenue of the two formats. In a commonthe value winner’s auction, curse therewin (WC). is will It also produce, arises anannouncement and when adverse of overbid bidders selection winning as neglect effectvalue, the a if the called auction result. not information leads The accountedregion a to for core 2 is when a of ex bidding. ante decrease the an That unbiased in WC estimator is, of is the even the that estimated value, though the the the largest of signal all in bidders’ auction when selling more thanraised one by unit. means In of this an auction, integerstart price-clock the from with price zero is full to gradually onewhen demand hundred, there and and players are yield onlypay units four the as units price that demanded the prevailed leftif when price in the rises. fifth-to-last the unit The auction, wasthe and auction surrendered. auction all Thus, ends winners was surrendered,π the profit-function is similar to the one above The phenomenon when bidders reduce demand in favor of a better price is buy as many items as the bidder showed demand for. Before the auction began, called demand reduction. This happens in a uniform auction since, with a the subjects got instructions and, in three trial periods, had the opportunity positive probability, bids may determine the price paid on all units. Thus, in to become familiar with the interface. The information a bidder got in advance every undominated equilibrium, bids other than on the first unit are lower of each round was: the (updated) monetary balance, the own signal (as well than the expected value. The hypothesis of which of the two formats trans- as its distribution), own demand, total supply, and how many bidders there mits more demand reduction than the other also hinges on how the dynamic were in the auction from the start. Moreover, the subjects were equipped with auction behaves relative to the static auction. If we use the theory for in- a starting balance of 50 experimental currency. Bidder instructions are find in terdependent values by Ausubel and Crampton (2002), the dynamic auction Appendix B and C. should, if there is no collusive behavior, diminish the demand reduction ten- dencies and thereby give smaller differences between the bids/dropouts. We In the static version, the subjects simultaneously submitted bids. Then, the will measure demand reduction as the spread in players’ bids/dropouts. software ranked them and made the necessary calculations. Following each auction period, bidders were provided with the true value, the price, the four Hypothesis 5 Demand reduction, or bid-spread (dropout-spread), is likely to highest bids along with adherent signals, the number of units won and own be in play, but to a lower extent in the dynamic auction. profit.

In the dynamic version, the price started with zero for 15 seconds and then increased at a rate of 1 per second. 6 Bidders responded with the quantities 4 Experimental design demanded at all prices, starting with full demand for all participants at time (price) zero. At any price, bidders were able to drop out on any number of Table 1 shows the design of the auction. Each format will have two group demanded units. When bidder i, say, dropped out on 1 or more units, the clock stopped for 5 seconds and increased at a rate of 1 per second thereafter. Any 3-player groups 6-player groups other bidder dropping out during this brief pause was regarded as having the same drop-out price as the first, but later in time. (This five-second delay of small large small large time was implemented for every dropout.) Moreover, a dropout was irrevocable. The auction ended when demand equaled supply. If a dropout produced Static auction x x x x excess supply, the price was rolled back one increment and the bidder (who Dynamic auction x x x x dropped out) got to buy as many units as were needed to clear supply and Table 1 demand. The information on the screen during the bidding process was the Auction configuration prevalent price and the number of active bidders and own dropout prices (so far). Then, following each period, the computer screen showed the true value, sizes, and each group size will have small and large demanders. The demand the price, the four highest dropout prices along with adherent signals, the configuration in 6-player groups is exactly twice that of the smaller group, number of units won and own profit. which, in turn, has two subjects who demand 2 units and one subject who demands 4 units. The software was developed in Asp.Net framework 2.0 using c# for back- end programming and MsSQL database. 7 The sessions lasted for about 40 In the experiment, students from KTH (the Royal Institute of Technology) to 70 minutes; the open format often took a little longer, but the number of were used as experimental subjects. They were from different Master of Engi- subjects in the session was also a time driver. After each session, all earnings neering programs, and the experiment took place in September 2011. were exchanged into real currency. Each subject earned the same amount in

The experiment is between subjects in a fixed matching procedure, i.e. they subjects would play differently knowing it was the last round. This could happen if, played against the same competitors and were placed in the same group and say, they had lost much money during the first nine rounds, and therefore wanted had the same demand throughout the session. The subjects were recruited for to gamble a bit. 6 computer sessions where the given auction mechanism was iterated (unknown This may seem fast, but it is not; according to the subjects themselves. Moreover, to the subjects) 10 times. 5 In each auction, the bidders had the opportunity to it seems to be the common rate in similar experiments. 7 We also use Ajax for front-end programming to improve the user experience and 5 The decision not to communicate the number of rounds was made on the basis that interact with the database for fast feedback of input/output.

10 11 ESSAY III # for back- c 0 using . The sessions lasted for about 40 7 Bidders responded with the quantities 11 6 i, say, dropped out on 1 or more units, the clock This may seem fast, but it is not; according toWe the also subjects use themselves. Ajax Moreover, for front-end programming to improve the user experience and to 70 minutes; thesubjects open in format the often sessionwere took was exchanged also a into a little real time longer, currency. driver. but Each After the subject each number session, earned of all the earnings same amount in demanded at all prices,(price) starting zero. with full At demand anydemanded for units. price, When all bidders bidder participants were at able time to drop out on any number of end programming and MsSQL database. stopped for 5 secondsother and bidder increased dropping at out asame during rate drop-out this of price brief 1 pause as pertime was the second was regarded first, implemented thereafter. as but for Any havingble. every later the dropout.) The in Moreover, auction time. a ended (Thisexcess dropout when five-second was supply, demand delay irrevoca- the of equaled pricedropped supply. was If out) rolled a got dropout back todemand. produced one buy The increment as information and many onprevalent the price units the bidder and as screen (who the were duringfar). number needed Then, the of following to active bidding each bidders clear process period,the and supply the was price, own computer and the dropout the screen pricesnumber showed four (so the of highest true units value, won dropout and prices own profit. alongThe with software adherent was signals, developed the in Asp.Net framework 2 it seems to be the common rate ininteract similar with experiments. the database for fast feedback of input/output. subjects would play differently knowingsay, it they was had the lost lastto round. much This gamble money could a during6 happen bit. the if, first nine rounds, and therefore wanted 7 buy as many items as thethe bidder subjects showed demand got for. Before instructionsto the and, become auction began, familiar in with three theof trial interface. The each periods, information round had a bidder was: theas got the opportunity in its (updated) advance distribution), monetary ownwere balance, in demand, the the total auction own supply, from signala and the starting (as start. how balance Moreover, well many of the 50 bidders subjectsAppendix experimental were there B currency. equipped and Bidder with C. instructions are find in In the static version,software the ranked subjects them simultaneouslyauction and submitted period, made bids. bidders the Then, werehighest provided necessary the bids with calculations. the along Following true withprofit. each value, adherent the signals, price, the the four number of unitsIn won the and dynamic own version,increased the at price a started rate with of zero 1 for per 15 second. seconds and then 10 3-player groups 6-player groups small large small large In each auction, the bidders had the opportunity to 5 Demand reduction, or bid-spread (dropout-spread), is likely to Static auctionDynamic auction x x x x x x x x The decision not to communicate the number of rounds was made on the basis that Table 1 Auction configuration 5 sizes, and each groupconfiguration size in will have 6-player smallwhich, groups and in is large turn, exactly demanders. hasdemands The twice 4 two demand that units. subjects of who the demand smaller 2 group, In units the and experiment, onewere students subject used from as who KTH experimental subjects.neering (the They programs, Royal were and Institute from the different of experiment Master Technology) took of place Engi- The in experiment September is 2011. betweenplayed subjects against the in same ahad competitors fixed the and matching same were demand procedure, placed throughoutcomputer i.e. the in sessions they session. where the The the same subjects givento group auction were the and subjects) recruited mechanism was 10 for times. iterated (unknown The phenomenon whencalled bidders reduce demand demand reduction.positive in This probability, favor bids happens of may inevery a determine a the undominated better price uniform equilibrium, price paid auctionthan bids is on the since, other all expected units. with than Thus, value.mits a on in The more the hypothesis demand first reduction ofauction than unit which behaves the of are other relative the lower alsoterdependent to two hinges values formats the on by trans- how static Ausubelshould, the and auction. if dynamic Crampton If there (2002), is wedencies the no use dynamic and collusive the auction thereby behavior, give theorywill diminish smaller measure the for demand differences demand in- reduction reduction between as ten- the the bids/dropouts. spreadHypothesis We in players’ 5 bids/dropouts. be in play, but to a lower extent in the dynamic auction. 4 Experimental design Table 1 shows the design of the auction. Each format will have two group SEK as the monetary balance on his/her screen. Subjects earned, in the mean, First unit bid SEK 253 ( 25) which included a show-up fee of SEK 100 ( 10). The minimum 100 earning was SEK 100 ( 10), and the maximum earning was SEK 659 ( 66). 80

5 Experimental results Bids 40

The data description is found in Table 2, which shows, for each format, the 20 number of subjects, how many rounds there were, and the number of unique 0 20 30 40 50 60 70 80 observations, and the average proﬁt. Signals

No. of subjects No. of rounds Unique observations Second unit bid 100 Static 64 140 626 80 Dynamic 65 149 653 Table 2 60

Data summary Bids 40

All comparisons below use statistic tests based on aggregated data over all auc- 20 tion periods, if not stated differently. The non-parametric Wilcoxon(-Mann- 0 Whitney) rank sum test has been the main tool, especially between treatments 20 30 40 50 60 70 80 Signals but also within treatments when there is no dependency between the variables. For some comparisons, within a treatment where there is dependency, the non- parametric Wilcoxon signed rank test is employed. We have also tested OLS Fig. 1. First and second unit bids in the static uniform auctions. and panel data (random effects) models with the profit and revenue (price) as the dependent variables. Profit is explained by signal, format, group-size, almost all sales and prices (even though the sales (prices) become slightly demand and round, while revenue is explained by value, format, group-size overestimated because of the missing 5 (20) percent). But the first impression and round. There was only a marginal change in the results presented below is nonetheless that there is substantial bidding above the signal for the first and, thus, the conclusions still hold. (The OLS regression on bidder profit can unit, with more than half of all first unit bids being greater than the signal. be found in Appendix A.) The second unit bid is, by its nature, lower, but it continues to be high for many subjects. When doing the econometric tests, one interesting result was that the region did not matter; only the signal. That is, first it did; the profit was significantly For the dynamic auction, the second to last dropout is when the aggregated lower (higher) in region 1 (3). But when controlling for the signal, the region demand in the auction shifts from six to five units. The last dropout is when became insignificant. Thus, we are using the whole set in the below analysis. the aggregated demand shifts to four units; that is, where the auction ends and, therefore, also the same as the clearing price in the auction. Thus, the To get a first impression of the data, we plot the bids/dropouts in a scatter dropouts are auction-specific, unlike the static auction, where the bids are diagram. Figure 1 shows the high bid and the second highest bid for the static subject-specific. uniform auction, while figure 2 shows the last and second-to-last dropout prices in the dynamic uniform auction. In the graph for the dynamic format, The first impression in the dynamic auction is that the clearing prices do not if a subject has not dropped-out on a unit, the price is registered. seem to be as high as in the static auction; a larger number of the last dropouts are below the signal, although there are quite a few above. Since 95 percent of all units won, and 80 percent of all clearing prices, in the static auction come from first and second unit bids, the figure comprises The auctions were not fully effective in that, in theory, the high signal holder(s)

12 13 ESSAY III 80 80 70 70 60 60 13 50 50 Signals Signals First unit bid Second unit bid 40 40 30 30 Fig. 1. First and second unit bids in the static uniform auctions. 20 20 0 0

20 80 60 40 40 20 80 60

100 100

Bids Bids almost all salesoverestimated and because of prices the (even missingis 5 though nonetheless (20) the that percent). But there salesunit, the is with (prices) first substantial more impression become bidding thanThe above slightly half second the of unit signal all bidmany for first subjects. is, the unit by first bids its being nature, greater lower, than butFor the the it signal. dynamic continues auction, todemand the be in second high the to auction for the last shifts dropout aggregated from is six demand to when shiftsand, five the to therefore, units. aggregated also four The the units; lastdropouts dropout same that are is as is, auction-specific, when the wheresubject-specific. unlike clearing the the price auction in static ends the auction, auction. where Thus,The the the first bids impression in are seem the to dynamic be auction as high isare as that in below the the the clearing static signal, auction; prices a although do larger there not number of are theThe quite last auctions a dropouts were not few fully above. effective in that, in theory, the high signal holder(s) 66). 10). The minimum 626 653 140 149 12 64 65 10), and the maximum earning was SEK 659 ( No. of subjects No. of rounds Unique observations 25) which included a show-up fee of SEK 100 ( Static Dynamic All comparisons below use statistic teststion based periods, on aggregated if data overWhitney) not all rank auc- stated sum test differently. has Thebut been non-parametric also the within main Wilcoxon(-Mann- treatments tool, when especially thereFor between some is treatments comparisons, no within dependency a between treatment theparametric where variables. Wilcoxon there is signed dependency, rank theand non- test panel is data employed. We (randomas have the effects) also dependent models tested variables. with OLS demand Profit the and is profit round, explained andand while by round. revenue revenue signal, (price) There is format, wasand, group-size, explained only thus, by a the value, conclusions marginal still change format,be hold. in found group-size (The the in OLS results Appendix regression presented on A.) bidder below profit can When doing the econometricdid tests, not one matter; only interesting the resultlower signal. was (higher) That that in is, the region first itbecame region 1 did; insignificant. (3). the Thus, But profit we when was are significantly controlling using for the the wholeTo signal, get set the in a region the firstdiagram. below impression Figure analysis. 1 of shows the theuniform data, high bid we auction, and plot while the theprices second figure bids/dropouts highest in bid the in 2 for dynamic a theif shows uniform static scatter a auction. the subject In last has the not and graph dropped-out for second-to-last on theSince a dropout dynamic 95 unit, format, the percent pricethe of is static registered. all auction units come from won, first and and 80 second unit percent bids, of the all figure clearing comprises prices, in SEK as the monetary balanceSEK on 253 his/her ( screen. Subjects earned,earning in was the SEK mean, 100 ( 5 Experimental results The data description isnumber found of subjects, in how Tableobservations, 2, many and rounds which the there shows, average were, profit. for and each the format, number the of unique Table 2 Data summary 5.1 Revenue and profit ranking Last dropout 100 The first hypothesis examines the (seller) revenue and the (buyer) profit col- 80 lected in each auction. Will the auction types give the same revenue in the

60 mean? Or, correspondingly, will they produce an equal proﬁt on average? We

Bids start with the profit. 40 Profit: 20 First, we look at winning bids only. Then, we have that, on average, the 0 20 30 40 50 60 70 80 mean of the signal minus the value is almost the same; it only differs at the Signals second decimal, it is 1.22 in the static and 1.29 in the the dynamic auction. Accordingly, in the mean, it was subjects with signals 1.22 (1.29) over the Second−to−last dropout 100 realized value who won the units.

80 But, even so, each column of table 3 shows the diﬀerence between the respec- tive, (realized) value and price, signal and price, and the ratio of the two. 60 Here, it is readily seen that the dynamical auction is superior in raising proﬁt

Bids (when we are looking at winning bids only). On average, when looking at the 40 s p v p s p − 20 v p − − − Static auction 1.22 1.99 1.63 0 20 30 40 50 60 70 80 Signals Dynamic auction 4.54 4.12 0.91 Table 3 Mean profit, pseudo profit and the ratio of the two. Fig. 2. Last and second-to-last dropout prices in the dynamic uniform auctions. values minus the prices (first column), it gives almost four times (3.72) as much profit as compared to the static auction. But the pseudo profit, which we define as the signal minus the price, was only 2 (2.07) times as large in the dynamic auction. (The p-values for the first two measures are below 0.01 between auctions.) should always win units. If the signal vs. units won relationship is more Thus, even though the value is on average 1.29 lower than the signal in the closely examined, the result becomes the following: Given all subjects with dynamic auction, the average profit is higher than the pseudo profit. Whereas, the (weakly) highest signal within each auction round, 86 percent in the static in the static auction, where the value is on average 1.22 lower than the signal, auction won some units, as compared to only 77 percent in the dynamic auc- the average profit is lower than the pseudo profit. The two formats go separate tion. Instead looking at the (weakly) lowest signal within each auction round, ways in this respect which, in turn, makes the dynamic auction perform better 28 percent in the static auction now won units, as compared to 48 percent in for bidders. The last column shows the ratio between the actual and the pseudo the dynamic auction. Hence, close to twice as many subjects with the weakly profit, where it is seen that the static auction has nearly twice (1.79) as high lowest signal won units in the dynamic auction as compared to the static auc- a ratio as compared to the dynamic auction. tion. Naturally, this affects the profit in the auction. But, it need not be exotic since it is often quite natural for the low signal holder to win units, e.g. when, Now we turn to all bids, not just winning ones. Each auction format comprises in 3-player groups, a large demander has the weakly lowest signal. (Then, if both 3-player and 6-player groups. And inside each group size, 2/3 of the the other two small demanders are engaged in demand reduction, the larger subjects demanded 2 units (small demanders) and 1/3 demanded 4-units (large demander has a big chance of winning; even though she has the lowest signal.) demanders). In table 4, we see a highly significant ranking between the group

14 15 ESSAY III 01 72) as . /3 of the 79) as high . 29) over the 22 (1. . 07) times as large in p 63 91 −p . . s− v 22 lower than the signal, . /3 demanded 4-units (large p 99 1 12 0 . . − s 29 in the the dynamic auction. 29 lower than the signal in the . p 22 1 54 4 − 1. 4. v 15 22 in the static and 1. Dynamic auction Static auction demanders). In table 4, we see a highly significant ranking between the group a ratio as compared to the dynamicNow auction. we turn to allboth bids, not 3-player just winning and ones.subjects 6-player Each demanded auction 2 groups. format units comprises (small And demanders) and inside 1 each group size, 2 values minus the prices (first column), it gives almost four times (3 Table 3 Mean profit, pseudo profit and the ratio of the two. 5.1 Revenue and profit ranking The first hypothesis examineslected the in (seller) each revenue and auction.mean? the Or, Will (buyer) correspondingly, the will profit auction theystart col- produce with types an the give equal profit. the profit same on average? revenue We Profit: in the First, we lookmean at of winning the signalsecond bids minus decimal, only. the it Then, value is is we 1. almost have the that, same;realized it on value who only average, won differs the the at units. the But, even so, each columntive, of (realized) table value 3 showsHere, and the it price, difference is between readily signal the seen(when and respec- that we price, are the dynamical looking and auction at the is winning superior bids ratio in only). of raising On profit the average, when two. looking at the much profit as comparedwe to define the as static thethe auction. dynamic signal But auction. minus the (The the pseudobetween p-values auctions.) profit, price, for which the was first only two 2 measures (2. areThus, below even 0. though the value is on average 1 dynamic auction, the average profitin is the higher static than the auction, pseudo where profit. the Whereas, value is on average 1 the average profit is lower thanways in the this pseudo respect profit. which, Thefor in two bidders. formats turn, The go makes last separate the column dynamic showsprofit, auction the where ratio perform between it better the is actual and seen the that pseudo the static auction has nearly twice (1 Accordingly, in the mean, it was subjects with signals 1 80 80 70 70 60 60 14 50 50 Signals Signals Last dropout Second−to−last dropout 40 40 30 30 20 20 0 0

80 60 40 20 80 60 40 20

100 100

Bids Bids Fig. 2. Last and second-to-last dropout prices in the dynamic uniform auctions. should always winclosely units. examined, If thethe result the (weakly) becomes highest signal signal the within vs.auction following: each won auction units Given some round, all 86 units, won percenttion. as subjects in relationship Instead compared the with looking to is static at only28 the more 77 (weakly) percent percent lowest in in signal the the withinthe static dynamic each dynamic auction auction auc- auction. now round, Hence, wonlowest close units, signal to as won twice compared units as in totion. many the 48 Naturally, subjects dynamic this percent affects auction with in the as thesince profit compared weakly it in to the is the auction. often static But, quitein auc- it natural 3-player need for groups, not the be a lowthe exotic signal large other holder demander two to has win smalldemander the units, demanders has e.g. weakly a are when, big lowest engaged chance signal. in of (Then, winning; demand even if reduction, though she the has larger the lowest signal.) All players 3-player groups 6-player groups But, players with larger demand should, especially in 3-player groups, win more units due to their greater demand. Table 6 shows the number of units Static auction 0.67 (8.53) 5.55 (10.80) 1.74 (5.82) − won, divided into group and demander size for the two auction formats pooled. Dynamic auction 2.85 (10.71) 8.26 (14.53) 0.28 (5.71) (There is no significant difference in either group size or demand size between − Table 4 the two formats when comparing units won.) This is only confirmed in 3- Mean bidder profit in each group size. (Standard deviations inside the brackets) 3-player groups 6-player groups sizes in each auction format. Regarding the ranking between auction formats, we lose some predicting power when we split them up because of the poor small large small large significance between the auctions in 3-player groups (p-value = 0.1354), but in 6-player groups the p-value is 0.0026 and even lower for all players. Continuing Static & Dynamic 1.07 1.80 0.63 0.77 Table 6 3-player groups 6-player groups Number of units won, pooled auctions. small large small large player groups, the large demander won 1.8 units in the mean, while the small demander won just over 1 unit. Static auction 5.05 (8.29) 6.51 (14.50) -0.95 (4.59) -3.25 (7.46) Dynamic auction 7.72 (12.05) 9.34 (18.62) -0.06 (5.30) -0.75 (6.50) Hence, large demanders in 3-player groups win 1.68 as many units as small demanders, but they do not earn more profit. If, then, profit per unit won is Table 5 examined on the pooled set of auctions, table 7 is finally obtained, where the Mean bidder profit for small and large demanders, in each group size. to table 5, there is no significant difference in the inter-auction comparison for 3-player groups 6-player groups large demanders in 3-groups; otherwise, the significance is at least on the 10% small large small large level (ranging from 1 to 10) between auctions. From the standard deviations in the two tables, we have a possible explantation for the poor significance Static & Dynamic 6.15 4.49 -0.76 -2.52 between the groups in question; it is quite high in those. Table 7 Thus, overall, we see that the dynamic auction was better at delivering profit Mean profit per unit won, pooled auctions. to the subjects. When the auctions were split into the two group sizes, the difference between mean profit per unit won is statistically significant for both ranking between formats became insignificant in 3-player groups, although the group sizes at the 1 percent level. ranking of group sizes inside each format was significantly distinct. But, when the groups were divided into even finer parts, large and small demanders, the Thus, in both group sizes, when the auction formats are pooled, the profit per ranking between the auctions was partly recovered. But, we must not forget unit won is significantly lower for large demanders, although it was only in 3- that there was some doubt about the effectiveness of the open auction since player groups that large demanders won significantly more units per auction. quite a large fraction of low signal holders won units, but it could also have As for the total profit per subject, it is only significantly lower in 6-player a natural explanation (presented in the above subsection). (Effectiveness is groups; but it holds for both auction formats. The rationale for this would not to be confused with efficiency, since all allocations are efficient in a CV be aggressiveness; large demanders act as (are) big participants and become auction.) price drivers. They outbid small demanders and thus, earn less profit per unit won, but, at least in 3-player groups, they win more units. (More on this in An interesting property from table 5 is that large demanders in 3-player groups subsection 5.2.) seem to get more profit than do small demanders, but this does not carry over to 6-player groups; in these groups, the large demanders get less profit. Looking The fact that only large demanders in 3-player groups won significantly more more closely at that phenomenon, only data from 6-player groups confirms units is probably explained by the fact that they were the sole large demanders that large demanders get less profit than small demanders (p = 0.0320). In in the auction, and thereby represented half of the aggregated demand. In 6- 3-player groups, even if the data is pooled, there is no statistical difference player groups, there were two large demanders, who together represented half between small and large demanders. of the demand, but alone only 1/4 of the aggregated demand. Consequently,

16 17 ESSAY III 68 as many units as small . 8 units in the mean, while the small . 17 /4 of the aggregated demand. Consequently, small large small large 3-player groups 6-player groups 3-player groups 6-player groups small large small large Static & Dynamic 6.15 4.49 -0.76 -2.52 Static & Dynamic 1.07 1.80 0.63 0.77 demanders, but they doexamined not on earn the more pooled profit. set If, of then, auctions, profit table per 7 is unit finally won obtained, is where the demander won just over 1 unit. Hence, large demanders in 3-player groups win 1 Table 7 Mean profit per unit won, pooled auctions. difference between mean profit pergroup unit sizes won is at statistically the significant 1 for percent both level. Thus, in both group sizes,unit when won the is auction significantly formats lowerplayer are for groups pooled, large that the demanders, large profit althoughAs demanders per it won for was significantly only the more ingroups; units total 3- per but profit auction. it perbe holds subject, aggressiveness; for it large both is demandersprice auction act drivers. only formats. They as significantly outbid The (are) lower smallwon, big rationale demanders in but, participants and for at 6-player thus, and this earn leastsubsection become less would in 5.2.) profit 3-player per groups, unit they win moreThe units. fact (More that on onlyunits large this is demanders in probably in explained by 3-player thein groups fact the won that significantly auction, they more were and theplayer thereby sole groups, represented large there demanders half were of twoof large the the demanders, aggregated demand, who demand. but together In represented alone 6- half only 1 player groups, the large demander won 1 Table 6 Number of units won, pooled auctions. But, players withmore larger units demand due should, towon, especially divided their into greater in group and demand. 3-player demander(There Table size groups, is for 6 no the win shows significant two auction difference thethe formats in number pooled. two either of group formats units size when or demand comparing size units between won.) This is only confirmed in 3- 0320). In . 82) . 1354), but in . 28 (5.71) 74 (5 −0. −1. 6-player groups 80) . 26 (14.53) 55 (10 16 71) 8. 53) 5. . . 0026 and even lower for all players. Continuing . 67 (8 3-player groups 85 (10 0. All players 3-player groups 6-player groups 2. small large small large Dynamic auction Static auction Static auctionDynamic auction 5.05 7.72 (8.29) (12.05) 6.51 9.34 (14.50) (18.62) -0.95 -0.06 (4.59) (5.30) -3.25 -0.75 (7.46) (6.50) Table 4 Mean bidder profit in each group size. (Standard deviations inside the brackets) to table 5, there islarge no demanders significant in difference 3-groups; in otherwise,level the the (ranging inter-auction significance comparison from is for 1 atin least to the on 10) the two between 10% auctions. tables,between the From we the groups have standard in a question; deviations possible it is explantationThus, quite for overall, high we the in see poor those. thatto significance the the dynamic subjects. auction wasranking When better between the formats at became auctions delivering insignificant profit ranking in were of 3-player split groups, group although into sizes the insidethe the each groups two format were was group divided significantly sizes,ranking into distinct. even But, between the finer when the parts, auctions largethat was and there partly small was recovered. demanders, somequite But, the doubt a we about must large the not fractiona effectiveness forget of natural of low the explanation signal opennot (presented holders auction to in won since be units, the confusedauction.) but above with it subsection). could efficiency, (Effectiveness since also is all have allocationsAn are interesting efficient property from in table 5 aseem is to CV that get large more demanders in profitto 3-player than 6-player groups groups; do in small these demanders, groups,more the but large this closely demanders does get at not less profit.that carry that Looking over large phenomenon, demanders only get3-player data groups, less from profit even 6-player if thanbetween groups small small the confirms and demanders data large (p is demanders. = pooled, 0 there is no statistical difference sizes in each auction format.we Regarding lose the some ranking betweensignificance predicting auction between power the formats, auctions when in6-player we 3-player groups groups split the (p-value p-value = them is 0 0 up because of the poor Table 5 Mean bidder profit for small and large demanders, in each group size. since the market is much tighter in the larger group, the large demanders do 3-player groups 6-player groups Pooled groups not have the same price influence as they had in the smaller group sizes. Static auction 0.23 0.71 0.47 Ergo, the dynamic auction is the choice for the players. It is naturally better Dynamic auction 0.19 0.45 0.31 to be in a small group than in a large one, due to the tighter cap to which Table 9 the bigger groups give rise. Furthermore, group size has more bearing than Fraction of auctions with negative profits. auction format; the 3-player groups in the static format give a significantly greater profit than 6-player groups in the dynamic auction. When it comes to the dynamic auction, as suggested by the hypothesis. Moreover, we saw the demand size, there is more ambiguity about the ranking. But solely looking at corollary that the profit was in favor of the dynamic auction. 6-player groups, subjects with small demands earned less negative profit than large demanders. And, overall, small demanders earned more profit per unit won, but, at least in 3-player groups, they won less units. 5.2 Winner’s curse

Revenue: There are two types of overbidding; (i) bids which result in prices above the Revenue is closely (negatively) affiliated to the profit in CV auctions, i.e. how expected value of the objects, that is the signal (E(v)=s), and (ii) bids much money each auction delivers to the auctioneer. The revenue is defined as that result in prices above the conditional expected value (E(v si >s i)= n 1 | − s 10 − ), but below or equal to the expected value (or signal), that is, the how much money each round delivers, i.e. the price times four (units). Thus, i − n+1 we now measure between auction rounds, not between subjects. winner’s curse (WC) interval. In the experiment, even though the equilibrium bids are unknown in the auctions, there is always a potential risk that a When using this definition, we do not see any significant differences; neither player’s bid becomes the price-setting bid. Thus, bidding above the conditional between formats, nor between group sizes. But when controlling for value, by expected value could be costly. dividing all prices by the value of the unit, the p-values goes down. Table 8 shows that, overall, the static auction hands over more revenue than does the Starting with the static auction format, 74 percent of the bids where subjects dynamic auction (p-value = 0.0169). The ranking also seems to extend down won one or more units were above E(v si >si). 44 of these were above | − to group sizes, as can be seen in the table, but it is only in 6-player groups E(v); hence, taking the difference between the two intervals, we have that 30 that the mean values differ significantly (p-value = 0.0090) from each other. percent of the winning bids were in the WC interval. The outcomes in the two group sizes were almost identical in bidding above the signal, category (i), but the outcomes for bids in the WC interval were significantly different. Both groups 3-player groups 6-player groups The outcome was 38 percent for 6-player groups as compared to only 25 for Static auction 1.05 0.92 1.18 the smaller group size, which is seen in table 10. Dynamic auction 0.93 0.86 1.02 There is much less overbidding in the dynamic form; as has been noted above as Table 8 fewer auctions with negative profits. 31 percent of all bids are above E(v s > Mean revenue, divided with value. | i s i) as compared to 74 in the static form. Bidding above E(v)=s is also − But, on the horizontal level, i. e. intra comparison, there is no doubt about the much lower, 21 percent as compared to 44 above. Thus, the percentage of bids natural hypothesis that the more bidders, the more revenue (p-values below in the WC interval is just 10 percent compared to 30 above for the static. 0.0001). Or the tighter the market, the larger (smaller) the revenue (profit). Moving to group sizes, we note that the proportion between the two group sizes is almost identical to the static auction; but the level in the dynamic For the revenue ranking hypothesis, the competitive effect had the largest auction is just one third of the level of the static auction (table 10). bearing in the experiment. Participants engaged in overbidding, generally in 6-player groups which, especially in the static auction, led to negative profits Bidding above E(v)=s results in a negative profit in the mean, while bidding in the greater part of all auctions. Table 9 shows the percentage of auctions above E(v si >s i) could, upon winning, give rise to a price that is greater | − with negative profits. than the estimated worth of the objects and, possibly, create negative profit. It is in this interval that the WC reigns. Looking more closely at winning bids Result 1 The static auction surrenders more revenue to the auctioneer than in the WC interval, we get that in the static auction, actually 18 percent fall

18 19 ESSAY III > i )= s i | − v is also ( E >s i s | 47 31 )=s v v 0. 0. ( ( E s), and (ii) bids E )= v ( E ). 44 of these were above i − 71 45 0. 0. >s i s | v ( E 19 23 19 0. 0. 3-player groups 6-player groups Pooled groups results in a negative profit in the mean, while bidding s ) could, upon winning, give rise to a price that is greater )= v −i ( E >s i s | ), but below or equal to the expected value (or signal), that is, the 1 v ( − +1 n n E Static auction Dynamic auction 10 ); hence, taking the difference between the two intervals, we have that 30 ) as compared to 74 in the static form. Bidding above v i − ( − i winner’s curse (WC) interval. Inbids the experiment, are even unknown thoughplayer’s the bid in equilibrium becomes the the price-setting bid. auctions,expected Thus, value bidding there could above the be is conditional costly. always aStarting potential with risk the static that auctionwon format, a one 74 or percent of more the bids units where were subjects above s the dynamic auction,corollary as that suggested the by profit was the in hypothesis. favor Moreover, of we the dynamic saw auction. the 5.2 Winner’s curse There are two typesexpected of value overbidding; (i) of bidsthat the which result objects, result in in that prices prices is above above the the the conditional signal expected ( value ( Table 9 Fraction of auctions with negative profits. much lower, 21 percent asin compared to the 44 WC above. Thus,Moving interval the to is percentage of group just bids sizes,sizes 10 we is percent note almost compared thatauction identical to is the to 30 just proportion the above one between static third for the of auction; the the two but static. Bidding level group above the of the level static inabove auction the (table dynamic 10). s than the estimated worthIt of is the in objects this interval and,in that possibly, the the create WC WC negative interval, reigns. we profit. Looking get more that closely at in winning the bids static auction, actually 18 percent fall E percent of thetwo winning group bids sizes were were(i), in but almost the the identical WC outcomes inThe interval. for outcome bidding The bids was above in outcomes 38 the thethe in percent WC smaller signal, the interval for group category were 6-player size, significantly groups which different. is as seen comparedThere in to is table much only less 10. overbidding 25 in thefewer for dynamic auctions form; with as has negative been profits. noted 31 above as percent of all bids are above 18 02 1. 1. 0090) from each other. 92 86 0. 0. 18 05 93 0169). The ranking also seems to extend down 1. 0. Both groups 3-player groups 6-player groups The static auction surrenders more revenue to the auctioneer than Static auction Dynamic auction 0001). Or the tighter the market, the larger (smaller) the revenue (profit). But, on the horizontal level,natural i. e. hypothesis intra that comparison, there the0. is more no doubt bidders, about the the more revenue (p-valuesFor below the revenuebearing ranking in hypothesis, the the experiment.6-player competitive groups Participants engaged which, effect especially in hadin in overbidding, the the the generally static greater in largest auction, partwith led of negative to profits. all negative auctions. profits Table 9 showsResult the 1 percentage of auctions since the market isnot much have tighter the in same the price larger influence group, as the theyErgo, large had the demanders in dynamic do the auctionto smaller is group be the sizes. choice in for athe the small bigger players. group groups It is thanauction give naturally in rise. format; better a Furthermore, the large groupgreater 3-player profit one, size groups than due has in 6-player to more groups thedemand in the size, bearing static there the tighter format than is dynamic cap more auction. give6-player ambiguity to groups, When a about subjects it which the significantly comes with ranking. to smalllarge But demands solely demanders. looking earned And, at less overall, negativewon, small profit but, than demanders at earned least in more 3-player profit groups, per theyRevenue: unit won less units. Revenue is closely (negatively) affiliatedmuch to money each the auction profit delivers inhow to CV the much auctions, auctioneer. money i.e. The each how revenuewe round is now delivers, defined measure as i.e. between the auction price rounds, times not four betweenWhen (units). subjects. using Thus, this definition,between we formats, do nor between not groupdividing see sizes. all any But prices significant when differences; byshows controlling neither the that, for overall, value, value the by of staticdynamic the auction auction unit, hands (p-value over the = moreto 0. p-values revenue goes group than does sizes, down. the Table asthat 8 can the mean be values seen differ in significantly (p-value the = table, 0. but it is only in 6-player groups Table 8 Mean revenue, divided with value. prey to the WC; 23 percent in 6-player groups and just 11 percent in 3-player The result shows that the dynamical auction shifts all bids downward toward groups. As for the dynamic auction, only 6 percent are accounted for in the WC more rational bidding even though the bids do not do fully converge to rational and 8 (4) percent in 6-player (3-player) groups. Table 10 summarizes bids in bidding. The subjects seem to better understand the laws of demand and this interval, where WCI is an abbreviation for the winner’s curse interval and supply in the open auction, and also seem to better grasp the idea of a pure common value. 3-player groups 6-player groups Both groups

Bids WCI WC Bids WCI WC Bids WCI WC ∈ ∈ ∈ Static 0.25 0.11 0.38 0.23 0.30 0.18 5.3 Equilibrium bidding Dynamic 0.08 0.04 0.12 0.08 0.10 0.06 Table 10 Frequency of (winning) bids in the WC interval, and actual WC. Even though the equilibrium strategies are unknown in this game, we know that all weakly undominated equilibria have players who bid the conditional WC for the actual winner’s curse; that is, the negative profit following from expected value on the first unit. And if we allow bids/drop-outs 1 of E(v si > ± | bids in WCI. The dynamic auction has approximately one third of the entires s i) to also count as correct bids, we have that only six percent are using this − of the static auction; hence, the format produces a much smaller number of strategy in the static auction (fifteen percent meet the naive expected value bids in the WCI as well as actual WC. In both auctions, the larger group sizes to bid 1 of s); in the dynamic form, the corresponding numbers are sixteen ± produce 3/2 more bids in the WCI, but approximately 2 times as many WC percent (twenty percent meet the naive EV). Thus, there is a relatively larger cases. This is quite natural since the 3/2 more bids in 6-player groups are amount of equilibrium play in the dynamic auction as compared to the static numerically larger than 3/2 more bids in 3-player groups. counterpart.

If the group sizes were replaced by small and large demanders in the above Result 3 Compared to the static auction, the dynamic auction had more table, all entries would be almost exactly the same, barely diﬀering in two dropouts coinciding with individual rationality, i.e bidding below the expected slots. This tells us that we have the same results, according to WC, as in value; much more for bidding below the conditional expected value. table 10 between small and large demanders if groups are pooled. Hence, it is through the large demanders, or 6-player groups, that 2/3 of all WC is encountered.

The dynamic auction behaved as hypothesized, it mitigated much of the WC 5.4 Bidding behavior in 3 vs. 6-player groups encountered by subjects in the static auction. It also lessened the pervasive bidding above both expected values. Hence, for bidders, it is the auction of choice, at least inside this model. Thus, the experimental literature has more The hypothesis of increasing the number of bidders instead of halving the bearing on behavioral prediction than the theoretical literature. The latter supply to construct a tighter market seemed to be correct, considering the accounts for those players that scale down bids and do not bid above the con- bidding behavior. Subjects’ bids did not decrease in response to the increased ditional expected value, but it does not happen in this experiment, especially number of bidders, contrary to the Nash equilibrium theory (for single unit not in the static auction, with frequent overly aggressive bidding. Theory does demand). The null hypotheses that the bids are independent samples from the not account for bidders who overbid as severely as do subjects in the static same distributions cannot be rejected between 3 and 6-player groups in any of auction; it assume equilibrium play, or play that gives a weakly positive proﬁt the auctions (p-values: 0.5097 and 0.3077.) Moreover, in the revenue-section in the mean. above, we saw that the more bidders, the higher the revenue. Result 2 There was three times as much bidding in the winner’s curse interval, as well as the experienced winner’s curse, in the static auction estimated Result 4 Increasing the number of bidders instead of halving the supply, to relative to the dynamic auction. The static format also had more than two create tighter markets, cannot be rejected as false. Moreover, the tighter the times as much bidding above the standard, naive, expected value. market, the larger the revenue.

20 21 ESSAY III > i s | v ( E 1 of ± 3077.) Moreover, in the revenue-section 21 -player groups 6 vs. 3 5097 and 0. ); in the dynamic form, the corresponding numbers are sixteen s Increasing the number of bidders instead of halving the supply, to Compared to the static auction, the dynamic auction had more 1 of ± ) to also count as correct bids, we have that only six percent are using this i − The hypothesis ofsupply increasing to the construct numberbidding a of behavior. tighter Subjects’ bidders market bids insteadnumber did seemed not of of to decrease bidders, halving in be contrarydemand). response the correct, The to to null considering the the hypotheses increased that the Nashsame the equilibrium distributions bids cannot are theory be independent (for rejected samplesthe from between single auctions the 3 unit and (p-values: 6-player 0. above, groups we in saw any of that the more bidders, the higherResult the 4 revenue. create tighter markets, cannotmarket, be the rejected larger as the revenue. false. Moreover, the tighter the 5.4 Bidding behavior in percent (twenty percent meet theamount naive of EV). equilibrium Thus, play there incounterpart. is the a dynamic relatively auction larger as compared to the static Result 3 dropouts coinciding with individualvalue; rationality, much i.e more bidding for below bidding the below expected the conditional expected value. s 5.3 Equilibrium bidding Even though the equilibriumthat strategies all are weakly undominated unknownexpected equilibria in value on have this the players first game, who unit. we And bid if know the we allow conditional bids/drop-outs The result shows thatmore the rational dynamical bidding auction even though shifts thebidding. all bids bids do The downward not toward do subjects fullysupply converge seem in to rational the to open bettercommon auction, value. understand and also the seem laws to of better demand grasp the and idea of a pure strategy in the staticto auction bid (fifteen percent meet the naive expected value WCI WC /3 of all WC is ∈ WC Bids WCI /2 more bids in 6-player groups are ∈ 20 WC Bids WCI ∈ 3-player groups 6-player groups Both groups Bids There was three times as much bidding in the winner’s curse inter- StaticDynamic 0.25 0.08 0.11 0.04 0.38 0.12 0.23 0.08 0.30 0.10 0.18 0.06 Table 10 Frequency of (winning) bids in the WC interval, and actual WC. prey to the WC; 23groups. As percent for in the 6-player dynamic auction, groupsand only and 8 6 just percent (4) are 11 accounted percent percentthis for in in in interval, the where 3-player 6-player WC WCI (3-player) is groups. an Table abbreviation 10 for summarizes the winner’s bids curse in interval and WC for the actualbids winner’s in WCI. curse; The that dynamicof is, auction the the has static approximately negative one auction; profitbids third in hence, following of the the from the WCI format as entires produce well produces 3/2 as a actual more WC. much bidscases. In smaller in both This number the auctions, is the of WCI, largernumerically but quite group larger approximately sizes natural than 2 3/2 since times more as the bids many 3 in WC 3-playerIf groups. the group sizestable, were all replaced entries byslots. would small be This and almost tells largetable exactly us demanders 10 the that in between same, the we smallis above barely have and through differing the large the in same demandersencountered. large two if results, demanders, groups according are or to pooled. 6-player WC, Hence, groups, as it The that in dynamic 2 auction behavedencountered as by hypothesized, it subjects mitigated inbidding much the of above the static both WC auction. expectedchoice, at It values. least also Hence, inside lessened forbearing this the bidders, model. on pervasive it Thus, behavioral the isaccounts prediction experimental for the literature those than auction has players the of that more ditional scale theoretical expected down value, literature. bids but and Thenot it do in does latter not the not bid static happen above auction,not the in with con- account frequent this overly experiment, for aggressive especially bidding. biddersauction; Theory it who does assume overbid equilibrium as play,in or severely the play as mean. that gives do a subjects weakly positive in profit theResult static 2 val, as well asrelative the to experienced winner’s the curse,times dynamic in as auction. the much The static bidding auction above static the estimated format standard, also naive, expected had value. more than two 5.5 Demand reduction 6 Discussion

The results of the experiment in the present paper both contradict and are in The last hypothesis concerns demand reduction, which, measured here, trans- line with existing theory. The first hypothesis of the revenue ranking contra- lates more into bid/dropout spread; that is, how large is the difference between dicted the existing theoretical literature contention that open formats should the first and the second bid, or, for large demanders, the difference between deliver more revenue, not less. The present experiment also comes up with the first and the mean of the three lower bids. This is a crude measure since the a different outcome than Ausubel et al. (2009) who established similar prices above result gave us that roughly just 1/6 of the first unit bid was equilibrium for the two formats. But, generally, it is in line with the experimental liter- bids, but it gives an indication of demand reduction. ature pointing at overly aggressive bidding in the static auction, manifested in that the better part of the auctions often ends up with negative profits for the subjects. And we have seen in this experiment that the dynamic auction In the comparison between the two auction types, there is a significant dif- cushions much of this bidding above value. Even if it still exists, it is more ference at the 1 percent level in that there is less bid spread in the dynamic than halved as compared to the static auction. auction; the mean of the spread is 6.43 in the dynamic auction, while it is 8.97 in the static auction. Moreover, it is of no importance if the formats are split Regarding the WC, we distinguished between bidding in the actual WC in- into 3- and 6-player groups, or into small and large demanders; the result is terval and just bidding above the conditional expected value. We have not approximately the same bid spread, and it is always significant at, at least, seen this before, since experiments often report the latter interval. In the WC the 1 percent level. interval, subjects experienced three times as much WC, i.e. negative profit, in the static auction as compared to the dynamic format. (Consistently, there were also three times as many bids in the actual interval.) This shows the This substantial difference partly originates from the fact (described above) superiority of the dynamic auction over the static auction in guiding subjects that there is considerably more overbidding in the static auction. But the to what the actual common value is in the auction. relatively lower spread in the dynamic auction is according to the theory of information dispersion by Wilson (1977); players need not take as much The first explanation of the WC should probably be that players in this ex- precaution as in the static format, since information about the common value periment were inexperienced. They came to the experiment without knowing is being updated during the bidding process. what to expect. Nevertheless, all players had three dry runs before the experiment, in addition to ten rounds in the experiment. Therefore, subjects at least gained experience along the way. Another explanation is limited liability, Further elaborating on the bid spread, there is a much greater spread in 6- meaning that subjects did not have to stand their own losses; they had their player groups as compared to 3-player groups. Pooling both formats provides starting balance of 50, and had to leave when they went bankrupt. (It only the mean value of 5.44 in 3-player groups and of 8.75 in the larger group sizes happened 5 times, 3 in the static and 2 in the dynamic auction.) However, (p-value < 0.001). Hence, not only does the bid spread chiefly emanate from Kagel and Levin (1991) and Lind and Plott (1991) provide an experimental the static auction, the larger part comes from the larger group size. But, there verification that limited liability forces did not account for the overly aggres- is no significant difference between small and large demanders. sive bidding reported in, at least, their set-up, which is similar to the set-up in the present paper, but with single-unit demand.

Consequently, subjects behave according to the theory of demand reduction. There was also much less demand reduction, or bid spread, in the dynamic The rationale is that bidders reduce their demand for a more favorable price. auction. This is according to theory, but, at the same time, since there was no considerable equilibrium bidding on the first unit, it is hard to evaluate the demand reduction. Still, the variance in the static auction is twice as high as Result 5 There is a widespread demand reduction in both formats, and as in the dynamic auction, and the standard deviation is also higher in the static predicted by the hypothesis, the spread was somewhat smaller in the dynamic auction. This tells us that subjects behave more uniformly in the dynamic format as compared to the static. Moreover, there was a significant difference form which is, probably, brought forth from the information revelation in the between 3- and 6-player groups. auction.

22 23 ESSAY III 23 6 Discussion The results of the experimentline in with the existing present theory. paperdicted The both the first contradict existing hypothesis and theoretical ofdeliver are literature the in more contention revenue that revenue, ranking open nota contra- formats different less. should outcome The thanfor present Ausubel the et experiment al. two also (2009) formats.ature comes who But, pointing up established generally, at similar with it overly prices in is aggressive that in bidding the line better inthe with part the subjects. of the static the And experimental auction, auctions wecushions liter- manifested often have much ends seen of up in with this thisthan negative experiment halved bidding profits as that above for compared the value. to dynamic Even the auction if static it auction. Regarding still the exists, WC, it weterval is distinguished and more between just biddingseen bidding in this above before, the the since actualinterval, experiments conditional WC often subjects expected in- report experienced the value.in three latter We the interval. times have In static as not the auction WC aswere much compared WC, also to i.e. three the negative dynamicsuperiority times format. of profit, as (Consistently, the there many dynamicto auction bids what over in the the static the actual auction common actual in value interval.) guiding is subjects This inThe the shows first auction. the explanation ofperiment the were inexperienced. WC should Theywhat came probably to to be the expect. that experimentperiment, players Nevertheless, without in in all knowing addition this players to ex- least had ten gained rounds three experience in along dry the themeaning runs experiment. way. Another that Therefore, before explanation subjects subjects is the did limited at starting liability, not ex- balance have of to 50, standhappened their and 5 own had times, losses; toKagel 3 they leave and had in when Levin their the they (1991)verification static went and that and bankrupt. limited Lind 2 (It liability andsive in forces only Plott bidding did the (1991) reported not dynamic provide account in,in auction.) an for at the However, experimental the least, present overly their paper, aggres- set-up, but with which single-unit is demand. similarThere to was the also set-up muchauction. This less is demand according reduction, toconsiderable theory, or equilibrium but, bid at bidding the spread, ondemand same in the reduction. time, since Still, the first there the dynamic unit,in was variance it no the in dynamic is the auction, hard static andauction. auction to the This standard is evaluate deviation twice tells the is asform us also high which higher that as is, in probably, subjects the broughtauction. static behave forth from more the uniformly information in revelation in the the dynamic 97 . 75 in the larger group sizes . /6 of the first unit bid was equilibrium 22 43 in the dynamic auction, while it is 8 . 44 in 3-player groups and of 8 -player groups. 6 001). Hence, not only does the bid spread chiefly emanate from 0. There is a widespread demand reduction in both formats, and as < 3- and in the static auction.into Moreover, 3- it is and of 6-playerapproximately no groups, the importance or same if into the bidthe small formats 1 spread, and are percent and large split level. it demanders; the is result always significant is at, at least, This substantial difference partlythat originates there from is therelatively considerably fact lower more (described spread above) overbiddingof in in information the the dispersion dynamic staticprecaution by as auction auction. Wilson in But the (1977); is staticis players the according format, being need since updated to information not during about the take the the theory bidding as common process. value much Further elaborating on theplayer groups bid as spread, compared therethe to is mean 3-player value groups. a of Pooling much 5. (p-value both greater formats spread provides inthe 6- static auction, the largeris part no comes significant from difference the larger between group small size. and But, large there demanders. Consequently, subjects behave accordingThe to rationale the is theory that of bidders demand reduce reduction. their demand for a more favorableResult price. 5 predicted by the hypothesis,format the as spread was compared to somewhat thebetween smaller static. in Moreover, there the was dynamic a significant difference 5.5 Demand reduction The last hypothesis concerns demandlates reduction, more into which, bid/dropout measured spread; here,the that trans- is, first how large and is the thethe difference first second between and bid, the mean or, ofabove for the result three gave large lower us bids. demanders, that Thisbids, the roughly is but just a difference crude 1 it between measure gives since an the indication of demand reduction. In the comparison betweenference the at the two 1 auctionauction; percent the types, mean level there of in is the that spread a is there significant 6 is dif- less bid spread in the dynamic In IPV settings, some lab and field experiments have showed the superiority 7 Conclusions of the static uniform auction over the dynamic form, and have also underlined the caution that is warranted in using open formats in multi-unit settings. This does not need to carry over to common value settings. While the static form In deciding which of the two auction formats of the uniform price auction that delivers more revenue than the dynamic form also in the present experiment, were used in our controlled laboratory experiment that is preferred, one has it comes at a pretty high cost for the subjects. to decide if (i) collecting the most revenue or (ii) avoiding the most negative bidder profit is the most important criterion in the choice process. The other side of the coin is that, in CV settings, the static auction seems to If revenue is the most important selection criterion, the static format is the bring forth an overbidding which is moderated in the dynamic auction. CV best choice. It generally collects a significantly greater revenue, particularly in auctions are known to produce allocations with negative profits and, in these, a bigger group size. As a result, the profit is greater in the dynamic auction, the dynamic form could be an excellent guide to price discovery. Why there which holds true (weakly) even if the two formats are split into finer parts; is this overbidding, especially in the static auction, is hard to tell; it seems first into large and small group sizes, and then, even finer, into large and small as if there is some myopic joy of winning, see Holt and Sherman (1994). The demanders. We also got the corollary that the tighter the market (or the more competitive effect takes over the rationality. Consequently, the dynamic form bidders), the greater the revenue. has nice properties for a common value auction, especially for inexperienced bidders. On the other hand, if avoiding negative profit is more interesting as the selection criterion, the dynamic auction is better. It only has 1/3 of the actual WC In both Kagel (1995) and Ausubel et al. (2009), we find strong advocates for of the static auction, and less than half of the bidding of the static auction the dynamical auction over the static one. The first is a survey of (partly) above the conditional expected value. Moreover, almost half of all auctions in single-unit, common value experiments establishing that the dynamic form the static form terminated with a negative profit for the subjects, as compared helps alleviate the overbidding in the static format. This also holds for expe- to 3/10 in the dynamic form. Moreover, not only were there more auctions rienced bidders. One of the problems, they argue, is that an increased number with a negative profit in the static form, the mean of the negative profit was 8 of bidders produces no change in bidding in second-price auctions ; which also greater in it. it should, according to the robust Nash equilibrium prediction. Subjects encountered the same problem in the present experiment, especially in the static The weak equilibrium strategy to bid (either one of the two) EVs on the first auction; they did not seem to understand that the more bidders in the auction, unit, was also better in the dynamic auction. But both only had a few bids the bigger the chance of being the price setter and/or bidding above value. on the (extended) target. No auction format (at either of the two targets) had better than 1/5 of the bid in the zone. In the paper by Ausubel et al. (2009), the experiment, which is similar to ours, As for the prevalence of demand reduction, we measured the bid spread and produces equal prices in the two formats, contrary to the present experiment, found that the subjects of both formats employed such strategies, but the but, at the same time, they find that the open format is less prone to bidding spread was larger in the static auction. Both the variance and the standard error and to deliver much higher payoffs. The rationale is that the open envi- deviation were significantly larger there. But since subjects in neither auction ronment helps subjects understand complicated settings and thereby reducing utilized the weak (individual rational) first unit bid strategy, and we know errors in finding, if not the optimum, a better outcome than the static auction that there was considerable overbidding in the static auction relative to the can contribute to. dynamic auction, it is hard to draw any conclusions.

Auction format is just one feature that determines the outcome of an auction, and our results also illuminate the importance of being in a smaller group size. No matter the format, being in a small group size counts more in terms of bidder profits. But, given the group size, there were different findings for 8 The static uniform price auction is the extension of the second-price auction small and large demanders; in 3-player groups, the large demanders earned for multi-unit auctions, in the same way as the dynamical uniform auction is the more profit than small demanders, whereas it was the other way around in extension of the English auction. 6-player groups.

24 25 ESSAY III /3 of the actual WC 25 7 Conclusions In deciding which of thewere two auction used formats in of our theto uniform controlled decide price laboratory if auction experiment that (i)bidder that collecting profit is the is most preferred, the revenue one most or has important (ii) criterion avoiding inIf the the most revenue choice is negative process. thebest most choice. It important generally selection collectsa a criterion, bigger significantly the group greater static revenue, size.which particularly As format in holds a is true result, the (weakly) thefirst into even profit large if is and greater small thedemanders. group in two We sizes, also the formats and got dynamic then, are the auction, evenbidders), corollary split finer, the that into into greater the large finer tighter the and the small parts; revenue. market (or the more On the other hand, iftion criterion, avoiding negative the profit dynamic is auction is more better. interesting It as only the has selec- 1 of the static auction,above the and conditional less expected thanthe value. static Moreover, half form almost of terminated half with the ofto a all bidding negative 3/10 auctions profit of in for in the thewith the subjects, static a dynamic as auction negative compared form. profitalso Moreover, in greater not the in only static it. form, were the there mean more of auctions The the weak negative equilibrium profit strategy was unit, to bid was (either also one betteron of the in the (extended) the two) target. EVs dynamic Nobetter on auction auction. than the format But 1/5 first (at both of either the of only the bid had two in targets) a the had As few zone. for bids the prevalencefound of that demand reduction, the wespread subjects measured was of the larger bid both indeviation spread the were formats and significantly static employed larger auction. there. suchutilized But Both strategies, the since the subjects but weak variance inthat (individual the and neither there rational) the auction was first standard considerabledynamic unit overbidding auction, bid in it strategy, is the and hard static to we auction draw know relative anyAuction to conclusions. format the is just oneand feature our that determines results thesize. also outcome No of illuminate matter an the auction, theof format, importance bidder being of profits. in being But, asmall small in given and group the a large size group smallermore demanders; counts size, more profit group in there in than 3-player were terms groups,6-player small different groups. demanders, the findings whereas large for it demanders was earned the other way around in ; which 8 24 The static uniform price auction is the extension of the second-price auction it should, according tocountered the the same robust problem Nash inauction; the equilibrium they present did prediction. experiment, not especially Subjects seem inthe to en- the understand bigger that static the the chance more bidders of in being the the auction, price setter and/orIn the bidding paper above by value. Ausubelproduces et equal al. prices (2009), in the experiment,but, the which at two is formats, the similar contrary same to to ours, error time, the they and present find to experiment, that deliverronment the much helps higher open subjects payoffs. format understand The is complicated rationaleerrors less settings in is prone and finding, that to thereby if the bidding reducing notcan open the contribute envi- optimum, to. a better outcome than the static auction for multi-unit auctions, inextension the of the same English way auction. as the dynamical uniform auction is the 8 In IPV settings, someof lab the static and uniform field auctionthe experiments over caution the have that dynamic is showed warranted form, the indoes and using superiority have open not also formats need underlined in to multi-unitdelivers settings. carry more This over revenue to than commonit the value comes dynamic settings. at form While a also the pretty in high static the cost present form for experiment, the subjects. The other side ofbring the forth coin is an that, overbiddingauctions in are which CV known is settings, to moderated the producethe static allocations in dynamic auction with the form seems negative dynamic could profits to is auction. and, be this in CV an overbidding, these, excellent especiallyas guide if in to there the price is staticcompetitive discovery. some effect Why auction, myopic takes there joy is over of hard thehas winning, rationality. nice to Consequently, see properties the tell; Holt dynamic for it andbidders. form a Sherman seems common (1994). The value auction, especially for inexperienced In both Kagel (1995)the and dynamical Ausubel et auction al.single-unit, over (2009), the common we static value findhelps strong one. experiments alleviate advocates The establishing the for first that overbiddingrienced in is the bidders. the a One dynamic static of survey form format. theof of problems, This bidders they also (partly) argue, produces holds is for no that expe- an change increased in number bidding in second-price auctions The bottom line is that, especially for inexperienced players, and for common Holt, C. A. and R. Sherman: 1994, ‘The Loser’s Curse’. The American Eco- value settings, the dynamic auction seems to be a better format for price nomic Review 84(3), 642–653. discovery, which mitigates the common overbidding that has been produced Kagel, J. H.: 1995, Auction: Survey of experimental research. In Kagel, J.H., in static auction formats. Roth, A.E. (Eds.), The Handbook of Experimental Economics. Princton University Press. There is still lack of knowledge in multi-unit settings in general, and in CV Kagel, J. H., R. M. Harstad, and D. Levin: 1987, ‘Information Impact and settings in particular. The present experiment was carried out with inexpe- Allocation Rules in Auctions with Affiliated Private Values: A Laboratory rienced subjects, even though they gained experience along the way. But it Study’. Econometrica 55(6), 1275–1304. would be interesting to see how experienced bidders performed in a similar Kagel, J. H. and D. Levin: 1991, ‘The Winner’s Curse and Public Information setting. The conclusion from earlier experiments is that they continue to have in Common Value Auctions: Reply’. The American Economic Review 81(1), problems with CV settings. 362–369. Kagel, J. H., D. Levin, and R. M. Harstad: 1995, ‘Comparative static effects of number of bidders and public information on behavior in second-price common value auctions’. 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Mathematics of Operations Re- search 6(1), 58–73. Ahlberg, J.: 2011, ‘Multi-unit common value auctions: A laboratory experi- Riley, J. G. and W. F. Samuelson: 1981, ‘Optimal Auctions’. The American ment with three sealed-bid mechanism’. Working paper. Economic Review 71(3), 381–393. Alsemgeest, P., C. Noussair, and M. Olson: 1998, ‘Experimental comparisons Vickrey, W.: 1961, ‘Counterspeculation, Auctions, and Competitive Sealed of auvctions under single- and multi-unit demand’. Economic Inquiry 36(1), Tenders’. The Journal of Finance 16(1), 8–37. 87–97. Wilson, R.: 1977, ‘A Bidding Model of Perect Competition’. The Review of Ausubel, L. M.: 2004, ‘An Efficient Ascending-Bid Auction for Multiple Ob- Economic Studies 44(3), 511–518. jects’. The American Economic Review 94(5), 1452–1475. Wilson, R.: 1979, ‘Auctions of Shares’. Quarterly Journal of Economics 93, Ausubel, L. M. and P. C. Crampton: 2002, ‘Demand Reduction and Ineffi- 675–689. ciency in Multi-Unit Auctions’. Mimeographed, Department of Economics, Wilson, R. B.: 1969, ‘Competitive Bidding with Disparate Information’. Man- University of Maryland. agement Science 15(7), 446–448. Ausubel, L. M., P. Cramton, E. Filiz-Ozbay, N. Higgins, E. Y. Ozbay, and A. J. Stocking: 2009, ‘Common-Value Auctions with Liquidity Needs: An Experimental Test of a Troubled Assets Reverse Auction’. Papers of peter cramton, University of Maryland, Department of Economics - Peter Cram- ton. Bazerman, M. H. and W. F. Samuelson: 1983, ‘I Won the Auction but Don’t Want the Prize’. Journal of Conflict Resolution 27(4), 618–634. Engelmann, D. and V. Grimm: 2009, ‘Bidding behaviour in multi-unit auctions - An experimental investigation’. Economic Journal 119(537), 855–882.

26 27 ESSAY III , 93 (1), Man- , 293– 81 Princton 24 Journal of The American The Review of The American Eco- Journal of Economic (1),81 335–346. Mathematics of Operations Re- Quarterly Journal of Economics The American Economic Review 27 (1),16 8–37. International Journal of Game Theory (2),61 304–323. (5),50 1089–1122. (1),16 169–189. (6),55 1275–1304. (3), 511–518. 44 The American Economic Review (3),71 381–393. (7),15 446–448. Auction: Survey of experimental research. In Kagel, J.H., (3),84 642–653. Econometrica The Journal of Finance Econometrica 6(1), 58–73. Bidding’. search Economic Studies nomic Review 362–369. of number ofcommon bidders value and auctions’. public information on behavior in second-price Behavior & Organization agement Science A Comparison of Static and Dynamic Mechanisms’. 675–689. Tenders’. Economic Review Roth, A.E. (Eds.), The Handbook of Experimental Economics. in Common Value Auctions: Reply’. University Press. Allocation Rules in AuctionsStudy’. with Affiliated Private Values: A Laboratory 319. 10.1007/BF01243157. Economic Perspectives and with Sellers’. Milgrom, P. and R. Weber: 1982, ‘A Theory of Auctions and Competitive Myerson, R.: 1981, ‘Optimal auction design’. Wilson, R.: 1979, ‘Auctions of Shares’. Riley, J. G. and W. F. Samuelson: 1981, ‘Optimal Auctions’. Wilson, R.: 1977, ‘A Bidding Model of Perect Competition’. Holt, C. A. and R. Sherman: 1994,Kagel, ‘The J. Loser’s H.: Curse’. 1995, Kagel, J. H., D. Levin, and R. M. Harstad: 1995, ‘Comparative static effects Manelli, A. M., M. Sefton, and B. S. Wilner: 2006, ‘Multi-Unit Auctions: Wilson, R. B.: 1969, ‘Competitive Bidding with Disparate Information’. Vickrey, W.: 1961, ‘Counterspeculation, Auctions, and Competitive Sealed Kagel, J. H. and D. Levin: 1991, ‘The Winner’s Curse and Public Information Kagel, J. H., R. M. Harstad, and D. Levin: 1987, ‘Information Impact and Klemperer, P.: 2002, ‘What Really MattersLind, in B. Auction and C. Design’. R. Plott: 1991, ‘The Winner’s Curse: Experiments with Buyers (1), 36 (537), 855–882. 119 (4), 618–634. Economic Inquiry 27 ardh for important help with Working. paper (5),94 1452–1475. 26 Economic Journal Mimeographed, Department of Economics, . Journal of Conflict Resolution The American Economic Review ciency in Multi-Unit Auctions’. University of Maryland A. J. Stocking: 2009,Experimental Test ‘Common-Value of Auctions a withcramton, Troubled Liquidity University Assets of Needs: Reverse Maryland, Auction’. An Departmentton. Papers of of Economics peter - Peter Cram- Want the Prize’. - An experimental investigation’. 87–97. jects’. ment with three sealed-bid mechanism’. of auvctions under single- and multi-unit demand’. Ausubel, L. M. and P. C. Crampton: 2002, ‘DemandAusubel, Reduction L. and M., Ineffi- P. Cramton, E. Filiz-Ozbay, N. Higgins, E. Y. Ozbay, and Bazerman, M. H. and W. F. Samuelson: 1983,Engelmann, ‘I D. and Won V. the Grimm: Auction 2009, ‘Bidding but behaviour Don’t in multi-unit auctions Ausubel, L. M.: 2004, ‘An Efficient Ascending-Bid Auction for Multiple Ob- The bottom line isvalue that, especially settings, for the inexperienceddiscovery, players, dynamic which and for mitigates auction common the seemsin common static to overbidding auction be that formats. has a been better produced formatThere is for still price lacksettings of in knowledge particular. in Therienced multi-unit present subjects, settings experiment even in waswould though general, carried be they and out interesting gained in with tosetting. experience CV The inexpe- see along conclusion how the from earlier experiencedproblems way. experiments with But bidders is CV it performed that settings. they in continue a to have similar 8 Acknowledgements We would like toments thank on the Lars paper. Hultkrantz, Also,econometrics. Jan-Eric thanks This to nilsson study Jan-Erik has for Sw¨ beenStudies valuable conducted (CTS). com- within The the author Centre is for Transport responsible for any remaining errors. References Ahlberg, J.: 2011, ‘Multi-unit common value auctions:Alsemgeest, P., A C. laboratory Noussair, experi- and M. Olson: 1998, ‘Experimental comparisons 9 Appendix A 10 Appendix B

Variable OLS Robust Bidder Instructions for the static, uniform, common value auc- standard error tion 3-player groups Reference 6-player groups -8.83*** 1.29 10.1 Introduction Dynamic auction 2.31*** 0.97 Signal 0.08*** 0.02 Hello and welcome. You will participate in an experiment on economic decision- making. The purpose is to study sales by bidding, i.e. through an auction. Large demander -0.71 1.36 Intercept 3.91*** 1.04 You have the opportunity to win money through participation. The show-up fee is SEK 100 ( 10), and by learning the rules of the game, you have the No of observations 1279 opportunity to earn more than that. On the other hand, you could also lose R2 0.149 in the process. To ensure that you walk away with at least SEK 100 in your pocket, we give you a starting balance of SEK 50. If you lose this money, you Notes: a; Dependent variable is profit. will be excluded from the experiment. Your winnings, and the show-up fee, b; ***, ** and * denote difference from zero at the one, will be paid in cash after the experiment. five and ten percent significance level respectively. A rule that applies at all times is that all communication between participants Table 11 is prohibited. If you have any questions, raise your hand and I will come to Regression on bidder profit you and you may ask your question in a whisper. If I believe the question must be answered, I will repeat it to everyone and give the answer.

10.2 Design

Rounds: The experiment consists of several rounds. In each round, 4 identical objects, or units, are to be sold through an auction. (How many rounds to actually play will be unknown to you.) The commodities: Each of you has a value associated with owning these units and would like to buy them. We call this the redemption value, which is the same for all units. How many units you want to buy, i.e. your demand, will be seen on your screen, and the number never changes during the game. The redemption value: Before the start of each round, the value of the units is randomly determined through the program. It draws an integer from an array of possible values. The value can never be less than 10, and the maximum is 90. Therefore, the (value) v belongs to the set 10, 11, , 89, 90 . (All values in this range have an equal probability.) However,{ you··· will not} know what this value is. Instead you will get private information about this value. Information: Even if you do not know the true value, you will receive information that limits the set of possible values. This will be done through a private information signal that is randomly chosen from a range of values

28 29 ESSAY III . 90} 89, , ··· 11, 10, { 29 Before the start of each round, the value of the Each of you has a value associated with owning these 10), and by learning the rules of the game, you have the Even if you do not know the true value, you will receive infor- The experiment consists of several rounds. In each round, 4 identical (All values in thisknow range what have this an value is. equalvalue. Instead probability.) you However, will you get will private not information about this objects, or units, areactually to play be will sold be through unknown an to auction. you.) (How many rounds to units and would like tois buy the them. same for We call allwill units. this be How the many seen redemption units on value, you your which want screen, to and buy, the i.e. number your never demand, units changes is during randomly the determined through game. thean array program. of It possible draws values. an The integerimum value from can is never 90. be less Therefore, than the 10, (value) and v the max- belongs to the set mation that limits theprivate set information signal of that possible is values. This randomly will chosen from be a done range through of a values Information: opportunity to earn morein than the that. process. On Topocket, the ensure we other that give hand, you you a walkwill you starting away could be with balance also excluded of at lose will SEK from least 50. be the SEK If paid 100 experiment. you in in Your lose cash your winnings, this after money, and the you A experiment. the rule show-up that applies fee, atis all times prohibited. is If that you allyou communication and have between you any participants may questions, askbe your raise question answered, in your I a hand will whisper. and repeat If I I it believe to will the everyone question come and must to give the10.2 answer. Design Rounds: The commodities: The redemption value: 10 Appendix B Bidder Instructions for thetion static, uniform, common value auc- 10.1 Introduction Hello and welcome. You will participate inmaking. an The experiment on purpose economic is decision- to study salesYou by have bidding, the i.e. opportunity throughfee to an win is auction. money SEK through 100 participation. ( The show-up 1.29 1.36 0.97 0.02 1.04 Robust standard error 28 OLS 0.149 0.08*** 3.91*** : a; Dependent variable is profit. 2 Variable Large demanderIntercept -0.71 No of observationsR 1279 Notes b; ***, ** and * denotefive difference and from ten zero percent at significance the level one, respectively. 3-player groups6-player groups Reference Dynamic auction -8.83*** Signal 2.31*** Table 11 Regression on bidder profit 9 Appendix A between the minimum value v 10 and the maximum value v + 10. There- (or more) person(s) less in the auction. But the auction continues as usual fore, (your signal) will belong to− the set v 10,v+ 10 . (All values in this without these people. range have the same probability.) Your signal{ − will also} be an integer. One round: After you have entered your bid in the fields, press the button Example: Suppose that the true value of the goods is 36, then your signal ”Add bids”. When everyone has pressed the button, the bids are ranked. will be in the set 36 10, 36 + 10 = 26, 46 . Those who have placed the highest bids purchase units at a price that is Opponents: You can{ either− have two or} five{ opponents.} Your group-size will determined by the maximum rejected bid. be seen on the screen. If there are more winning bids than units for sale, the program randomizes Bids: After receiving your information, that is, after you have seen your sig- the winners. The balance is recalculated and a new round starts. On the nal, you should decide what you want to bid for those units that you de- screen, you will see what the value of the units was, the price, the winning mand. It is permissible to place equal or different bids for the units. bids (as well as the signals from those with winning bids in parenthesis), the units won, and own profits/losses. The end: After a certain number of rounds, the experiment will end. Then 10.3 Instructions press the logout button, and you will come to a page showing what you have earned in the experiment. Buy: Those who have placed the four highest bids purchase the units. This may be the same person or different people. If there are ties among the (winning) bids, the program will randomly choose the winner(s). 10.5 Summary Price: The winners pay a price equal to the highest bid that did not win. That is, the highest bid that was rejected. Thus, all winners pay the same You will play a certain number of rounds and in each round, 4 identical • price for the units. units are for sale. Example: 4 units are sold. Five people (A, B, C, D, E) have the five highest You will play against two or five opponents. You will see the number of • bids: 25 (A), 23 (B), 19 (C), 15 (D), 12 (E). A, B, C and D purchase the opponents on the screen. units and everyone pays 12. In each round, all players in an auction will have the same redemption value • Gain/Loss: The winners make a profit equal to the difference between the for all demanded units. (redemption) value and the price. If the difference is negative, you make a However, each player only gets an informational signal about the true value. • loss. Subjects may or may not see the same information as their opponents. Example of profit: You won one unit, and the price was 42. The value of One can place bids for as many units as one demands, one for each unit. It • the unit was 50. You made a profit of 8 (50 42 = 8). is permissible to place equal or different bids for the units. − Example of a loss: You won one unit, and the price was 65. The value of You start with SEK 50. If you lose this, the experiment is finished for you, • the unit was 61. You then made a loss of 4 (61 65 = 4). and you are excluded from the experiment. But you can also earn more, − − Note If you do not have one of the highest bids, nothing happens. The profit depending how you and your opponents act. is zero.

10.4 Practical execution

Bidding: You will come to a (web)page where you will see the signal you received, how many units you demand, and how many opponents you have. On basis of this, you place your bids. You bid in the empty boxes and each box represents one unit. Only integer bids from 0 and up to 100 are possible. Money: You will see what your current balance is before every game starts on the screen. The starting balance is 50. If you lose your starting balance, the auction is over for you. Lost starting balance: If someone (or some) lose her starting balance, she/they will no longer participate in the auction. This means that there will be one

30 31 ESSAY III 31 After you have entered your bid in the fields, press the button After a certain number of rounds, the experiment will end. Then If there are more winning bids than units for sale, the program randomizes You will play aunits are certain for number sale. ofYou will rounds play and againstopponents in on two each the or round, screen. In five 4 each opponents. round, identical all You players willfor in an all see auction demanded will the units. haveHowever, number the each same player of only redemption gets value anSubjects informational may signal or about may theOne true not can value. see place the bids sameis for information permissible as as to many their units place opponents. You as equal start one or with demands, different SEK one bidsand 50. for for If you each the you unit. are units. lose It depending excluded this, how the from you experiment the and is experiment. your finished opponents But for act. you, you can also earn more, press the logouthave button, earned and in you the experiment. will come to a page showing what you (or more) person(s) lesswithout in these the people. auction. But the auction continues”Add as usual bids”. When everyoneThose has who pressed have the placeddetermined button, the by the the highest maximum bids bids rejected are purchase bid. ranked. units atthe a winners. price The thatscreen, balance is you is will recalculated seebids and what (as a the value well new of asthe round the units the starts. units won, signals was, On and the from the own price, those profits/losses. the with winning winning bids in parenthesis), 10.5 Summary One round: The end: • • • • • • + 10. There- v 4). − . (All values in this 65 = + 10} − . 42 = 8). − 46} 10,v − {26, {v = 30 10 and the maximum value − 36 +} 10 v , 10 − If someone (or some) lose her starting balance, she/they You won one unit, and the price was 42. The value of You won one unit, and the price was 65. The value of {36 Suppose that the true value of the goods is 36, then your signal 4 units are sold. Five people (A, B, C, D, E) have the five highest You can either have two or five opponents. Your group-size will The winners make a profit equal to the difference between the You will come to a (web)page where you will see the signal you You will see what your current balance is before every game starts The winners pay a price equal to the highest bid that did not win. After receiving your information, that is, after you have seen your sig- If you do not have one of the highest bids, nothing happens. The profit Those who have placed the four highest bids purchase the units. This will be in the set the unit was 50. You made a profit of 8 (50 bids: 25 (A), 23units (B), and 19 everyone (C), pays 15 12. (D), 12 (E). A, B, C and D purchase the the unit was 61. You then made a loss of 4 (61 fore, (your signal) will belongrange to have the the same set Example: probability.) Your signal will also be an integer. between the minimum value is zero. That is, the highestprice bid for that the was units. rejected.Example: Thus, all winners pay the same (redemption) value and theloss. price. If the differenceExample is of negative, profit: you make a be seen on the screen. nal, you should decidemand. what It is you permissible want to to place bid equal for or those different units bids for that the you units. de- may be the(winning) same bids, person the program or will different randomly people. choose the If winner(s). there are ties among the will no longer participate in the auction. This means that there will be one received, how many units youOn demand, basis and of how this, manybox you opponents represents place you one your have. unit. bids. Only You integer bid bids from in 0 the and emptyon up boxes to the and 100 screen. are each The possible. the starting auction balance is is over 50. for If you. you lose your starting balance, Example of a loss: Note 10.4 Practical execution Bidding: Gain/Loss: Opponents: Bids: 10.3 Instructions Buy: Price: Money: Lost starting balance: 11 Appendix C between the minimum value v 10 and the maximum value v + 10. There- fore, (your signal) will belong to− the set v 10,v+ 10 . (All values in this range have the same probability.) Your signal{ − will also} be an integer. Bidder Instructions for the dynamic, uniform, common value Example: Suppose that the true value of the goods is 36, then your signal auction will be in the set 36 10, 36 + 10 = 26, 46 . Opponents: You can{ have− either two or} five{ opponents.} Your group-size will be seen on the screen. 11.1 Introduction

Hello and welcome. You will participate in an experiment on economic decision- making. The purpose is to study sales by bidding, i.e. through an auction.

You have the opportunity to win money through participation. The show-up fee is SEK 100 ( 10), and by learning the rules of the game, you have the 11.3 Instructions opportunity to earn more than that. On the other hand, you could also lose in the process. To ensure that you walk away with at least SEK 100 in your Auction Procedure: This auction is not a so-called price auction, i.e. an pocket, we give you a starting balance of SEK 50. If you lose this money, you auction where you place bid(s) for the units. This is a quantity auction; will be excluded from the experiment. Your winnings, and your show-up fee, that is, there is a price clock starting from 0 and ticking up to 100, and the will be paid in cash after the experiment. players themselves choose when they want to yield their units. Each player A rule that applies at all times is that all communication between participants starts with demanding all his/her units, but may yield one or more units at is prohibited. If you have any questions, raise your hand and I will come to any time during the game. you and you may ask your question in a whisper. If I believe that the question Auction Time: The price clock starts at 0 for 15 seconds, then increases at must be answered, I will repeat it to everyone and give the answer. a rate of 1 unit per second. Every time someone gives up one or more units, the price clock stops for 5 seconds. If someone else gives up one or more units during this short break, the same price is registered but later in time. 11.2 Design The clock also stops for an additional 5 seconds. Auction Stop: When the number of non-yielded units is equal to the supply Rounds: The experiment consists of several rounds. In each round, 4 identical of units, the auction automatically ends and all those who still demand objects, or units, are to be sold through an auction. (How many rounds to units will win them. They will pay the price that cleared the market, for actually play will be unknown to you.) each unit won. That is, the last registered price. The commodities: Each of you has a value associated with owning these Example: 3 players are asking for 2 units each; the supply is 4. Then, as units and would like to buy them. We call this the redemption value, which soon as 2 units are yielded, the market clears, since demand is then equal is the same for all units. How many units you want to buy, i.e. your demand, to supply. The price that everyone pays for each of their units won is equal will be seen on your screen, and the number never changes during the game. to the price that cleared the auction; that is, the price that prevailed when The redemption value: Before the start of each round, the value of the the second unit was yielded. units is randomly determined through the program. It draws an integer from Excess Supply: If a bidder yields more than one unit, and thus gives rise to an array of possible values. The value can never be less than 10, and the max- an oversupply, the clock will be rolled back one increment, and the player imum is 90. Therefore, the (value) v belongs to the set 10, 11, , 89, 90 . who made this happen may purchase the same number of units to clear the (All values in this range have an equal probability.) However,{ you··· will not} auction. All players who have won units may then also buy at the new price. know what this value is. Instead, you will get private information about this Example: Suppose that the price clock is at 49, and 5 demanded units value. remain in the auction. If a player then yields 2 units when the price clock Information: Even if you do not know the true value, you will receive infor- turns to 50, the aggregated demand drops to only 3 units, while the supply mation that limits the set of possible values. This will be done through a is 4. Then, the player who yielded 2 units may only yield 1 unit, but the private information signal that is randomly chosen from a range of values price is rolled back to 49. This price applies to everyone who won units.

32 33 ESSAY III + 10. There- v . (All values in this + 10} . 46} 10,v − {26, {v = 33 10 and the maximum value − 36 +} 10 v , 10 − This auction is not a so-called price auction, i.e. an {36 If a bidder yields more than one unit, and thus gives rise to The price clock starts at 0 for 15 seconds, then increases at When the number of non-yielded units is equal to the supply Suppose that the true value of the goods is 36, then your signal You can have either two or five opponents. Your group-size will Suppose that the price clock is at 49, and 5 demanded units 3 players are asking for 2 units each; the supply is 4. Then, as will be in the set remain in the auction. Ifturns a to 50, player the then aggregated yields demandis 2 drops 4. units to only Then, when 3 the the units,price price player while is clock who the supply rolled yielded back 2 to units 49. may only This yield price 1 applies unit, to but everyone who the won units. soon as 2 units areto supply. yielded, The the price market that clears, everyoneto pays since the for demand price each that is of cleared then theirthe the units equal auction; won second that is unit is, equal the was price yielded. that prevailed when fore, (your signal) will belong to the set range have the sameExample: probability.) Your signal will also be an integer. between the minimum value be seen on the screen. auction where youthat place is, bid(s) there for isplayers a the themselves price choose units. clock when starting This theystarts from is with want 0 to demanding and a all yield ticking quantity his/her their upany auction; units, to time units. but 100, Each during may and player the yield the game. one or more units at an oversupply, the clockwho will made be this rolled happenauction. may back All purchase one players the increment, who same have and number wonExample: the of units may units player then to also clear buy the at the new price. a rate of 1 unitthe per price second. clock Every timeunits stops someone during for gives this up 5 short oneThe break, seconds. or the clock more If same also units, someone price stops is else for registered an gives but additional up later 5 in one seconds. time. or more of units, theunits auction will automatically win ends them.each unit and They won. will all That payExample: those is, the the who last price still registered that price. demand cleared the market, for 11.3 Instructions Auction Procedure: Opponents: Auction Time: Excess Supply: Auction Stop: . 90} 89, , ··· 11, 10, { 32 Before the start of each round, the value of the Each of you has a value associated with owning these 10), and by learning the rules of the game, you have the Even if you do not know the true value, you will receive infor- The experiment consists of several rounds. In each round, 4 identical objects, or units, areactually to play be will sold be through unknown an to auction. you.) (Howunits many and rounds would to like tois buy the them. same for We call allwill units. this be How the many seen redemption units on value, you your which want screen, to and buy, the i.e. number your never demand, units changes is during randomly the determined through game. thean array program. of It possible draws values. an The integerimum value from can is never 90. be less Therefore, than(All the 10, (value) and values v the in max- belongs thisknow to range what the have this set value an is. equalvalue. Instead, probability.) you However, will you get private will information not about this mation that limits theprivate set information signal of that possible is values. This randomly will chosen from be a done range through of a values The commodities: The redemption value: Information: 11 Appendix C Bidder Instructions forauction the dynamic, uniform, common value 11.1 Introduction Hello and welcome. You will participate inmaking. an The experiment on purpose economic is decision- to study salesYou by have bidding, the i.e. opportunity throughfee to an win is auction. money SEK throughopportunity 100 participation. to ( The earn show-up morein than the that. process. On Topocket, the ensure we other that give hand, you you a walkwill you starting away could be with balance also excluded of at from lose will SEK least the 50. be SEK experiment. If paid 100 Your you in in winnings, lose cash your and this after money, your the you show-upA experiment. rule fee, that applies atis all times prohibited. is If that you allyou communication and have between you any participants may questions, askmust your raise be question your in answered, hand a I whisper. and will If repeat I I it will believe to that come everyone the to and question give11.2 the answer. Design Rounds: 11.4 Practical execution When demand equals supply, i.e. when there are only four units left, the • auction ends automatically. Auction start: You will come to a (web)page where you will see how many You start with SEK 50. If you lose this, you are bankrupt, and you are • units you demand, how many opponents you have, and your balance. When excluded from the auction. But you can also earn more, depending how you the auction starts you will also get your signal. From then on, you can yield and your opponents act. units. You also have 15 seconds to think before the price clock starts. Money: On the screen you will see your up-dated balance after each round. The starting balance is fifty. If you lose your starting balance, the auction is finished for you. Lost starting balance: If someone (or some) loses her starting balance, she/they will not participate in the auction any more. This means that there will be one (or more) person(s) less in the auction. But the auction continues as usual without them. One round: After each round, the balance is re-calculated and a new round starts. On the screen you will see what the true value of the units was in the round before, the price of the units, what price the price-clock registered for the four most recent (highest) yielded units (as well as the signal these players had in parenthesis), units won, and the profit/loss in the round. Gain/Loss: The winners make a profit equal to the difference between the (redemption) value and the price. If the difference is negative, you will make a loss. Example of profit: You won one unit and the price was 42. The value of the unit was 50. You made a profit of 8 (50 42 = 8). − Example of a loss: You won one unit and the price was 65. The value of the unit was 61. You then made a loss of 4 (61 65 = 4). − − Note If you yield all your units, nothing happens. The profit is zero. The end: After a certain number of rounds, the experiment will end. Then, press the logout button, and you will come to a page displaying what you have earned in the experiment.

11.5 Summary

You will play a certain number of rounds, and in each round 4 identical • units are for sale. You will play against two or ﬁve opponents. You can see the number of • opponents on the screen. In each round, all players in an auction will have the same redemption value • for all units. However, each player only gets an informational signal about the true value. • Subjects may or may not see the same information as their opponents. When you see the signal, the auction has started. Then you demand all your • units, but can yield a unit at any time. You can yield one, or more units, depending on what you think is the best. After ﬁfteen seconds, the price clock starts.

34 35 ESSAY III 35 When demand equals supply,auction i.e. ends automatically. when thereYou are start only with fourexcluded SEK from units 50. the left, auction. If the Butand you you your can lose opponents also act. this, earn more, you depending are how you bankrupt, and you are • • 4). − 65 = − 42 = 8). − 34 If someone (or some) loses her starting balance, You won one unit and the price was 42. The value of You won one unit and the price was 65. The value of You will come to a (web)page where you will see how many After each round, the balance is re-calculated and a new round The winners make a profit equal to the difference between the After a certain number of rounds, the experiment will end. Then, On the screen you will see your up-dated balance after each round. If you yield all your units, nothing happens. The profit is zero. the unit was 50. You made a profit of 8 (50 the unit was 61. You then made a loss of 4 (61 she/they will notthere participate will in be the onecontinues as (or auction usual more) any without person(s) them. more. less This in means thestarts. auction. that On the But screen the youround will auction see before, what the the truefor price value the of of four the the most unitsplayers units, was recent had in what (highest) the in yielded price parenthesis), units units the (as won, price-clock well and registered as the(redemption) the profit/loss value signal in and these the the price. round. a If the loss. difference is negative,Example you will of make profit: press the logout button,have and earned you in will the experiment. come to a page displaying what you You will play aunits are certain for number sale. ofYou will rounds, play and againstopponents in on two each the or round screen. In five 4 each opponents. round, identical all You playersfor can in an all see auction units. will the haveHowever, the number each same player of only redemption gets value anSubjects informational may signal or about may theWhen true not you value. see see the the signal, sameunits, the information auction but as has can started. their Then yield opponents. depending you a demand on all unit what your atclock you any starts. think time. is You can the yield best. one, After or fifteen more seconds, units, the price units you demand, how manythe opponents auction you starts have, you and will yourunits. also balance. You get When also your have signal. 15 From seconds then on, to you think canThe before yield starting the balance price clock isis starts. fifty. finished If for you you. lose your starting balance, the auction Example of a loss: One round: Gain/Loss: Note The end: 11.5 Summary 11.4 Practical execution Auction start: Money: Lost starting balance: • • • • •

Publications in the series Örebro Studies in Economics

1. Lundin, Nannan (2003) International Competition and Firm- Level Performance. – Microeconomic Evidence from Swedish Manufacturing in the 1990s. Licentiate thesis. 2. Yun, Lihong (2004) Productivity and Inter-Industry Wages. Licentiate thesis. 3. Poldahl, Andreas (2004) Productivity and R&D. Evidence from Swedish Firm Level Data. Licentiate thesis. 4. Lundin, Nannan (2004) Impact of International Competition on Swedish Manufacturing. Individual and Firm-Level Evidence from 1990s. 5. Karpaty, Patrik (2004) Does Foreign Ownership Matter? Evidence from Swedish firm Level Data. Licentiate thesis. 6. Yun, Lihong (2005) Labour Productivity and International Trade. 7. Poldahl, Andreas (2005) The impact of competition and innovation on firm performance. 8. Karpaty, Patrik (2006) Does Foreign Ownership Matter? Multinational Firms, Productivity and Spillovers. 9. Bandick, Roger (2005) Wages and employment in multinationals. Microeconomic evidence from Swedish manufacturing. Licentiate thesis. 10. Bångman, Gunnel (2006) Equity in welfare evaluations – The rationale for and effects of distributional weighting. 11. Aranki, Ted (2006) Wages, unemployment and regional differences – empirical studies of the Palestinian labor market. 12. Svantesson, Elisabeth (2006) “Determinants of Immigrants’ Early Labour Market Integration” (Essay 1). “Do Introduction Programs Affect the Probability for Immigrants getting Work?” (Essay 2). 13. Lindberg, Gunnar (2006) Valuation and Pricing of Traffic Safety. 14. Svensson, Mikael (2007) What is a Life Worth? Methodological Issues in Estimating the Value of a Statistical Life. 15. Bandick, Roger (2008) Multinationals, Employment and Wages. Microeconomics Evidence from Swedish Manufacturing. 17. Krüger, Niclas A. (2009) Infrastructure Investment Planning under Uncertainty. 18. Swärdh, Jan-Erik (2009) Commuting Time Choice and the Value of Travel Time. 19. Bohlin, Lars (2010) Taxation of Intermediate Goods. A CGE Analysis. 20. Arvidsson, Sara (2010) Essays on Asymmetric Information in the Automobile Insurance Market. 21. Sund, Björn (2010) Economic evaluation, value of life, stated preference methodology and determinants of risks. 22. Ahlberg, Joakim (2012) Multi-unit common value auctions: Theory and experiments.