Essays on Multi-Unit Auctions Theory and Experiment

Essays on Multi-unit Auctions: Theory and Experiment

Dissertation

Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy

in the Graduate School of The Ohio State University

Jinsoo Bae, M.A.

Graduate Program in Economics

The Ohio State University

2020

Dissertation Committee

John H. Kagel, Advisor

Dan Levin

Paul J. Healy

Copyrighted by

Jinsoo Bae

2020

Abstract

This dissertation contains three essays on the topic of multi-unit auctions. In Chapter 1, I experimentally investigate implications of weak and strong budget constraints on two share auctions, the uniform price and the proportional share auction, compared to the first price auction. Under the strong budget constraint, the two share auctions raise higher revenue than the first price auction, in contrast to studies without budget constraints.

More surprisingly, the proportional share auction achieves higher efficiency than the first price auction despite its inherently inefficient allocation rule. The proportional share auction performed better than the uniform price auction under the strong budget constraint, but under the weak budget constraint, the uniform price auction performed the same or better than the proportional share auction.

In Chapter 2, We experimentally investigate the Generalized Second Price (GSP) auction used to sell advertising positions in online search engines. Two contrasting click through rates (CTRs) are studied, under both static complete and dynamic incomplete information settings. Subjects consistently bid above the Vikrey–Clarke–Grove’s (VCG) like equilibrium favored in the theoretical literature. However, bidding, at least qualitatively, satisfies the contrasting outcomes predicted under the two CTRs. For both CTRs, outcomes under the static complete information environment are similar to those in later rounds of the dynamic incomplete information environment. This supports the theoretical ii

literature that uses the static complete information model as an approximation to the dynamic incomplete information under which advertising positions are allocated in field settings.

Chapter 3 studies a pro-competitive effect of joint bidding in multi-unit uniform price auctions where bidders have private values and demand different quantities of units. I analyze a simple model with three identical items for sale, two small bidders each demanding a single unit, and a big bidder demanding two units. I show that joint bidding of the two small bidders, which recovers the symmetry of bidders, enhances competition among the bidders and increases efficiency and revenue of the auction.

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Dedication

To my parents

Acknowledgments

The completion of this dissertation would not have been possible without the continual support and encouragement from John Kagel. I thank Dan Levin for his wisdom, insight and advice. I would also like to thank Paul J. Healy for his guidance and his efforts to maintain the HAKK reading group where I received tremendous feedback. I am also grateful to Yaron Azrieli, Lixin Ye, James Peck, Huanxing Yang, Ian Krajbich, John

Rehbeck and other faculty members in the Department of Economics for helpful comments.

Chapter 2 of this dissertation was written in collaboration with John Kagel. I have benefited immensely from working with him.

I received funding from National Science Foundation (SES-1919450), the Journal of

Money, Credit, and Banking, and Alumni Grants for Graduate Research and Scholarship for research in Chapter 1. The research in Chapter 2 was supported by National Science

Foundation (SES-1630288) and the Journal of Money, Credit, and Banking. I thank John-

David Slaughter, Rick Tobin, Tiffany Garner, and other staff members for their help to run experiment sessions.

I would like to extend a special thanks to my family and dear friends for your care and encouragement throughout my educational career. I thank my parents, Euiyeol Bae and

Ilsoon Shin, for their unending love and support. I also thank my sister, Soyoon Bae for v

her continual care and love. I am also grateful to Sangjun Yea, Ben Casner, Renkun Yang and OSub Kwon for being with me during my Ph.D program. Last but not the least, I thank God for enabling me to reach this far.

Vita

2012 ...... B.A. Economics, Yonsei University

2014 ...... M.A. Economics, Yonsei University

2015 ...... M.A. Economics. The Ohio State University

2015-2018 ...... Graduate Teaching Associate, Department

of Economics, The Ohio State University

2018-2020 ...... Graduate Research Associate, Department

of Economics, The Ohio State University

Publications

Bae, J. and Kagel, J. H. (2019). An experimental study of the generalized second price auction. International Journal of Industrial Organization, 63, 44-68.

Fields of Study

Major Field: Economics

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Table of Contents

Abstract ...... ii Dedication ...... iv Acknowledgments...... v Vita ...... vii List of Tables ...... xi List of Figures ...... xii Chapter 1. Selling Shares to Budget Constrained Bidders: An Experimental Study of Two Share Auctions ...... 1 1.1. Introduction ...... 1 1.2. Literature review ...... 6 1.3. Theoretical framework ...... 9 1.4. Experimental design and hypotheses ...... 12 1.5. Experimental procedures ...... 17 1.6. Experimental results...... 20 1.6.1. Efficiency and Revenue ...... 21 1.6.2. Bidding behavior ...... 25 1.6.2.1. First price auction ...... 25 1.6.2.2. Proportional share auction ...... 28 1.6.2.3. Uniform price auction ...... 30 1.6.2.4. Uniform price auction – modified rule ...... 36 1.6.3. Dominated strategy ...... 39 1.6.3.1. Proportional share auction ...... 39 1.6.3.2. Uniform price auction ...... 41 1.6.3.3. Uniform price auction – modified rule ...... 44 1.7. Summary and conclusion ...... 47

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Chapter 2. An Experimental Study of the Generalized Second Price Auction ...... 50 2.1. Introduction ...... 50 2.2. Theoretical framework ...... 55 2.3. Experimental design and procedures ...... 59 2.4. Predicted Outcomes and Propositions to be Investigated ...... 62 2.5. Experimental results...... 64 2.5.1. Efficiency and revenue ...... 64 2.5.2. Bidding over Time ...... 70 2.5.3. Nash Equilibrium Bidding and Best Responding ...... 78 2.6. Summary and Conclusions ...... 84 Chapter 3. A Pro-Competitive Effect of Joint Bidding in Multi-Unit Uniform Price Auction with Asymmetric Bidders ...... 87 3.1. Introduction ...... 87 3.2. The Model ...... 89 3.2.1. Without Joint Bidding ...... 89 3.2.2. With Joint Bidding ...... 90 3.2.3. An Example of the Pro-Competitive Effect of Joint Bidding ...... 92 3.3. Efficiency, Incentive to Merge, Generalizability ...... 95 Bibliography ...... 97 Appendix A. Chapter 1 Appendix ...... 103 A.1. Characterization of an equilibrium in UPA ...... 103 A.2. An example of UPA equilibrium...... 104 A.3. Efficiency and revenue predictions with n=5 and n=10 ...... 105 A.4. Dominated bids in PSA ...... 106 A.5. Dominated bids in UPA and UPA-M ...... 107 A.6. Regression analysis for efficiency and revenue ...... 108 Appendix B. Chapter 2 Appendix ...... 110 B.1. Ratio Values Required to Maintain Contrasting Predictions Between Treatments...... 110 B.2. Value bidding as a Nash Equilibrium ...... 111 Appendix C. Chapter 3 Appendix ...... 112 C.1. Proof of Example 3.2...... 112 C.2. Proof of Proposition 3.1...... 113 ix

C.3. Proof of Example 3.3 ...... 114

List of Tables

Table 1.1. Benchmark predictions ...... 13 Table 1.2. Predicted and realized efficiency ...... 21 Table 1.3. Predicted and realized revenue ...... 23 Table 2.1. Efficiency ...... 65 Table 2.2. Percentage Differences between Observed and Predicted VCG-like Revenuea ...... 68 Table 2.3. Regressions: High and mid-value bids under SC ...... 75 Table 2.4. Regressions: High and mid-value bids under DI ...... 76 Table 2.5. Frequency of bid profile in Nash equilibria ...... 79 Table 2.6. Average Losses Relative to Best Responding ...... 81 Table 2.7. Median percentage deviation of mid-value bidders from VCG prediction ..... 83 Table A.1. Benchmark predictions (n=5) ...... 105 Table A.2. Benchmark predictions (n=10) ...... 105 Table A.3. Efficiency and revenue with auction format dummy variables ...... 108

List of Figures

Figure 1.1. Bid function and interim allocation of FPA and PSA ...... 15 Figure 1.2. Predicted and realized interim allocation in FPA ...... 26 Figure 1.3. Bid plots in the first price auction ...... 26 Figure 1.4. Predicted and realized interim allocation in PSA ...... 28 Figure 1.5. Bid plots in the proportional share auction...... 29 Figure 1.6. Predicted and realized interim allocation in UPA ...... 30 Figure 1.7. Bid Plots in the uniform price auction ...... 32 Figure 1.8. Predicted and realized interim allocation of UPA-M ...... 36 Figure 1.9. Bid Plots in uniform price auction – modified rule ...... 37 Figure 1.10. Frequencies of dominated strategy by budget ...... 40 Figure 1.11. Likelihood of negative payoff under w=50 ...... 41 Figure 1.12. Frequencies of dominated strategies by the number of dominated bids in UPA...... 42 Figure 1.13. Likelihood of negative profit by the number of dominated bids (UPA) ...... 43 Figure 1.14. Frequencies of the number of dominated bids submitted – Stack graph ..... 45 Figure 1.15. Likelihood of negative profit by the number of dominated bids (UPA-M) . 46 Figure 2.1. Payoff table used in the instructions ...... 61 Figure 2.2. Frequency of Truthful, Under and Over Bidding Relative to Valuations...... 67 Figure 2.3. Percent of each bundle’s allocation in relationship to valuations ...... 68 Figure 2.4. Median bids with over time: Static complete information ...... 71 Figure 2.5. Median bids across rounds: Dynamic incomplete information ...... 73 Figure 2.6. Frequency of Nash Equilibria over Time: SC (top panel) and DI (bottom panel)...... 80 Figure 3.1 Bid functions ...... 93

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Chapter 1. Selling Shares to Budget Constrained Bidders: An Experimental Study of Two

Share Auctions

1.1. Introduction

Many auctions sell items that could be sold in parts: shares of a company, mineral rights, electricity of a generator, and shares of facilities. If buyers are willing to buy the whole item on offer and are not budget constrained, then single-unit standard auctions where the highest bidder wins the whole item are known to allocate the item efficiently and raise the highest revenue (Myerson, 1981). However, when bidders are operating with a limited budget, auction practitioners have suggested that selling shares to many bidders can be more profitable than selling the whole item to a single highest bidder. However, little is known about revenue or efficiency implications of using share auctions with budget constrained bidders.1

In this paper, I experimentally studied two share auctions, the uniform price auction and the proportional share auction 2 which have been suggested by auction practitioners

1 Previous studies on share auctions mostly focused on bidders without budget constraints (more details in section 1.2) 2 The proportional share auction is also sometimes referred to as the voucher auction (Krishna, 2009). In this paper, I use the proportional share auction following Dobzinski et al. (2012). Brooks and Du (2019) refer to this mechanism as the proportional auction. 1

when bidders have substantial budget constraints.3 The uniform price auction is a prevalent format of share auction widely used to sell goods with large economic values

(e.g. US treasury bills and initial public offerings). Phillips' Plan for Outer Continental

Shelf (OCS) oil lease auctions is an example where practitioners have suggested using the uniform price auction to accommodate financially constrained bidders (Office of

Minerals Policy and Research Analysis and Anderson, 1979). Since OCS tracts are often too large compared to bidders’ ability to pay, Phillips' Plan suggested to allow bidders to submit bids on fractional working-interest shares and collect revenues from all bidders who receive shares of an OCS tract.

The proportional share auction is another type of share auction used in the privatization of Russia’s state-owned enterprises and more recently in cryptocurrency crowd-sale markets. When Russia privatized its state-owned enterprises, the private sector did not have much wealth for the enterprises, and Russia’s credit market could not sufficiently address this constraint. Therefore, economists who designed the privatization suggested the proportional share auction, effectively dividing enterprises into shares and allocating them to broader segments of the population (Boycko et al, 1994). More recently, the proportional share auction has been used in cryptocurrency crowd-sale markets where

3 Although these are not the only formats of share auctions, I focus on these two auctions since they have been used by auction practitioners in prominent examples and the two auctions have a desirable property that every bidder pays the same price per share (i.e. they are “non- discriminatory”). 2

most participants are individual investors who are more budget constrained than institutional investors.4

Two main research questions are investigated in this study. First, under budget constrained environments, can the uniform price auction or the proportional share auction raise more revenue or be more efficient than the first price auction? Second, between the uniform price auction and the proportional share auction, which auction would perform better under budget constraint environments? And to what extent do they differ?

In order to answer these questions, I designed an experiment where a seller sells a divisible good and three buyers participate in an auction.5 Each bidder is assigned a private value (1, 2, 3, …,100) with a budget constraint w. The budget constraint is common to all bidders so that no bidders can pay more than the budget constraint. Two budget constraints were employed: w=20 representing a strong budget constraint and w=50 representing a weaker budget constraint. In the first price auction (FPA), bidders submit a single bid that does not exceed w and the highest bidder wins the whole item. In the proportional share auction (PSA), each bidder submits a single bid up to w and pays what they bid upfront, with each bidder receives a fraction of the item equal to their bids divided by the sum of all bids.

In the uniform price auction (UPA), to keep things simple, the item for sale is offered in five equal shares. In general, an item could be offered in a larger number of shares in an

4 For example, block.one, a blockchain company used the proportional share auction to raise fund in 2017 (Stergiou, 2019). 5 Theoretical analysis hardly provides answers to the questions because the uniform price auction admits diverse multiple equilibria and calculating an equilibrium strategy is intractable even in simple examples (Krishna, 2009). 3

UPA, for example, 19.6 million shares were offered in the UPA used for Google’s initial public offering (IPO). In the current experiment, however, the item is offered in five shares to simplify the subjects’ bidding decision yet to allow considerable flexibility for bidding behavior.6 Each bidder can submit bids on as many shares as they want, and after all bids are submitted, the five highest bids each get a share and pay the market price, which is set by the sixth highest bid.7

In UPA, the experiment employed two different ways to enforce the budget constraint.

The first rule restricts the sum of all bids to be less than or equal to the budget constraint.

This is similar to the advice offered to bidders in Google’s IPO prospectus: “Do not submit bids that add up to more than the amount of money you want to invest in the IPO.

This is a very important point.” Thus, in the first rule, the experiment does not let bidders submit bids that add up more than the budget constraint w. However, this rule is unnecessarily restrictive to enforce a budget constraint as winning bidders pay a price equal to the highest losing bid. The alternative rule restricts each bidder’s kth bid from exceeding w/k, which is a more relaxed restriction than the original rule while maintaining the overall budget constraint.8 This alternative uniform price auction rule

6 Thus, In the experiment, there is lumpiness in bidding decision in that bidders had to bid the same price for 1/5 of the item and another same price for the next 1/5 of the item. UPAs in field settings also exhibit lumpiness to some extent. For example, Google IPO required the minimum size of a bid to be five to a hundred shares. In UPAs for Treasury bills of the Czech government, bidders are allowed to submit up to ten bidpoints (price–quantity pairs) in any given auction, and the average number of bidpoints submitted by a bidder in an auction is less than three (Kastl, 2009). 7 In case of ties, the ties are broken randomly. 8 For example, suppose a bidder has budget constraint 100 and wishes to bid on three shares. Under the first rule the bidder can submit 100 for the 1st share, but then he has no more budget left for the other two shares. Under the second rule however, the bidder can bid 100 for the 1st share and 50 for the 2nd share and (approximately) 33 for the 3rd share. Yet, the bidder will never 4

will be referred to as UPA-M, and will be compared to the simpler budget constraint where the sum of all bids must be less than or equal to w, referred to as UPA.

The experiment shows three main results. First, under the strong budget constraint, all share auctions (PSA, UPA, UPA-M) raised more revenue than the FPA, offering an alternative to an FPA with highly budget constrained bidders. In addition, PSA achieved even higher efficiency that FPA under the strong budget constraint. This is surprising since PSA inherently allocates items inefficiently to all bidders, even including low-value bidders. The higher efficiency under the strong budget constraint can be attributed to FPA essentially offering the item randomly to the set of bidders whose value is greater than or equal to the budget constraint. That share auctions raise higher revenue or efficiency than

FPA contrasts with the results of previous theoretical studies without budget constraints.

Second, there were no strict performance rakings between PSA and UPA. Under the strong budget constraint, PSA performed better than UPA in terms or revenue and efficiency. However, under the weak budget constraint, UPA raised more revenue than

PSA. The UPA was more frequently constrained than PSA under both weak and strong budget constraints. However, UPA being more susceptible to the budget constraints does not result in strict dominance of PSA over UPA. This is partly because UPA has a better allocation rule than PSA: PSA allocates shares to all participating bidders, but UPA only allocates shares to winning bidders who tend to be the highest value bidders. The better allocation rule in UPA encourages bidders to submit more aggressive bids to win,

pay more than 100 under the second rule. For example, if he wins three shares, the market price must be equal or less than his third bid which is at most 33. Thus, his payment, (3 × the market price), will not exceed 100. This logic applies to whatever number of shares he gets. 5

enabling UPA to raise more revenue than PSA when bidders are not strongly budget constrained.

Third, UPA-M uniformly performed better than UPA under both weak and strong budget levels for both revenue and efficiency. In particular, the relaxed rule significantly increased the revenue of UPA under w=20 where the UPA suffered from a strong budget constraint effect. This result is intuitive since the modified rule only relaxes the budget constraint of UPA but does nothing else. This indicates that auction practitioners can design a budget enforcement rule to achieve a more desirable outcome under uniform price auctions.

1.2. Literature review

The uniform price auction and the proportional share auction

The uniform price auction has been widely studied in the economics literature, but mostly without budget constraints. Wilson (1979), in response to the Philips Plans for OCS auction, showed that the uniform price auction would result in lower revenue than a single-unit standard auction under a pure common-value setting.9 Ausbel et al. (2014) showed that bidders in UPA has a generic incentive to submit bids less than their values

(demand reduction), which could result in poor revenue performance such as zero-

9 He also inexplicitly assumed that bidders can only use continuous bidding strategies, which is a crucial assumption for the low revenue result. Kremer and Nyborg (2004) showed that continuous bidding strategy assumption distorts the set of equilibria, since it forces demand to be equated with supply at the clearing price. They showed that allowing discontinuous bids with pro-rata rule for excess demand could eliminate low revenue equilibria. See also Kremer and Nyborg (2003) 6

revenue equilibrium. Kagel and Levin (2001) demonstrated that experimental subjects exercise strong demand reduction, the force behind the lower revenue prediction. List and

Reiley (2000) reported strong demand reduction in a field experiment. However, all these studies have not taken budget constraints into account. The current study demonstrates that under a strong budget constraint, the uniform price auction (particularly with the modified budget enforcement rule) can raise more revenue than the first price auction.

The proportional share auction has been relatively scarcely studied and its equilibrium behavior has not been fully explored (Krishna, 2009). Although the proportional share auction where each bidder pays what they bid and receives a fraction of an item equal to their bids divided by the total bids appears to be similar to the widely studied Tullock contest (Tullock 1980), they are dramatically different in terms of information structure and allocation rule.10,11 The current study contributes to the literature of the proportional share auction as it is one of the first to study equilibrium bidding behavior and efficiency and revenue implications, in particular, with budget constrained bidders.

10 In the proportional share auction, bidders do not know other bidders’ values for the item (incomplete information) while in the Tullock contest, the value of the prize is publicly known and the same for everyone (complete information). Further in the proportional share auction everyone gets a share of the item (proportional reward) whereas in the Tullock contest, only the winning contestant gets the prize (winner-takes it all). If assuming risk neutrality, the difference in the allocation rules would not matter for equilibrium predictions. However, the equilibrium behavior in practice could be different between the two (see experimental studies comparing proportional reward and winner-takes it all. Chowdhury et al, 2014; Fallucchi et al, 2013; Cason et al, 2018; Masiliunas et al, 2014). 11 The proportional share auction is also similar to lotteries for raising public funds (Morgan, 2000; Lange et al., 2007; Schram and Onderstal, 2009). The lottery literature, however, assumes a common prize value to all bidders, winner-takes it all, and that bidders receive an additional utility from the total fund raised by the lottery. 7

Auctions with budget constrained bidders.

Theoretical literature on auctions with budget constrained bidders has established strict efficiency and revenue rankings for single-unit auctions. It is known that, with binding budget constraints, the all-pay auction performs better than the first price auction, which in turn performs better than the second price auction (Che and Gale, 1998; Che and Gale,

1996). Also, the all-pay auction is known to be the optimal auction when bidders have a common budget constraint (Laffont and Robert, 1996; Maskin, 2000).12 However, the all- pay auction is not a practical selling mechanism as it requires losers to pay their bids, which can be off-putting to potential bidders and hard to enforce. Thus, this paper restricts attention to winner-pay auctions and study selling shares of an item to bidders as a practical alternative to the first price auction with budget constrained bidders.

To my best knowledge, I am aware of only a few experimental studies on auctions with budget constrained bidders. Pitchik and Schotter (1988) studied two-stage sequential auctions where two budget-constrained bidders strategically deplete the other bidder’s budget in the first stage. Kotowski (2020) and Kotowski (2011) theoretically and experimentally studied the first-price auction where two bidders have private values and private budget constraints. Ausbel et al. (2017) experimentally studied the first and second price auctions where the budget is endogenously set by a financial manager. I expect that the current study would contribute to experimental literature on auctions with

12 If bidders have private budgets, a modified all-pay auction is the optimal mechanism (Pai and Vohra, 2014). 8

budget constrained bidders as this paper studies share auctions which have not been experimentally studied under budget constrained environments.

1.3. Theoretical framework

A seller sells a divisible good without a reserve price and n bidders wish to buy the good.

All bidders are risk neutral and each bidder has a private value of the good on sale. Let vi be the value of the item for bidder i when the bidder receives the whole item. If the

13 bidder receives a share x ∈ [0,1] of the good, the payoff for the share equals to vi x.

Values are independently drawn from a distribution F(v). I assume that each bidder has a common budget constrain w, which is the most amount they can spend.

First price auction

In the first price auction (FPA), each bidder submits a single bid less than or equal to their budget constraint w. The bidder with the highest bid wins the item and pays what he bids. In case of ties, one of the highest bidders is randomly selected as the winner. It is known that FPA has a unique equilibrium when bidders have a common budget constraint, (Milgrom, 2004). The bid function has two parts with a cutoff value discontinuously dividing the two. Bidders whose value is lower than the cutoff submit bids according to a typical FPA without budget constraint, and bidders whose value is

13 If the utility is diminishing in the quantity a bidder receives, it is even more profitable to allocate shares of an item to many bidders instead of using a single-unit auction since bidders have higher marginal values in their earlier consumption. Thus, if share auction can perform better than single-unit auction under constant marginal utility environment, the same result will hold under diminishing marginal utility environments as well. 9

higher than the cutoff submit the budget w. The following proposition characterizes the equilibrium bid function.

Proposition 1.1. (Milgrom, 2004) Let 훽̂퐹푃퐴(푣) be the unique equilibrium bid function in

FPA without a binding budget constraint. With a binding budget constraint w, the unique bid function 훽퐹푃퐴(푣, 푤) of FPA is

훽̂퐹푃퐴(푣), 푣 < 푣∗ 훽퐹푃퐴(푣, 푤) = { 푤 , 푣 ≥ 푣∗ where 푣∗ is the solution of the following equation.

푣∗ 1 − 퐹(푣∗)푛 (푣∗ − 푤) = ∫ 퐹(푠)푛−1푑푠 푛[1 − 퐹(푣∗)] 0

Proportional share auction

In the proportional share auction (PSA), each bidder submits a single bid up to the budget constraint w and pays what they bid. Then each bidder receives a fraction of the item equal to their bids divided by the sum of all bids. It is known that the proportional share auction has a unique equilibrium. (Wasser, 2013; Ewerhart, 2014).14 Let 훽푃푆퐴(푣, 푤) denote the unique equilibrium in PSA with a budget constraint w. 훽푃푆퐴(푣, 푤) is characterized by the following equation.

푏 푃푆퐴 푖 훽 (푣푖, 푤) = argmax푏 ≤푤[ E푣−푖 [ 푃푆퐴 ] 푣푖 − 푏푖] 푖 푏푖 + ∑−푖 훽 (푣−푖, 푤)

14 Their results easily apply to PSA with a budget constraint. 10

It is intractable to get an analytical solution 훽푃푆퐴(푣, 푤) since the expected payoff does not have a closed form expression as it has random variables (other bidders’ values) in the denominator. Therefore, I numerically approximate the equilibrium bids of PSA.15

The uniform price auction

In the uniform price auction (UPA), a seller offers m equal shares of an item and runs the uniform price auction. Since the item is divided into m shares, the per-unit value of a share is vi/m to bidder i. Each bidder can submit up to m bids. After all bids are submitted, the bids are ranked from highest to lowest, and the m highest bids each get a share, with ties are randomly broken. The market price for a share is set by the m+1 highest bid and bidders winning any number of shares pay the market price for each share they get.

I study two different ways to enforce the budget constraint. First, bids must add up to no

푚 푘 more than the budget constraint (∑푘=1 푏푖 ≤ 푤). This rule is similar to Google’s advice for their IPO: “Do not submit bids that add up to more than the amount of money you want to invest in the IPO”. This rule can be easily implemented by asking security deposits for all bids submitted, without asking each bidder’s budget level. However, this rule is unnecessarily restrictive. The alternative rule restricts each bidder’s kth bid from exceeding w/k, which is a more relaxed restriction than the original rule but never violates the budget constraint.16 I refer to the modified rule as UPA-M.

15 The uniqueness of the equilibrium ensures that the numerical solution will be close enough to the unique theoretical equilibrium. I used a modified version of Wasser (2013)’s Matlab program to calculate the equilibrium bids. I thank him for kindly sharing his program. 16 See footnote 7 11

Characterizing an equilibrium in UPA (UPA-M) is involved and deferred to the appendix

A.1. Bidders are predicted to exercise demand reduction (bidding lower than the per-unit value) for the second and subsequent shares in an equilibrium, but it is intractable to specify an exact theoretical prediction for the general case since UPA (UPA-M) admits diverse multiple equilibria (Wilson, 1979; Engelbrecht-Wiggans and Kahn, 1998;

Ausubel et al. 2014) and a closed-form expression for equilibrium bids is not available

(Krishna, 2009).17 Thus, this study does not try to theoretically pin down an equilibrium of UPA (UPA-M). Rather, it focuses on differences in outcomes under the different budget enforcement rules, compared to the other auction formats. In what follows, predictions offered for UPA (UPA-M) assume a naïve bidding behavior that bidders submit their per-unit values until they are constrained by the budget constraint.18

Although the assumed naïve bidding behavior does not represent an equilibrium behavior, it will serve as a benchmark to evaluate subjects’ bidding behaviors in the lab.

1.4. Experimental design and hypotheses

The experiment adopts a 4 by 2 design with 4 auction formats (FPA, PSA, UPA, UPA-

M) and two levels of budget constraint (w=50 and w=20), which simulate weak and

17 In addition, budget constraints may affect existence of equilibria of multi-unit auctions. For example, Ghosh (2015) showed that, with a common budget, multi-unit simultaneous first-price auction has no equilibrium in pure-strategy. In the appendix A.2, however, I show an example of equilibrium in UPA with a common budget constraint. 18 For example, if m=5, a bidder’s per unit value (v/5) is 12, and the budget constraint is 20, then the naïve bidding behavior assumes that the bidder submits (12, 8, 0, 0, 0) in UPA and (12, 10, 6.6, 5, 4) in UPA-M. 12

strong budget constraints. In each auction, three bidders (n=3)19 participate in the auction, and each bidder is assigned a private value v which is independently drawn from a uniform distribution on (0, 1, 2, …, 100). In UPA and UPA-M treatments, the item is offered in five equal shares and bidders can submit up to five bids.

I set up benchmark predictions to establish comparative statics between auction formats and the two budget levels. FPA and PSA each have a unique theoretical equilibrium which serve the benchmark predictions. Since UPA and UPA-M have multiple equilibria and characterizing an equilibrium is intractable, I calculate efficiency and revenue benchmark prediction by assuming a naïve bidding behavior that bidders submit their per-unit values until they are constrained by the budget constraint.

Table 1.1. Benchmark predictions FPA PSA UPA UPA-M w=20 Efficiency 81.1% 87.4% 69.2% 86.1% Revenue 19.9 30.4 17.4 33.8 w=50 Efficiency 97.5% 87.6% 89.6% 97.2% Revenue 45.5 31.0 47.3 49.5 * FPA and PSA predictions are calculated by the unique equilibrium. UPA and UPA-M predictions are calculated by simulating a naïve bidding strategy: submitting per-unit value until constrained by budget20. * Average efficiency is defined as Srealized /Smax, where Srealized is the realized surplus in 21 auctions and Smax is the maximum possible surplus.

19 In the appendix A.3, I discuss comparative statics between the four auctions with different number of bidders. The comparative statics remains to be similar. 20 The predictions for UPA and UPA-M were calculated by using 107 times of simulated auctions. 21 Since the focus of this paper is to compare performances between auction formats, any other commonly used efficiency measure will draw the same comparison. 13

Table 1.1 shows benchmark predictions. Following hypotheses are established based on the benchmark predictions.

Hypothesis 1.1. PSA performs better than FPA in terms of efficiency and revenue under w=20 while a reversal occurs under w=50

An intuition behind hypothesis 1.1 can be explained by figure 1.1. Figure 1.1 shows bid functions and expected interim allocations22 of FPA and PSA under both weak and strong budget constraints. Under the weak budget constraint (w=50, upper panels), the bid functions show that only FPA is bound by the budget constraint. However, the interim allocations show that PSA allocates more shares to low-value bidders than FPA, which means PSA performs worse in terms of efficiency.23 This is because PSA suffers heavier efficiency loss due to its inherently inefficient allocation rule, which allocates an item to all participating bidders, including low-value bidders. Under the strong budget constraint

(w=20, lower panels), however, the interim allocations show that FPA allocates more frequently to low-value bidders than PSA. This is attributable to FPA being heavily affected by the strong budget constraint and essentially allocating the item randomly to the set of bidders who are pooled at the budget level as shown in the bid function. Thus,

22 Interim allocation refers to expected shares or probability of getting the item given one’s own value, but not knowing others’ values. 23 Revenue is closely tied with efficiency since in any constant marginal utility environment without a reserve price, revenues are maximized by allocating the item efficiently (Ausbel and Crampton 1999). 14

under the strong budget, the inefficiency in FPA due to the strong budget constraint outweighs the inefficiency of PSA’s allocation rule.

Figure 1.1. Bid function and interim allocation of FPA and PSA

Hypothesis 1.2. PSA performs better than UPA in terms of efficiency and revenue under w=20, but under w=50, UPA achieves similar efficiency and raises higher revenue.

Since PSA and UPA have different strategy spaces, it is not feasible to directly compare bid functions of the two auctions. However, the intuition used to explain hypothesis 1.1 can be helpful to understand hypothesis 1.2. When it comes to budget constraints, UPA is more likely to be affected by budget constraints than PSA. In PSA, bidders pay what they bid upfront, so they bid cautiously, but bidders in UPA are likely to bid more 15

aggressively since only winning bidders pay the market price which is even typically lower than their bids..24 Therefore, under the strong budget constraint (w=20) where the budget effect is strong, UPA is predicted to perform worse than PSA. However, UPA has a better allocation rule than PSA; UPA allocates an item to only winning bidders, while

PSA allocates an item to all participating bidders. This better allocation rule in UPA encourages high-value bidders to submit aggressive bids to win, enabling UPA to raise higher efficiency and revenue than PSA under the weak budget constraint when bidders are not strongly budget constrained.

Hypothesis 1.3 UPA-M uniformly improves the performance of UPA under both w=20 and w=50.

Hypothesis 1.4 UPA-M achieves similar performances with PSA under w=20 but higher efficiency and revenue under w=50.

Hypothesis 1.3 is an obvious prediction since UPA-M only relaxes the budget constraints of UPA but does nothing else. Hypothesis 1.4 predicts PSA and UPA-M would achieve similar level of efficiency and revenue under the strong budget constraint. This is because both auctions handle the strong budget constraint well, so their efficiency and revenue approach to those of the optimal auction (91.8%, 36.3).25 Under the weak budget

24 This is analogous to the second price auction being more frequently affected by a budget constraint than the all-pay auction. 25 The optimal auction is the all-pay auction (see section 1.2). That PSA and UPA-M are predicted to perform close to the optimal auction under a strong budget constraint implies that the 16

constraint, however, UPA-M is predicted to perform better since it has a better allocation rule, which encourages high-value bidders to bid more aggressively to win.

1.5. Experimental procedures

A total of 12 sessions were run, three sessions for each auction format. 231 subjects participated in one of the 12 sessions. In FPA/PSA, subjects participated in 10 auctions under w=20 then participated in 10 auctions under w=50. In each auction, subjects were assigned integer values randomly drawn from [0, 100] and three subjects were randomly matched in a group. In FPA, each bidder could submit an integer bid (including 0) up to the budget constraint w, and the bidder with the highest bid won the whole item and paid his bid. In case of ties, one of highest bidders was randomly selected to be the winner.

Similarly, in PSA, bidders could submit an integer bid up to the budget constraint. After all bids are submitted, each bidder paid what they bid and received a fraction of the item equal to their bids divided by all bids. In extremely rare events where all bidders submitted 0, all bidders paid nothing and received nothing26. In the decision screen, subjects were provided with an on-screen calculator where they could enter hypothetical bids, which then would calculate their expected payoff.27

two auctions can be practical alternatives to all-pay auction when bidders are strongly budget constrained. 26 In the Tullock contest, it is assumed that one of the contestants randomly receives the prize when no one put positive efforts. However, In the PSA, it is natural to assume that no one can receive the item if all bidders submit 0 since PSA models a selling mechanism rather than a rent- seeking behavior. Moreover, in the unique equilibrium of PSA auction, all bidders are predicted to submit positive bids. In the experiment, only two auctions in PSA ended up with all bidders submitting 0 bids. 27 The experiment provided this calculator since PSA auction rule involves time-consuming calculations. 17

In UPA and UPA-M the item was offered in five shares (m=5), so each bidder’s per-unit value was supposed to be randomly drawn from a set of [0, 0.2, 0.4, … 20]. However, since these per-unit values were in two-digit decimal, which potentially could complicate the experiment, I multiplied all nominal values by five and divided the conversion rate from experimental currency units to dollars by five, keeping the monetary incentive unchanged. Therefore, subjects in UPA and UPA-M auctions were assigned integer per- unit values which were randomly drawn from [0, 100], and the per-unit values were the same for all five shares auctioned off. The budget constraints were also multiplied by five, so let w5 denote the multiplied budget constraints. The subjects in UPA and UPA-M first participate in 10 auctions under w5=100 then participated in 10 auctions under w5=

250.

In each auction, the subjects could submit up to five integer bids (including 0) but were not mandated to submit all five bids. In UPA, the sum of bids could not exceed the budget constraint. In UPA-M, under w5=100, subjects could submit up to (100, 50, 33, 25, 20)

st nd rd th th for (1 , 2 , 3 , 4 , 5 share), respectively. Under w5=250, the restriction was (250, 125,

83, 62, 50)28. In reporting the experiment results, I divide all nominal outcomes in UPA and UPA-M by five to make them comparable to FPA and PSA.

In all sessions, prior to the main experiment, subjects participated in three practice rounds where all subjects were assigned the same values and bid against computer bidders who submitted predetermined bids. This setting ensured the same learning experience across

28 In addition to this, bidders were not allowed to bid more than 100. I believe this restriction was not necessary and would not make much differences to the results. 18

subjects in the practice rounds. In the main experiment, the same set of values were used across auction formats and across the two different budget constraint levels. This was designed to minimize the effects of random realization of values on comparing the outcomes across the auction formats and across the budget levels. However, the values in the set were randomly assigned to the subject to prevent subjects from being assigned the same values under w=20 and later under w=50.

In all auctions, bidders only knew their own valuations but not others. In each auction, 1 min (1.5 min for the first three auctions) was given to subjects to make decisions. In the instructions, bidders were informed of the possibility of losing money, but no restrictions on bids were imposed to prevent loss of money. In the case of bankruptcy, bidders would no longer be able to participate in the experiment and be asked to leave with the show-up fee. However, no bankruptcies occurred. In all treatments. the change in the budget constraint was notified only after the 10 auctions of earlier budget treatment were completed.

Feedback following each auction consisted of a table reporting values, bids, allocations, payments and earnings of all bidders. Earnings were in terms of experimental currency units (ECUs). Subjects were provided with starting capital balance 100ECUs in FPA/PSA and 500ECUs in UPA/UPA-M, with earnings from each auction added to or subtracted from this. Final earnings were converted into dollars at the rate of 20ECUs=$1 in

FPA/PSA and 100ECUs=$1 in UPA/UPA-M. In addition to this, subjects were paid a $4 show-up fee designed to give some money even if a subject would go bankrupt. Earnings

averaged $19.82 per subject in FPA, $19.50 in PSA, $17.75 in UPA, $17.36 in UPA-M for sessions lasting in average 1.5 hours.

The experiment was run in the Ohio State University Experimental Economics

Laboratory between Mar 2019 and Sep 2019. Subjects participating in the experiment were generally undergraduate students drawn from all disciplines and were recruited through ORSEE (Greiner, 2004). Each subject participated in single experimental session. The experiment was computerized, programmed using z-Tree (Fischbacher,

2007). Sessions were run with between 15-21 subjects in each session.29

1.6. Experimental results

The analysis presented in this section is based on 210/180/190/190 auctions for

FPA/PSA/UPA/UPA-M treatments. The same number of auctions were conducted for both w=20 and w=50. While testing differences between treatments, I used Mann–

Whitney–Wilcoxon (MWW) tests for the session averages, assuming that session averages are independent.30

29 FPA sessions had 21,21,21 subjects, PSA had 18, 21,15, UPA had 21, 15, 21, and UPA-M had 21,18,18 subjects. 30 Since the number of sessions are small, statistical tests using session averages as observations may lose great amount of information. Alternatively, I run regressions on all auction outcomes with dummy variables for auction formats, clustering errors at session level and run F-tests for the dummy variables. The F-test results are essentially the same as the MWW test results and are reported in the appendix A.6. 20

1.6.1. Efficiency and Revenue

Table 1.2. Predicted and realized efficiency Realized Predicted FPA PSA UPA UPA-M FPA PSA UPA UPA-M w=20 81.2% 87.0% 75.2% 79.8% 81.6% 87.7% 69.0% 86.7% (1.7%) (0.6%) (0.9%) (0.9%) (0.8%) (0.4%) (1.1%) (0.7%) w=50 96.2% 90.2% 80.8% 83.2% 98.1% 87.8% 89.6% 97.7% (0.6%) (0.6%) (1.1%) (1.0%) (0.3%) (0.4%) (0.7%) (0.2%) * Parentheses are standard errors of the mean. * FPA and PSA predictions are calculated by the unique equilibrium. UPA and UPA-M predictions are calculated by assuming a naïve bidding strategy: bidding one’s per-unit value until constrained by budget. All predictions are calculated using values realized in the experiment.

Table 1.2 shows realized and predicted average efficiencies of all four auction treatments.

The realized efficiencies are broadly consistent with the benchmark predictions.

Hypothesis 1.1 clearly holds as PSA achieves significantly higher efficiency (87.0%) than

FPA (81.2%) under w=20 (Mann-Whitney test, p<0.05), while FPA achieved significantly higher efficiency (96.2%) than PSA (90.2%) under w=50 (p<0.05). The realized efficiencies for FPA and PSA were remarkably close to the theoretical predictions as well; all within 3 percent points of the predicted efficiencies.

As hypothesis 1.2 predicts, under w=20, UPA achieved lower efficiency than PSA, averaging 75.2% and 87.0%, respectively (p<0.05). Moreover, UPA achieved the lowest efficiency among the four auctions as the benchmark outcomes predicted. Under w=50,

UPA again achieved lower efficiency than PSA, averaging 80.8% and 90.2%, respectively (p<0.05). However, this outcome is against hypothesis 1.2 which predicts that UPA and PSA would achieve about the same or slightly higher efficiency under w=50. Particularly, the realized efficiency in PSA (90.2%) was higher than the prediction

(87.8%) but the realized efficiency in UPA (80.8%) was lower than the prediction

(89.6%).

UPA-M achieved uniformly higher efficiency than UPA under both w=20 and w=50 as hypothesis 1.3 predicted. Under w=20, UPA-M and UPA achieved 79.8% and 75.2%, respectively and 83.2% and 80.8% under w=50. This is a natural consequence since UPA-

M relaxed budget constraints of UPA but does nothing else. However, only the difference under w=20 is statistically significant (p<0.05) and the difference under w=50 is not statistically significant (p=0.14). This implies that the efficiency improvement caused by the relaxed budget enforcement rule in UPA-M was more effective when the budget constraint was stringent.

Finally, the efficiency of UPA-M was uniformly lower than PSA under both w=20 and w=50. Under w=20, UPA-M achieved 79.8% and PSA achieved 87.0% (p<0.05). Under w=50, UPA-M achieved 83.2% and PSA achieved 90.2% (p<0.05). This result is against hypothesis 1.4 which predicted that UPA-M would achieve a similar level of efficiency to

PSA under w=20 and higher efficiency under w=50. Particularly, the realized efficiencies of UPA-M were lower than its prediction.

Table 1.3. Predicted and realized revenue Realized Predicted FPA PSA UPA UPA-M FPA PSA UPA UPA-M W=20 19.8 29.7 21.2 27.6 19.9 30.1 17.4 33.6 (0.100) (0.905) (0.635) (0.893) (0.074) (0.922) (0.682) (0.703) W=50 46.6 37.8 42.8 43.2 45.2 30.4 47.5 49.5 (0.554) (1.556) (1.541) (1.445) (0.646) (0.945) (1.350) (1.551) * Parentheses are standard errors of the mean. Predictions are based on values used in the experiment * FPA and PSA predictions are calculated by the unique equilibrium. UPA and UPA-M predictions are calculated by assuming a naïve bidding strategy: bidding one’s per-unit value until constrained by budget.

Table 1.3 shows predicted and realized revenues. The realized revenues are broadly consistent with the benchmark predictions. First, as hypothesis 1.1 predicted, the revenue reversal between FPA and PSA under the strong and weak budget constraint was observed. Under w=20, PSA raised significantly higher revenue (29.7) than FPA (19.8, p<0.05), while under w=50, FPA raised significantly higher revenue (46.6) than PSA

(37.8, p<0.05). The realized revenues of FPA and PSA were remarkably close to the theoretical predictions except PSA achieved noticeably higher revenue (37.8) than the prediction (30.4) under w=50. Similarly, revenue reversal is observed between PSA and

UPA under the weak and strong budget constraint as hypothesis 1.2 predicts. Under w=20, PSA raised significantly higher revenue (29.7) than UPA (21.2, p<0.05), while under w=50, UPA raised significantly higher revenue (42.8) than PSA (37.8, p<0.05).

UPA-M raised uniformly higher revenue than UPA under both w=20 and w=50 treatments as hypothesis 1.3 predicted. Under w=20 the improvement was substantial as UPA-M raised 27.6 and UPA-M raised 21.2 (p<0.05). Under w=50, however, the improvement was only marginal (and not statistically significant) as UPA-M raised 43.2 and UPA

raised 42.8 (p=0.413). This shows that the relaxed budget enforcement rule of UPA-M was more effective in an environment where UPA suffered a strong budget constraint effect.

Finally, UPA-M raised slightly lower revenue (27.6) than PSA (29.7) under w=20 and the difference was not significant (p=0.26). Under w=50, UPA-M raised higher revenue

(43.2) than PSA (37.8) and the difference was significant at 10 percent level (p=0.06).

This outcome is quantitively off from the benchmark predictions but qualitatively consistent with hypothesis 1.4 which predicted similar revenue between the two under w=20 and higher revenue of UPA-M under w=50.

It is worth noting that under the strong budget (w=20), all share auctions (PSA, UPA,

UPA-M) raised revenue higher than 20, which could not be raised by a winner pay auction such as FPA. This result implies that share auctions could be more profitable than single-unit auctions as practitioners have suggested if the budget constraint is strong enough. However, under w=50, FPA raised the highest revenue. Therefore, the absence of strict ranking in revenue implies that a seller must carefully choose an auction format under a budget constraint environment, considering the extent to which the budget constraint would bind.

conclusion 1.1: The realized outcomes were broadly consistent with the four hypotheses. Between PSA and FPA, efficiency and revenue reversals were clearly observed between the strong and the weak budget constraints as hypothesis 1.1 predicts. Between PSA and UPA, PSA performed better than UPA in terms of both efficiency and revenue under w=20, while UPA raised higher revenue than PSA

under w=50 as hypothesis 1.2 predicts. However, PSA achieving higher efficiency than UPA under w=50 is against hypothesis 1.2. UPA-M uniformly performed better than UPA under all budget levels, confirming hypothesis 1.3, with the improvement substantially being larger under the strong budget constraint. UPA-M achieved a similar revenue to PSA under w=20 and higher revenue under w=50, being consistent with hypothesis 1.4. UPA-M achieved uniformly lower efficiencies than PSA under all budget level which disagrees with hypothesis 1.4.

1.6.2. Bidding behavior

1.6.2.1. First price auction

Figure 1.2 shows predicted and realized interim allocations of FPA treatment. The realized interim allocations of FPA are largely consistent with the theoretical prediction.

In FPA, the two different budget constraint resulted in drastic differences in allocations.

Under w=20, FPA was strongly constrained by the budget and allocates the item almost randomly to bidders with value 22-100, while under w=50, FPA allocates the item mostly to high-value bidders, with bidders whose values are higher than 63 receiving the item

64.1% of the time.

Figure 1.2. Predicted and realized interim allocation in FPA

Figure 1.3. Bid plots in the first price auction

Figure 1.3 plots submitted bids in FPA sessions under w=20 and w=50. The solid lines are equilibrium bid functions and the dashed lines are identity functions. Under w=20, bidders with values equal or higher than 22 (constrained bidders) are expected to pool at the budget constraint and bidders with values below 22 (unconstrained bidders) are expected to bid two-thirds of their values. The realized bids were largely consistent with the predictions with a slight overbidding tendency. 87.3% of constrained bidders pooled

at the budget level. 37.0% of unconstrained bidders submitted bids within 1ECU of the predicted bid, 44.2% were overbidding and 18.8% were underbidding.

Under w=50, bidders with values equal of higher than 63 are predicted to pool at the budget constraint and bidders with values below 63 are predicted to bid two-thirds of their values.

The realized bids were consistent with the predictions with a noticeable overbidding tendency. 88.6% of constrained bidders pooled at the budget level. Among bids submitted by unconstrained bidders, 22.7% were within 1ECU of the predicted bid, 56.9% were overbidding and 20.4% were underbidding.

The overbidding tendency relative to the (risk-neutral) equilibrium bid under both w=20 and w=50 is congruent with previous experimental studies on the first price auction

(without budget constraints) as bidding above the equilibrium is the most common outcome in single-unit first price auctions. (Kagel, 1995).31 Unlike the previous studies, however, the realized efficiency and revenue were not deviating from the predictions. This is because the efficiency and revenue were mostly determined by high-value bidders who were constrained by budget and pooled at the budget level, which coincides with the prediction.

Playing a dominated strategy (bidding more than one’s value) was exceedingly rare, measuring 0.63% under w=20 and 0.79% under w=50. Since it is obvious that bidders will

31 This overbidding behavior could be rationalized with some additional factors such as risk aversion, joy of winning, anticipated regret, but I do not further investigate the overbidding behavior in FPA since the focus of this study is to compare performances and bidding behavior across different auction formats. See Cox et al. (1988), Goeree et al. (2002), Filiz-Ozbay and Ozbay (2007), Kagel and Levin (2016) for more details.

lose money if they bid more than their values and win the item, subjects rarely played dominated strategies even without experiencing losses.

1.6.2.2. Proportional share auction

Figure 1.4. Predicted and realized interim allocation in PSA

Figure 1.4 shows predicted and realized interim allocations of PSA treatment. The predicted interim allocations between w=20 and w=50 are about the same as PSA is rarely bound by both budget constraints (see figure 1.1). Under w=20, the realized interim allocation closely followed the prediction. Under w=50, bidders with values below 30 received less shares than the prediction and bidders with values above 70 received more shares that the prediction. This deviation means PSA achieved higher efficiency than the prediction under w=50 (90.2% vs 87.8%, see table 1.2)

Figure 1.5 shows bid plots of PSA under the two budget constraints. The solid lines are theoretical equilibrium and the dashed lines are one-fourth of values. Proposition 1.2 shows that bidding more than one-fourth of value is a dominated strategy in PSA. In

PSA, increasing one’s bid means stealing portions of the item from other bidders but at

the same time increasing the effective price of the item. For any realization of others’ bids, if a bidder submits a bid higher than 1/4 of values, the price effect always dominates the stealing effect, and reducing bid increases the bidder’s payoff. The proof of proposition 1.2 can be found in the appendix A.4.

Proposition 1.2. In PSA, bidding more than ¼ vi is dominated by bidding ¼ vi regardless of n and F(v).

Figure 1.5. Bid plots in the proportional share auction.

Bids are remarkably close to the equilibrium bids under w=20; 55.9% were within 1ECU of the theoretical predictions, 24.8% were underbidding and 19.3% were overbidding.

This shows that the realized efficiency and revenue were close to the predictions under w=20 because the bidders play closely to the equilibrium.

Under w=50, realized bids are more scattered from the prediction than w=20 treatment.

Bidders with values below 40 submitted bids close to the predicted bids with a slight tendency of under bidding: 57.4% were within 1ECU of the prediction, 13.5% overbid, 29

and 29.0% underbid. Bidders with values higher than 40 tend to overbid, with 16.9% being within 1ECU of the prediction, 58.3% overbid, and 24.8% underbid. The tendency of low-value bidders underbidding, and high-value bidders overbidding explains how

PSA achieved higher efficiency and revenue than the prediction under w=50 (90.2% and

37.75 vs 87.8% and 30.37).

Two pronounced patterns in playing dominated strategies (bidding more than dashed line) are observed. First, dominated strategies were played more frequently in w=50 treatment.

This observation is clearly due to the restriction how much they could bid under w=20; bidders could not bid more than 20, which suppressed bidding higher than the dashed line. Second, most dominated strategies are played by bidders whose values are relatively high (roughly higher than 50). In section 1.6.3.1, I further investigate bidding behavior on dominated bids.

1.6.2.3. Uniform price auction

Figure 1.6. Predicted and realized interim allocation in UPA

Figure 1.6 shows predicted and realized interim allocations of UPA. Under w=20, naïve bidding behavior predicts that high-value bidders received less shares than mid-value bidders since they exhaust their budget on earlier shares and could not get the later shares. There seems to be a weak evidence of high-value bidders getting less shares than the mid-value bidder: bidders with values higher than 90 received 37.1% of the item while bidders with values between 45-65 received 40.1%. However, the extent of decreasing shares is very subtle, and the realized outcome was more efficient than the benchmark prediction (75.2% vs 69.0%). Under w=50, bidders with values below 40 received more shares than the prediction and bidders with values above 70 received less shares that the prediction. This deviation illustrates that UPA achieved lower efficiency than the prediction under w=50 (80.8% vs 89.6%).

Figure 1.7. Bid Plots in the uniform price auction

* Since the item was offered in five shares, per-unit values for a share were between 0 and 20.

Figure 1.7 shows bid plots in UPA in decreasing order of shares. Upper panels are bids under w=20 and lower panels are bids under w=50. Since the item was offered in five shares, per-unit value of a share was v/5 and ranged between 0 and 20.32 Under w=20, an immediate finding is that mid to high-value bidders (per-unit values above 8) were heavily affected by the strong budget constraint and could not submit meaningful bids for the 4th and 5th shares; They submitted 0 bids for 67.5% and 80.5% of the times for 4th and

5th shares, respectively. Among these 0 bids, 72.1%, 74.1% were attributed to bidders exhausting their budget. This heavy effect of budget constraint prevented mid to high- value bidders from getting more shares of the item, decreasing efficiency of UPA. As a result, the efficiency of UPA under w=20 was the lowest among all four auction formats

(see table 1.2).

Another interesting finding is that mid to high-value bidders submitted bids lower than per-unit values for the first share. This is a strategic behavior taking the budget constraint into account; they save budget on their earlier shares to use the budget for their later shares. If the bidders had not been budget-constrained, there is no incentive to bid below for the first share since it is a dominated strategy (Krishna 2009).33 When bidders are budget constrained, however, it is often more profitable to save budget on the first share and bid more for later shares to win more shares.34 Unlike mid and high value bidders,

32 Remind that, in the experiment, all nominal values were multiplied by five, so the subjects submitted bids as if their per-unit values were between 0 and 100. 33 It will only reduce the chance of winning the first share but never lower the marker price that a bidder has to pay. If a bidder wins any quantity of shares, his first bid is never the market price. 34 For example, under w=20, submitting 20 for the 1st share wins one share for sure, but submitting 10 for 1st and 2nd shares could be more profitable since it wins one share for sure (since it is not possible for the other two bidders to submit five bids equal or higher than 10) and 33

low-value bidders (per-unit values below 8) mostly submitted their per-unit values since their values are so low that the budget constraint would not bind. The budget saving behavior of mid and high-value bidders counteracted the strong budget constraint to some extent, enabling UPA to achieve higher efficiency than the naïve benchmark prediction

(75.2% vs 69.0%).

Under w=50 (lower panels) where the budget constraint is weak, the budget saving behavior by and large disappeared: bids were much more aligned with bidders’ per-unit values for the first share than w=20 treatment, except high-value bidders (per-unit values above 15) reduced their demand to some extent. This is because the bidders now have more budget so that they have less incentive to save up budget for the later shares. Since the budget constraint was weaker under w=50, mid and high-value bidders (per-unit value above 8) less frequently submitted 0 bids than w=20 treatment: 38.3% and 59.7% for 4th and 5th shares, respectively (67.5% and 80.5% under w=20). Among these 0 bids,

54.5%, 60.7% were attributed to budget exhausting for 4th and 5th shares, which were less frequent than w=20 treatment (72.1% and 74.1%). This implies that substantial amount of 0 bid under w=50 is not attributed to the budget effect but to demand reduction behavior, which attempts to set a lower market price.

Under w=50, an interesting bidding behavior, which was absent under w=20, was observed: a strong overbidding behavior for the first share regardless of values. A

wins the second share with a very high chance. Note that however, the market price could be higher in the latter case, though it is unlikely that the higher market price would make winning two shares less profitable than winning one share. 34

substantial number of bidders (17.0%) tender maximum overbids (20) for the first share.35 At the first glance, this seems to be irrational bidding behavior. However, this bidding behavior is not as risky as it seems. Since they submit only one (or at most two) such bids which have little chance of affecting the market price, it is unlikely that they wind up paying a higher price than their per-unit values (more in section 1.6.3.2).

Therefore, this type of overbidding prevailed throughout under w=50 treatment. This observation is not at all new in the literature of UPAs. Journals and academic research studies (Delaney and Sidel, 2004; Jagannathan et al., 2015) reported such incentive to bid high for a small number of units in UPAs in the field settings and the current experiment is replicating the behavior observed in the field setting in the lab.36 This extreme overbidding behavior has a clear adverse effect on the efficiency and partly contributed lower efficiency than the benchmark (80.8% vs 89.6%)

35 I counted 19.6, 19.8 and 20 as extreme bids, which were 98, 99, 100 in the experiment. 36 “… disruption can result when some bidders place noncompetitive (i.e. arbitrarily high-priced) bids. In a uniform price auction, such ‘‘free riding’’ places the bidder first in line for shares but may have little effect on the clearing price. However, each such bid reduces the pool of shares available to investors who actively participate in price discovery. …” (Jagannathan et al., 2015). “Because anyone bidding below the clearing price won't get any shares, there will be an incentive to bid high.” (Delaney and Sidel, 2004, WSJ - Google IPO Aims to Change the Rules). 35

1.6.2.4. Uniform price auction – modified rule

Figure 1.8. Predicted and realized interim allocation of UPA-M

Figure 1.8 shows predicted and realized interim allocations of UPA-M. Under w=20, bidders with values below 30 received more shares than the prediction. This deviation caused UPA-M to achieve less efficiency than the naïve benchmark prediction (79.8% vs

86.7%). Under w=50, bidders with values below 50 received more than the prediction and bidders with values above 60 received less than the prediction. Due to the deviation throughout all types of bidders UPA-M achieved substantially lower efficiency than the benchmark (83.2% vs 97.7%).

Figure 1.9. Bid Plots in uniform price auction – modified rule

Figure 1.9 shows bid plots in UPA-M in decreasing order of shares. Upper panels are bids under w=20 and lower panels are bids under w=50. Look at the bids for the first share under w=20. Unlike UPA, budget saving behavior does not appear in UPA-M: bids below per-unit values are not as frequently observed as UPA/w=20. Since bids for each share has its own bid cap (kth bid cannot exceed w/k) under UPA-M, there is no incentive to save budget on earlier shares to bid more later shares.37 Moreover, the relaxed budget enforcement rule enables mid-and higher value bidders to submit non-zero bids for 4th or

5th units unlike UPA/w=20. However, the relaxed budget brought an unwelcomed effect that substantial number of bids (20.7%) were maximum overbids for the first share, the same bidding behavior observed under UPA/w=50. Overall, the relaxed rule in UPA-M under w=20 increased the efficiency by mitigating the strong budget effect in UPA, but also decreased efficiency by introducing extreme overbidding behavior, together slightly increasing efficiency than UPA/w=20 (79.8% vs 75.2%). However, since both changes in bidding behavior encouraged bidders to bid higher, the revenue was substantially higher than UPA/w=20 (27.64 vs 21.16).

Under w=50, the bidding behavior was essentially the same with w=20 for the first share.

Budget saving behavior was not observed as almost no bids were below per-unit values for the first share. Extensive overbidding for the first share was observed, with 24.6% bids being maximum overbidding. However, since the budget constraint was higher than w=20, high value bidders could submit higher bids for the later share (3th-5th share) than

37 Thus, in UPA-M, bidding below per-unit value for the first share is a dominated strategy regardless of budget constraint. In UPA, this is true only if there is no budget constraint. 38

w=20. Thus, the efficiency and the revenue were higher than w=20 treatment (83.2% and

43.2 vs 79.8% and 27.6). However, compared to the UPA/w=50, the improvement was only marginal (83.2% and 43.2 vs 80.8% % and 42.8) since bidders under UPA/w=50 already had enough budget.

Conclusion 1.2: Bidding behavior in FPA closely followed the equilibrium predictions under both w=20 and w=50 with a notable tendency of overbidding relative to the equilibrium. Bidding behavior in PSA was close to the equilibrium prediction under w=20, but substantial number of dominated bids (bid higher than ¼ of values) were observed under w=50. In UPA, under the strong budget, mid and high-value bidders saved budget from earlier share to counteract the strong effect of the budget constraint. Under the weak budget, the budget saving behavior disappeared, and bidders submitted extreme overbids for a small number of units (mostly for 1st share). In UPA-M, no budget saving behavior observed, and bidders submit extreme overbids for earlier shares under both strong and weak budget constraints.

1.6.3. Dominated strategy

In this section, I will take a close look on dominated strategies played in PSA, UPA,

UPA-M.

1.6.3.1. Proportional share auction

In PSA, bidding above 1/4 of one’s value (dashed line in figure 1.5) is a dominated strategy, being dominated by bidding 1/4 of value. Figure 1.10 shows frequencies of

dominated bids under w=20 and w=50 treatments. Under w=20, only 12.8% 38 bids were dominated strategy, while 35.4% bids were dominated under w=50. Moreover, under w=50, bidders with values above 50 mostly submitted dominated bids, averaging 59.3% bids were dominated. The higher frequency of dominated bids on the part of high-value bidders under w=50 contributed higher realized efficiency (90.2%) than the theoretical prediction (87.8%). Moreover, high frequencies of dominated bids under w=50 contributed higher realized revenue (37.8) than the theoretical prediction (30.4).

Figure 1.10. Frequencies of dominated strategy by budget

* Under w=20, bidders with values higher than 80 could not play dominated strategies due to the budget constraint.

Figure 1.11 shows the likelihood of getting negative payoff under w=50 disaggregated by playing un-dominated and dominated strategy. bidders with values below 50 almost surely lose money when playing dominated strategy (right panel) while the likelihood of

38 Bidders with values above 80 could not submit dominated bids due to the budget constraints (w=20). If only consider bids submitted bidders with values below 80, 15.9% were dominated bids. 40

losing money is significantly lower if playing un-dominated strategy (left panel).

However, bidders with values above 50 rarely lose money regardless of playing dominated strategy or un-dominated strategy. Therefore, bidders with values below 50 could learn a lesson of not playing dominated strategy, but the same lesson was not taught to bidders with values above 50.

Figure 1.11. Likelihood of negative payoff under w=50

Lastly, I calculated opportunity cost of playing dominated strategy compared to the strategies of bidding 1/4 of values. Bidding dominated strategy loses in average 0.82

ECUs ($0.041) under w=20 and 3.19ECUs ($0.159) under w=50. The loss was higher under w=50 since bidders more wildly overbid (relative to ¼ of values) under w=50, while the extent of overbidding was restricted under w=20.

1.6.3.2. Uniform price auction

In UPA, bidding above one’s per-unit value is a dominated strategy, being dominated by bidding per-unit value. Since each bidder could submit up to five bids in per auction in 41

UPA, I disaggregate dominated strategies by the number of dominated bids submitted: dominated strategies with one dominated bid to five dominated bids. Figure 1.12 shows frequencies of dominated strategies by the number of dominated bids. Under w=20,

25.4% were dominated strategies, with 14.7% being with a single dominated bid (bid for

1st share), 6.0% with two dominated bids (bids for 1st-2nd shares), and only 4.8% with three or more dominated bids. Most dominate strategy were played with bidders with values less than 50. This is because the bidders faced a strong budget constraint (w=20) that they could not overbid if their values were high. Under w=50, dominated strategies were much more frequently played, averaging 45.1%. Dominated strategies played under w=50 mostly involved a small number of dominated bids: 15.6% with a single dominated bid, 15.8% with two dominated bids, and 13.7% with three or more dominated bids. This finding is consistent with the bidding behavior described in 6.2.2: submitting extreme overbids for one or two shares.

Figure 1.12. Frequencies of dominated strategies by the number of dominated bids in UPA

Figure 1.13 shows the likelihood of getting negative profit by the number of dominated bids submitted under w=20 and w=50. 39 Submitting one dominated bid rarely resulted in negative profit under both w=20 and w=50 treatments. Submitting two dominated bids mostly resulted in negative payoff under w=20, while under w=50, it rarely resulted in negative payoff if bidders have values higher than 60. This is consistent with higher frequency of submitting two dominates bids under w=50 (15.8%) than w=20 (6.0%) and substantial frequency of submitting two dominated bids by bidders with higher values under w=50 (figure 1.12). Bidding three dominated bids almost surely ended up with negative payoff under both budgets.

Figure 1.13. Likelihood of negative profit by the number of dominated bids (UPA)

39 Submitting 4 or 5 dominated bids are omitted since they rarely occurred. 43

I calculated opportunity cost of dominated strategy compared to value bidding strategy.

Bidding one dominated bid resulted in nearly no loss averaging -0.77 ECUs (-$0.007) under w=20 and -1.98ECUs (-$0.019) under w=50. Submitting two dominated bids resulted in substantially larger loss than bidding one dominated bid: -6.86ECUs (-$0.07) under w=20 and -9.11ECUs ($0.09) and under w=50. Bidding three dominated bids ended up with considerable losses under both treatments: -20.35 ECUs ($0.20) under w=20 and -39.05ECUs ($0.39) under w=50.

1.6.3.3. Uniform price auction – modified rule

In UPA-M, bidding above one’s per-unit value is a dominated strategy, being dominated by bidding per-unit value.40 Figure 1.14 shows frequencies of dominated strategies by the number of dominated bids. Under w=20, 54.0% were dominated strategies, with 40.9% being with a single dominated bid, 7.9% with two dominated bids, and 5.3% with three or more dominated bids. Under w=50, 56.1% were dominated strategies, with 26.1% being with a single dominated bid, 18.9% with two dominated bids, and 11.1% with three or more dominated bids. Under both budget constraint levels, dominated strategies were played across all value bidders mostly with one or two dominated bids.

40 Under UPA-M, bidding lower than per-unit value for the first share is also a dominated strategy. However, I focus on overbidding to make the analysis comparable to UPA analysis. In UPA-M, for the first share, 21.4% bids were lower than per-unit value under w=20, and 15.4% under w=50. 44

Figure 1.14. Frequencies of the number of dominated bids submitted – Stack graph

Figure 1.15 shows the likelihood of getting negative profit by the number of dominated bids submitted under w=20 and w=50. Like UPA treatments, submitting one dominated bid rarely resulted in negative profit under both w=20 and w=50 treatments. Submitting two dominated bids mostly resulted in negative payoff under w=20, while under w=50, submitting two dominated bids has a low chance or negative payoff if the bidders have values higher than 60. Bidding three dominated bids almost surely ended up with negative payoff under both budget levels.

Figure 1.15. Likelihood of negative profit by the number of dominated bids (UPA-M)

I calculated the opportunity cost of playing a dominated strategy compared to a value bidding strategy. Bidding one dominated bid rarely resulted in any loss, averaging -0.47

ECUs (-$0.005) under w=20 and -3.07ECUs (-$0.031) under w=50. Submitting two dominated bids resulted in higher loss than bidding one dominated bid. Average loss of bidding two dominated bids was -7.78ECUs (-$0.08) under w=20 and -7.05ECUs ($0.07) and under w=50. Bidding three dominated bids ended up with substantial losses under both treatments: -24.58 ECUs ($0.25) under w=20 and -21.90ECUs ($0.22) under w=50.

This pattern of profit loss is almost identical to UPA treatment.

1.7. Summary and conclusion

This paper studies the uniform price auction, the proportional share auctions and the first price auction under both weak and strong budget constraints. Under the weak budget constraint, the first price auctions performed the best in terms of efficiency and revenue, the uniform price auction raised the second highest revenue and the proportional auction raised the lowest revenue. However, when the budget constraint was strong, the proportional share auction performed the best both for efficiency and revenue, the uniform price auction raised the second highest revenue and the first price auction raised the lowest revenue; a complete reversal occurred. These findings contribute to auction literate in two different ways. First, previous studies on share auctions, which had not taken budget constraint into account, have suggested that share auctions would perform worse in terms of revenue or efficiency than standard auctions (Wilson,1979;

Krishna,2009). However, the current study shows that, when bidders have strong enough budget constraints, share auctions such as the uniform price auction or the proportional share auctions can perform better than the first price auction. Second, previous studies on auctions with budget constrained bidders have focused on single-unit auctions, but this study extends the effect of budget constraints on share auctions.

No strict revenue/efficiency rankings of the three auctions gives some insight to understand why all three auctions are observed in the field settings. For example, if a start-up company wishes to sell its own shares to raise fund, it may search for a venture capital firm that offers the highest amount as if it is running the first price auction. Since a start-up company typically has relatively small value compare to venture capitals’ 47

financial resources, such effort for searching the highest bidder will likely to raise more revenue than offering the companies’ shares to public who might not be much interested in the start-up. However, if a huge company such as Google would like to sell shares to raise fund, it would be a better idea to offer its shares to public in a uniform price auction rather than looking for a single venture capital firm. In the case of cryptocurrency crowd- sale market where most participants are individual investors who cannot (or does not want to) invest a large amount, collecting some revenue from all interested participants could be more profitable than using a selling method that select a few buyers to pay a lot.

Thus, the proportion share auction could be a reasonable choice in their place.

Another interesting finding of this paper is that a modified rule to enforce budget constraint in the uniform price auction raised higher revenue and efficiency than the uniform price auction that used a similar budget enforcement rule what Google’s IPO advised to its potential bidders. This result indicates that auction practitioners must carefully design budget enforcement rules to achieve a more desirable outcome when bidders are budget constrained.

The current study could be extended with different assumptions and focuses. First, future study can investigate how share auction would perform when bidders have different utility structure. For example, bidders may demand only a portion of an item or bidders may need at least a certain fraction of an item to operate. Second, different number of bidders could be studied to investigate the effects of increased participation on share auctions. Third, the effects of different budget constraints could be studied, for example

an intermediate level budget constraint. Lastly, share auctions other than the uniform price auction or the proportion share auction could be studied.

Chapter 2. An Experimental Study of the Generalized Second Price Auction

2.1. Introduction

Search engines such as Google, Yahoo and Microsoft sell advertisement slots on their search result pages through auctions, among which the Generalized Second Price (GSP) auction is the most prevalent format. Under GSP auctions, advertisers submit a single per-click bid. These bids are raked from highest to the lowest, with ad-slots assigned according to the ranking, with each advertiser paying a per-click price equal to the bid submitted by the next-highest bidder.

Edelman et al. (2007) and Varian (2007) were the first to characterize the Nash equilibrium of the GSP auction using a static complete information model about competitors’ per-click values. They showed that truthful bidding is not a dominant strategy, that multiple Nash equilibria exist and that that these equilibria need not to be efficient. Edelman et al. (2007) proposed a refinement for the Nash equilibrium referred to as locally envy-free equilibria (LEFE), where no bidder would prefer another’s slot to her own, given the ad-slot prices.41 The LEFE predicts efficient allocations of ad-slots but still admits multiple equilibria.

41 Varian (2007) independently discovered locally envy-free equilibria, referring to them as symmetric Nash equilibria. 50

Edelman et al. (2007) further proposed that the LEFE with the lowest possible bids would be the most likely equilibrium to emerge as the long-run outcome in GSP auctions, based on an ascending clock version of the GSP auction. Under this outcome, the allocation of ad-slots and the associated payments coincide with those of the dominant strategy equilibrium in the Vikrey-Clarke-Grove’s (VCG) auction. In what follows, this refinement will be referred to as the VCG-like equilibrium.

Behavior in the GSP auctions is explored here in an experiment with three bidders and two ad-slots under two contrasting CTRs that result in distinctly different bidding behavior. Specifically, in one treatment the CTRs are relatively far apart, where the first slot gets 11 clicks and the second slot 3 clicks. In this treatment the VCG-like equilibrium predicts mid-value bidders will employ modest reductions in bids relative to their valuations, while facing minimal competition from the high-value bidder.42 In the second treatment, CTRs are very close to each other, 11 clicks for the first slot and 10 for the second. In this treatment the VCG-like equilibrium predicts that mid-value bidders engage in sharp price cutting due to the first and second positions being close to each other, so that the high-value bidder has an incentive to compete for the second position at a favorable price. The experiment is conducted in a static complete (SC) information environment, and in a dynamic incomplete information (DI) environment, closer in structure to how GSP auctions are conducted in practice.

42 VCG-like equilibrium predicts low-value holders always bid their values regardless of CTRs. On the other hand, the high-value holders' bids are not pinned down by VCG-like prediction since the highest bids do not determine any prices. Thus, the bidding behavior of the mid-value holders is crucial to examine the GSP auction outcomes under the two different CTRs in relation to the VCG-like equilibrium predictions. 51

An additional feature of the experimental design is that while per-click values are assigned randomly across auctions, a fixed ratio is maintained between values, which is required to ensure that the contrasting predictions between the two CTRs are maintained.43 The result is that for the mid-value bidder in the 11-3 treatment the upper bound for an LEFE in undominated strategies (LEFEU) coincides with value bidding, compared to sharp price reductions in the 11-10 treatment. There are also contrasting differences in mid-value bids needed to achieve the VCG-like equilibrium, with bids just above that of the low-value 11-10 treatment, compared to much more modest bid shaving under 11-3.

Behavior is broadly consistent with the predictions of the theory in that there is minimal bid shaving with 11-3 and substantial price shaving with 11-10. The contrast is particularly strong in bidding over rounds in the DI treatment: under 11-3 bids hover around bidders’ values, compared to sharp price cutting over time with 11-10.

Efficiency, as traditionally defined in auction experiments, is consistently high averaging over 90% under 11-3 and around 75% under 11-10. Outcomes for mid-value bids under

11-3 lie within the range of LEFEU, but above the upper bound of the LEFEU for 11-10.

This is a consequence of the fact that value bidding is at the upper bound of the LEFEU under 11-3, but well above it for 11-10. Bidders consistently bid higher than the VCG prediction under both CTRs, substantially less so under 11-3 than 11-10. Differences between mid-value bids relative to the VCG prediction are the same under SC compared

43 Subjects are unaware of this so that under DI there is no way they can deduce other bidders’ values based on their own valuation. 52

to the last round under DI. This provides support for the idea that SC auctions can serve as a model for bidding in GSP auctions, as actually practiced. However, the predicted

VCG-like revenue is not likely to be achieved. Part of this has to do with low-value bidders bidding above value, a common outcome in single unit second-price, private value auctions.44 Median deviations of mid-value bids from the VCG-like equilibrium average around 25% and 150% under 11-3 and 11-10 respectively, under both SC and DI.

Reasons for these marked differences are discussed below.

There have been a number of empirical studies of GSP auctions. Börgers et al. (2013) used a revealed preference approach to infer the per-click values of bidders, reporting that in a number of cases the revealed click values do not correspond to a NE. Athey and

Nekipelov (2012) developed and estimated a model that allows uncertainty in quality score and bidder entry, and show that efficiency of GSP auctions is slightly less than

Vickery auctions, but revenue effects are ambiguous. Varian (2007) showed that if bids are part of an LEFE, the expenditure profile must be increasing and convex, and that the data from Google’s auctions often consistent with this.45 Empirically investigating whether GSP auction outcomes are a VCG-like equilibrium is difficult, since advertisers' click values are not observable. In contrast, in an experiment one can induce click values, providing a clean environment to investigate this question.

44 See Kagel et al., (1987), Kagel and Levin (1993), Andreoni et al., (2007), and Cooper and Fang, (2008). Garratt et.al, (2012) show that this result holds in a field setting, using subjects who regularly participate in eBay auctions for Morgan silver dollars. In addition, Andreoni et al. (2007) show that with complete information about valuations, bidders’ exhibit rivalrous bidding, where low value bidders bidding above their value. 45 The expenditure profile is a function that maps CTR to expenditure (CTR x Price). 53

There have been several experimental studies comparing outcomes of GPS auctions in relation to the VCG-like equilibrium, with mixed results. Fukuda et al. (2013) and

McLaughlin and Friedman (2016) compared revenues of GSP to sealed bid VCG auctions run as a control treatment, concluding that revenue is indistinguishable between the two. However, Noti et al. (2014), in comparing the two found higher revenue in the

GSP compared to VCG auctions. In this experiment subjects were instructed to bidding their values in the VCG auctions was optimal.46 All three studies employed a limited set of pre-determined valuations, which restricts the potential variation in outcomes that can be observed with random valuations such as those employed here.

The paper that is closest to ours is Che et al. (2017). They use new random valuations in each auction and two contrasting CTR treatments with characteristics similar to the 11-3 and 11-10 treatments employed here. However, unlike here, they use unrestricted random values which create different equilibrium outcomes between auctions, some of which may deviate from the general characteristics of the CTR treatments under study. In what follows, we employ the same fixed ratio for values that avoids this, while still employing random realization valuations. (More on this below). Further, the fixed ratio employed creates very narrow bounds for LEFEU outcomes, and quite demanding VGC-like equilibrium outcomes for mid-value bidders. Nevertheless, we view the two papers as complements, with each providing a similar structure for studying GSP auctions, but with some significant differences in experimental design.

46 Still bidders consistently bid above their values, with greater overbidding than in the GSP auctions. Fukuda et al (2013) also report bids above value in their VCG auctions for lower valued bidders, but with these bids typically below the value of the next highest valued bidder. 54

2.2. Theoretical framework 47

There are three bidders and two advertising positions, with CTRs c1 and c2, where c1 > c2.

Each bidder submits a single per-click bid, with these bids ranked from highest to lowest.

The bidder with the kth highest bid wins the kth highest slot and pays the (k+1)th highest bid per click. The bidder with the lowest bid wins nothing and pays nothing. To streamline notation, renumber the bidders in the decreasing order of bids so that b1>b2>b3 and let vk be the per-click value of the bidder assigned ad-slot k. The payoff for getting

th the k highest slot is ck × (vk – bk+1)

Definition 2.1. A Nash equilibrium of the GSP auction is a bid profile b = (b1,b2 b3) that satisfies the following inequalities

c1(v1-b2) ≥ c2 (v1-b3)

≥ 0

c2(v2-b3) ≥ c1 (v2-b1)

≥ 0

48 0 ≥ (v3-b2) c2

The first two conditions mean the highest bidder has no incentive to deviate to win the bottom position or to lose all positions. The next two conditions mean the second-highest bidder has no incentive to deviate to win the top position or to lose all positions. The last

47 The results here are based on Edelman et al. (2007). Also see Varian (2007). 48 This implies 0 ≥ (v3-b1) c1 55

condition means the lowest bidder has no incentive to bid high enough to get one of the two ad-slots. There will typically be a range of bids that satisfy these inequalities, and that these allocations need not be efficient (i.e., assortative). These two properties are not surprising since the Nash equilibrium for single unit second-price auctions (SPA) has the same properties under complete information.49 However, the GSP auction has a clear disadvantage compared to an SPA as truthful bidding is not a dominant strategy.50

Definition 2.2. A bid profile b = (b1,b2 b3) is a locally envy-free equilibrium (LEFE) if it satisfies the following inequalities

c1(v1-b2) ≥ c2 (v1-b3)

≥ 0

c2(v2-b3) ≥ c1 (v2-b2)

≥ 0

51 0 ≥ (v3-b3) c2

In a NE, bidders may envy a higher position and its associated price, but not so under an

LEFE. For example, suppose that

49 Under complete information, a SPA admits multiple Nash equilibrium, including inefficient equilibria. Imagine a case where a lower value holder bids more than the value of the highest value holder, all others bid 0. This constitutes an inefficient Nash equilibrium in which bidders fail to delete weakly dominated strategies. 50 The GSP auction does not necessarily elicit truthful bids since a bidder may bid below his value to get a lower position at a cheaper price if the CTR of the lower position is high enough (Edelman and Ostrovsky, 2007). Bidding above value is a dominated strategy in the GSP auction. 51 This implies 0 ≥ (v3-b2) c1 56

c1 (v2-b2) ≥ c2 (v2-b3) ≥ c1 (v2-b1)

This can be a NE but not an LEFE. The second-highest bidder envies the highest bidder who gets CTRs c1 and pays b2 per click (left inequality), but has no incentive to increase her bid to get the top position, as she would earn less than her current earnings (right inequality).

52 LEFE have several desirable properties: (i) any LEFE is efficient (v1>v2>v3), and (ii) any LEFE has a stable assignment of ad positions in that no bidder wishes to exchange his ad-slot and payment with another bidder’s ad-slot and their payment.

Definition 2.3. An LEFE with the lowest bids is obtained by reclusively choosing the lowest bid that satisfies the LEFE.

b3 = v3

b2 = (1-c2/c1) v2 + (c2/c1) v3

53 b1 > b2

Among the set of LEFE, Edelman et al. (2007) suggest that the lowest LEFE is the most plausible outcome, where allocations and payments coincide with the dominant bidding strategies in a VCG auction. 54 Hence the name for this equilibrium – a VCG-like

52 However, not all efficient equilibria belong to the LEFE. The LEFE is a small subset of the efficient equilibria. 53 Note that b1 does not appear in any conditions of LEFE. Thus, the only condition for b1 is to be greater than b2 as assumed. 54 Edelman et al. (2007) showed that the Lowest LEFE is the unique perfect Bayesian equilibrium in an ascending clock version of GSP auction. 57

equilibrium. The VCG-like equilibrium corresponds to a socially efficient allocation of ad-slots despite the fact that truthful bidding is not a dominant strategy.

Note the VCG-like equilibrium predicts a certain amount of bid shaving for the second highest bidder (b2 < v2). However, the amount of bidding shaving depends critically on the ratio of c2/c1, with the amount of bid shaving increasing as c2/c1 increases. This comparative static prediction is used to determine the experimental treatments – the two

CTR sets (c1, c2) = (11, 3) and (11, 10).

While in general there will be sharp price cutting under 11-10 compared to 11-3, there is one complicating factor in maintaining these contrasting predictions across random realizations of click values. For example, if v2 is close to v3, there would not be much bid shaving in the 11-10 treatment, contrary to the predicted outcome. On the other hand, if v2 is much higher than v3, the amount of bid shaving will be substantial even with 11-3, where minimal bid shaving is predicted. The relative closeness between v2, v3 needs to be

3 푣1−푣2 10 restricted by < < to maintain the contrasting prediction between the two 11 푣1−푣3 11

푣1−푣2 treatments (see the appendix B.1).55 Specifically, we use a fixed ratio = 0.58 to 푣1−푣3 maintain these contrasting predictions between the two CTRs throughout the experiment.56 Keeping the same ratio in values also enables normalizing the auction

55 k th We use v to denote k highest value. This coincides with vk in an LEFE, which is efficient. Since the two, however, in general could be different, we use vk whenever the order of values needs to be emphasized. 푣1−푣2 56 Since the values in the experiment are integers, they cannot exactly satisfy = 0.58. So 푣1−푣3 푣1−푣2 that draws satisfied 0.57 < < 0.59. 푣1−푣3 58

outcomes across auctions, which enables comparing outcomes across auctions as if subjects played the same auction.

While theorists’ focus on a VCG-like equilibrium being achieved under SC, a key question is the extent to which this can be achieved, over time, with DI. Appropriate levels of bid shaving on the part of the mid-value bidder are critical to achieving a VCG- like equilibrium. Assuming that bidders do not use dominated strategies (i.e., do not bid above their values), given the normalization in the ratio of values employed, mid-value bids satisfying an LEFE are bounded as follows for 11-3 and 11-10, respectively:

8 3 푣 + 푣 ≤ 푏 ≤ 푣 11 2 11 3 2 2 1 10 1 10 푣 + 푣 ≤ 푏 ≤ 푣 + 푣 11 2 11 3 2 11 1 11 3

Where in both cases the lower bound corresponds to the VCG-like equilibrium. In what follows we look at whether mid-value bids satisfy these inequalities for achieving an

LEFE in undominated strategies (which will be referred to as LEFEU) and the degree to which both SC and DI auctions correspond to the VCG-like outcome.

2.3. Experimental design and procedures

Our experiment considers GSP auctions where three bidders compete for two ad-slots.

The number of bidders and the ad-slots employed here are minimal, yet large enough to investigate the question of whether outcomes correspond to a VCG-like equilibrium or other equilibrium outcomes. In experiments it is important to start with the minimum

design that captures the essential behavioral questions under investigation. Two ad-slots and three bidders serve to do this.

Each experimental session started with 10 static complete information (SC) auctions, with new values randomly drawn from the interval [1, 100] in each auction.57 These were followed by 8 dynamic incomplete information (DI) auctions with 10 rounds per auction, with new values randomly drawn from the interval [1, 100]. CTRs remained the same in each experimental session – 11-3 or 11-10. Instructions for the SC and DI parts were read separately prior to the start of the treatment in question. Subjects were randomly reassigned to three-person groups prior to the start of each auction.

In the SC auctions information regarding all three bidders’ valuations were posted on subjects’ screens, with each bidder submitting a single bid for the two ad-slots (named bundles A and B), with the number of items (the CTR value for each bundle), displayed at their computer screens. Bids were ranked from highest to lowest, with the highest bidder getting the larger of the two bundles, with the second-highest bidder getting the smaller bundle. Payoffs in each bundle were equal to the number of items in the bundle multiplied the winners’ unit value, at a cost equal to the number of items in the bundle multiplied by the next highest unit-value bid. The lowest value bidder got no items and paid nothing. This was explained to subjects using Figure 2.1.

57 These draws, done in advance, were repeated until three values satisfied the ratio on valuations discussed earlier. 60

Figure 2.1. Payoff table used in the instructions Bidder Ranked Unit Unit Bundle Payoff Bids Valuations Prices Earned 2 ub2 V2 ub1 A (11) (V2 - ub1) x 11= xx 1 ub1 V1 ub3 B (3) (V1 - ub3) x 3 =zz 3 ub3 V3 - - -

Feedback following each auction consisted of a table reporting values, bids, prices paid per unit, the bundle assignment and earnings of all bidders. Subjects were provided with an on-screen calculator where they could enter what they believed others would bid and their own bid, which would calculate their expected rank and earnings. Subjects could use the calculator as many times as they wanted within a 90 second time interval for calculating.58

The DI auctions followed similar procedures in each of the 10 rounds, except that bidders only knew their own valuations, and payoffs were computed based on one, randomly drawn round of the auction. Subjects had 60 seconds to work the calculator, composing as many hypothetical scenarios as time permitted. Feedback following each auction round consisted of a table reporting back bids, prices paid per unit, the bundle assignment and what own earnings would be if that was the payoff round. 59

Earnings were in terms of experimental currency units (ECUs). Subjects were provided with starting capital balances of 500 ECUs, with earnings from each auction added to, or

58 See http://econ.ohio-state.edu/kagel/Insts%20with%20screenshots.pdf for a complete set of instructions and screen shots. 59 Not bidders’ valuations in the last round of the DI auctions. 61

subtracted, from this. For the SC auctions earnings were converted into dollars at the rate of rate of 200ECUs = $1 in 11-3 treatment and 320ECUs in the 11-10 treatment, the latter on account of the higher CTRs in 11-10. Earnings for the randomly chosen round for payment in the DI auctions were converted into dollars at a higher rate to compensate for the fact that they were paid for one of the 10 rounds. Earnings averaged $24.19 per subject for sessions lasting 2 hours.

The experiment was run in the Ohio State University Experimental Economics

Laboratory between March 2017 and April 2017. Subjects in the experiment were primarily undergraduate students drawn from all disciplines and recruited through

ORSEE (Greiner, 2004). Each subject participated in single experimental session. The experiment was computerized, programmed using z-Tree (Fischbacher, 2007). There were three sessions with the 11-3 treatment and three with the 11-10 treatment. Sessions were run with between 12 and 24 subjects in each session.60

2.4. Predicted Outcomes and Propositions to be Investigated

Based on the fixed ratio between click values (v1-v2)/(v1-v3) = 58%, predicted outcomes can be discussed in terms of the following normalized valuations - v1 =100, v2 = 42, and v3 = 0. Normalized bids are used since otherwise results can be misleading.61 Using

60 11-3 sessions had 24, 12, and 18 subjects respectively, and 11-10 sessions had 24, 15, and 15 subjects. 61 For example, consider two realization of values (100, 42, 0) and (49, 42, 37). In the first case, the VCG-like prediction for the mid-value bidder under 11-3 treatment is 30.5, 40.6 in the second case. So if the mid-value bidder bids his own value 42, the percentage deviation is 31.7% in the first case and 3.4% in the second case. With normalization, the percentage deviation is the same, 31.7%. 62

these normalized valuations in place of the general restrictions on the bounds for mid- value bids constituting on LEFEU, are reduced to:

30.5 ≤ 푏2 ≤ 42 for 11-3 and

3.81 ≤ 푏2 ≤ 9.09 for 11-10.

Several observations are worth noting with respect to these bounds: First, for 11-3 the upper bound for mid-value bids corresponds to value bidding, which may also serve as a focal point. In contrast, for 11-10 the upper bound involves substantial price shaving, cutting bids just over 75% relative to v2. Second, for 11-10, the lower bound on bids, the

VCG-like bid, is just above the normalized value for v3, whereas it is well above that for

11-3. Among other things, this means that mid-value bids will be much more sensitive to low-value bids above value, as would be anticipated based on single unit second-price auctions.62 Third, for 11-3 the range of bids satisfying these bounds is twice that of 11-10.

Based on the above observations we expect to see substantially greater price cutting under 11-10 compared to 11-3. And that deviations from the VCG-like equilibrium will be greater for 11-10 than 11-3, along with greater deviations from the NE and the LEFEU equilibria.

62 See footnote 43 for references 63

2.5. Experimental results

2.5.1. Efficiency and revenue

Table 2.1 reports average efficiency across treatments along with the frequency of fully efficient outcomes. For the DI auctions average efficiency is reported for the first, last and all rounds.63 Several things stand out: First average efficiency is significantly higher under 11-3 compared to 11-10, under both SC and DI. Under 11-3, the average efficiency is remarkably high, averaging over 90% for both SC and DI treatments. In contrast, average efficiency under 11-10 is far less than fully efficient in both cases, ranging between 73.5%-74.7%. Similar differences hold with respect rank order efficiency – the frequency with which bidders with valuations 1, 2 and 3, obtained CTRs ranked 1, 2, and

3 respectively. As will be shown below, these differences are a result of the sharp price cutting over time in 11-10 compared to 11-3. With the downward adjustment of prices in over time in 11-10, it is not uncommon for the v2 bidder to displace the v1 bidder, along with sporadic bidding above value on the part of the v3 displacing v1 and/or v2.

63 Average efficiency is defined as (Sactual – Srandom)/ (Smax-Srandom), where Sactual is the actual realized surplus from the auction, Srandom is the mean surplus for all possible allocations, and Smax is the maximum possible surplus. 64

Table 2.1. Efficiency Frequency of rank CTRs Information Average efficiency order efficiency SC 91.7% (2.91) 86.8% (2.85) DI (Round 1) 84.6% (3.16) 78.4% (3.43) 11-3 DI (All Rounds) 92.0% (2.19) 86.7% (2.82) DI (Round 10) 94.1% (1.63) 88.9% (2.62) SC 73.5% (3.88) 56.7% (3.69) DI (Round 1) 71.4% (4.46) 62.5% (4.04) 11-10 DI (All Rounds) 75.8% (4.17) 57.9% (4.13) DI (Round 10) 74.7% (5.67) 54.8% (4.17) (standard errors of the mean in parentheses)

Not reported are tests for differences in efficiency between the SC and DI treatments.

These show that for 11-3, both average and rank order efficiency are significantly lower in the first round of DI (p < 0.01) than under SC, as might be expected given the absence of information about bidders’ valuations. But there are no significant differences between the two averaged over all DI rounds or DI in round 10. For 11-10, the only significant difference is that rank order efficiency is marginally higher (p < 0.10) in the first round of DI bidding compared to SC. However, there are no significant differences in rank order efficiency or average efficiency between the two averaged over all rounds of DI, in DI round 10.

One reference point against which to judge these efficiency outcomes is to compare them to outcomes for budget constrained, zero intelligence (ZI) bidders (Gode and Sunder,

1993) who bid randomly between 0 and their valuations. Outcomes for ZI bidders have been shown to track efficiency outcomes quite well in continuous double auction experiments. ZI average efficiency values are 55.4% under 11-3 and 53.6% under 11-10,

both well below the levels actually achieved.64 ZI rank order efficiency averages 47.9% for both CTRs, well below the 11-3 levels, but only marginally lower than under 11-10.65

Conclusion 2.1: Efficiency levels are substantially higher under 11-3 compared to 11-10, which is expected given the sharp price cutting over time with 11-10 compared to 11-3. As reported on below, the sharp price cutting under 11-10 results in some reversals of winning bids relative to valuations. Minimal price cutting under 11-3 precludes this. There are no significant differences between efficiency in the last round with DI compared to the SC for both 11-3 and 11-10. Average efficiency is substantially higher than would have been achieved with zero intelligence bidders.

Figure 2.2 shows the frequency of truthful, under and over bidding relative to valuations under SC (left panel) and for the last round under DI (right panel). With SC, the frequency of truthful bidding for both high and mid-value bidders is substantially higher under 11-3 compared to 11-10. These differences are even greater in the last round for

DI. As shown below, under 11-10 bid shaving increases over time under DI, as bidders adjust their bids based on information from previous rounds and competition based on previous rounds bids. The limited bid shaving under 11-3 results in efficiency close to the

100% level reported in Table 2.1. In contrast, bid shaving under 11-10 often results in an inefficient allocation as high-value bidders try to under-cut mid-value bidders, as a result of strategic uncertainty about their rival’s action, as well as their rivals valuations.

64 Simulations employed 10,000 observations per auction. 65 The CTRs do not affect the frequency of fully efficient outcomes as it only considers the order of bids. 66

Although bidding above value is a dominated strategy, it is present, to some extent, for all valuations, consistent with the non-negligible frequency of bidding above value in single unit second-price auctions. Bidding above value for low-value bidders is more common under DI than SC, and tends to be more prevalent for 11-10.

Figure 2.3 reports the percent of each bundle’s allocation relative to valuations. For 11-3 bidders’ valuations are highly correlated with bundles obtained for both SC and DI. For example, under SC, the frequency with which bundles A, B, and no-bundle were assigned to the high, mid, and low-value bidders was 94.45, 87.5%, and 88.1%, respectively.

Under 11-10 the results are quite different. In this case, under SC, bundles A, B and no bundle were assigned to high, medium and low bidders 68.9%, 58.9% and 75.6%, respectively, with most of the missed allocations occurring between the high and mid- value bidders. Similar patterns are reported under the DI treatments.

Figure 2.2. Frequency of Truthful, Under and Over Bidding Relative to Valuations.

*Truthful bidding includes bids within ± 2 ECUs

Figure 2.3. Percent of each bundle’s allocation in relationship to valuations

Table 2.2. Percentage Differences between Observed and Predicted VCG-like Revenuea Lower Bound CTR Information (VCG-like revenue) 1.15 SC (2.62) DI 11.7*** Round 1 (2.6) 11-3 DI 12.4*** All Rounds (3.4) DI 12.2*** Last Round (4.0) 61.0*** SC (12.2) DI 50.3*** Round 1 (10.2) 11-10 DI 42.7*** All Rounds (10.9) DI 42.6*** Last Round (13.5) a Mean differences. *** Significantly different from zero at p = 0.01 (standard errors of the mean in parentheses)

Previous studies have focused on the extent to which revenue deviates from a VCG-like equilibrium, particularly with DI information. Table 2.2 reports these deviations in

percentage terms. For the 11-3 SC auctions, average revenue is slightly higher than the

VCG-like equilibrium. However, this does not mean that a VCG-like equilibrium has been achieved. Rather, as the data in figure 2.2 indicates, the VCG-like revenue achieved in this case is a fortuitous combination of value bidding, in conjunction with bidding below and above value. Bidding above value should not be observed in a VCG-like equilibrium. For the 11-3 DI auctions, revenue averages around 12% higher than the

VCG-like equilibrium (p < 0.01), as this fortuitous combination breaks down a bit.

For 11-10 revenue is substantially higher than the VCG like equilibrium under SC, indicating that this should not be taken for granted, as much of theory does. The interesting thing here is that although revenue is still much higher than the VCG like outcome in the last round of DI bidding, it is lower compared to SC, at just under 43% in the last round of DI bidding (p < 0.05 under a two-tailed Mann-Whitney test). While part of this is a result of the close to value bidding in one 11-10 SC sessions (see below), this difference remains statistically significant even after dropping this session (p <0.05).

Conclusion 2.2: Bidding reflects the contrasting predictions with respect to equilibrium outcomes under the two CTRs: Significant price cutting under 11-10 compared to minimal price shaving under 11-3. In both cases, aside from the notable exception of the 11-3 SC sessions, revenues are significantly higher than the VCG- like equilibrium. In 11-3 SC this is a result of fortuitous combination of value bidding, bid shaving, and bidding above value. The latter should not be observed in a true VCG-like equilibrium.

2.5.2. Bidding over Time

Figure 2.4 plots median bids separately for high, mid and low-valued bidders under SC in terms of normalized values, along with the VCG-like prediction for mid-value bids. For

11-3, with experience, high-value bids converge to their normalized value in all three sessions. In two of the three session’s mid-value bids converge close to their values, above the VCG-like outcome. The exception is session 3 where median mid-value bids converge close to the VCG-like equilibrium. Median low-value bids tend to be at or below the zero normalized value. However, a closer look at the data shows that 28.9% of these low value bids were above value, with an average absolute deviation of 24.6 in terms of normalized bids. Obviously, this tends to discourage mid-value bids converging to the VCG-like prediction.

Figure 2.4. Median bids with over time: Static complete information

Results are different for 11-10. In sessions 2 and 4, over time high-value bidders cut their bids substantially, relative to their values, with mid-value bids below their normalized value in session 2, and occasionally so in session 4. The price cutting by high-value bidders results in some reversals of winning positions, with the high-value bidder getting bundle B and the mid-value getting A. Session 6 has a different pattern, with high-value bidders reducing their bids substantially in the first several auctions, only to revert to close to value bidding after that. The apparent reason for this is that in auctions 3 and 4 of session 6, 60% of low value bids were above value, with an average absolute deviation of 32.6 in normalized bids.66 This seems to have spooked high value bidders to cut their bid shaving, reverting to value bidding, taking the pressure off of mid-value bidders to bid much below their values. As shown in Table 2.3 below, low-value bids above value have a strong, statistically significant effect on both high and mid-value bidders for 11-

10, but not so for 11-3.

66 For example, in session 6 subject 12 was assigned the high-value (100) in auctions 2 and 3, submitting bids of 55.3 and 27.9, respectively. However, in auction 3, the bidders assigned value 0, wildly overbid at 202.3, with the mid-value holder bidding 41.9, so the high value bidder was not assigned an ad-slot even though he had the high value. From that auction on, subject 12 reverted to value bidding, rarely deviating from it.

Figure 2.5. Median bids across rounds: Dynamic incomplete information

Figure 2.5 plots median bids across arounds under DI. Unlike with SC, under 11-3 there are only minor deviations from value bidding. One explanation for this is that under SC, complete information about the other bidders’ values leaves room to bid below value without risking adverse consequences. However, under DI, given the incomplete 73

information, there is little room to bid below one’s value, without risking an adverse outcome. This is in line with Gomes and Sweneey (2014) who show that under incomplete information, as the CTR of the second ad-slot gets smaller relative to the first, the Bayesian Nash equilibrium results in bids closer to value.

Under 11-10, there is sharp price cutting on the part of high-value bidders over rounds,with some criss-crossing with mid-value bids over the last 3 rounds. In sessions 2 and 4, there is also sharp price cutting on the part of mid-value bidders, but still above the VCG-like equilibirum, with minimal price cutting in session 6. What is interesting here is that unlike with SC in session 6, under DI high-value bidders engage in sharp price cutting over rounds. This suggests that as information is revealed across rounds, high-value bidders see the opportunity for higher earnings, and are more comfortable going for it. Here too bids above value on the part of low-value bidders result in sharp increases in mid and high-value bids (Table 2.4 below).

Tables 2.3 and 2.4 report regressions for bidding across auctions under SC and across rounds within auctions under DI. As with the figures, the regressions are based on normalized values and the corresponding normalized bids. All regressions employ session dummies that have been normalized to sum to zero. These have been suppressed in the Tables to save space.67 Standard errors are clustered at the subject level and outliers (bids over 300 in normalized bid) were removed. Bidding above value on the part of low-value bidders will no doubt impact both mid and high-value bidders,

67 For 11-3 there are no significant session level effects for SC, and a marginally significant effect (p = 0.08) for DI mid-value bids. For 11-10 there are significant differences (p < 0.05) between intercepts for both high and mid-value bids, for both SC and DI. 74

particularly for 11-10 auctions. Two dummy variables are employed to account for this:

Modest bids above value are defined as bids above value but less than or equal to the median overbid on the part of low-value bidders (wOBt-1). Strong bids above value

(sOBt) are defined as bids above value that are greater than the median overbid. For high

(mid-value) bidders right hand side variables include the lagged mid-value (high-value) bids. A time trend is added to the DI auctions to capture the obvious time trends reported in the Figure 2.5. Bid estimation specification is as follow:

Ht=Constant+wOBt-1+sOBt-1+Mt-1+t (only in DI)+ϵt

Mt=Constant+wOBt-1+sOBt-1+Ht-1+t (only in DI)+ϵt

Table 2.3. Regressions: High and mid-value bids under SC 11-3 treatments 11-10 treatments VARIABLES High Bid Mid Bid High Bid Mid Bid 0.602 0.740 12.37 3.852 wOB t-1 (5.33) (1.76) (8.39) (4.05) 0.524 -1.051 33.85** 26.48*** sOB t-1 (6.93) (8.60) (12.77) (7.44) 0.152 0.047 Mt-1 (0.153) (0.101) -0.000 -0.011 Ht-1 (0.001) (0.017) 93.1*** 37.1*** 58.6*** 32.4*** Constant (6.94) (1.76) (12.3) (3.96) Observations 151 151 154 154 Standard errors, in parentheses, clustered at the subject level. *** p<0.01, ** p<0.05, * p<0.1

For the SC 11-3 sessions (left hand side of Table 2.3) none of the right hand side variables, other than the intercept, are statistically significant, nor are these variables

significant as a whole.68 The constant for the mid-value bidder (37.1) is within the range for LEFE bidding in undominated strategies (29.09-42). In contrast, in the 11-10 SC sessions (right hand side of Table 2.3), strong bids above value in the previous auction prompt higher bids for both high and mid-value bidders. The constant for the high-value bidder (58.6) is well below the value reported for 11-3, consistent with the sharp price discounting compared to 11-3. The constant for mid-value bidders is well above the range for LEFE bidding in undominated strategies – 32.4 compared to a normalized high 9.09.

Table 2.4. Regressions: High and mid-value bids under DI 11-3 treatments 11-10 treatments VARIABLES High Bid Mid Bid High Bid Mid Bid -4.56 0.300 6.416 6.954*** wOB t-1 (4.79) (0.434) (3.96) (1.70) 8.52 1.041* 20.68*** 13.85*** sOB t-1 (12.4) (0.558) (4.10) (1.75) 0.019 0.455*** Mt-1 (0.014) (0.034) 0.008 0.193*** Ht-1 (0.006) (0.007) -0.408 0.025 -2.085*** 0.154 T (0.334) (0.065) (0.553) (0.237) 96.2*** 40.0*** 29.8*** 7.66*** Constant (4.34) (0.75) (4.00) (1.76) Observations 1,280 1,280 1,294 1,294 Standard errors, in parentheses, clustered at the subject level. *** p<0.01, ** p<0.05, * p<0.1

Table 2.4 shows the regression results for the DI treatment. For 11-3, the constants are just below value bidding for both high and mid-value bidders. Strong bids above value have a modest impact on mid-value bids, and both time trends are small and not

68 F values of .88 and .45 for high and mid-value bidders respectively. 76

significant.

11-10 is markedly different. Bids of both high and mid-value bidders are responsive to each other’s past bids, as the two undercut each other over time. They both increase significantly in response to strong bids above value on the part of low-value bidders.

Mid-value bids are also increasing in response to weak bids above value on the part of low-value bidders, as a consequence of the sharp bid shaving on their part.69 The negative time trend for high-value bidders reflects the sharp price cutting over time in efforts to get the second ad-slot at a favorable price. Constants under 11-10 are sharply lower compared to 11-3, as well as to the constants under 11-10 with SC, with the constant for the mid-value bid within the range for an LEFEU.

Note that low-value bidders typically do not bid above v2, doing so 6.1% on average under SC for both treatments, and typically less than this under DI. Bids of this sort are suggestive of rivalrous behavior, but commonly do not result in losses (8.3% and 12.3% of the time under SC for 11-3 and 11-10) with losses, conditional on winning, averaging of $0.65 and $0.45 per auction. Losses were less common under DI (4.7% and 8.3% of the time under 11-3 and 11-10) with losses conditional on winning higher - $1.47 and

$1.04 – reflective of the higher conversion rate to dollars under DI. Rivalrous bidding of this sort has been documented in experiments (Andreoni et al., 2007), as well as in field settings (Cramton, 1997). Nevertheless, these bids clearly discourage sharp discounting of bids for mid-value bidders, as required for an LEFE or VCG-like equilibrium.

69 Of course, high and mid-value bidders do not know how far low bids are above value under DI. But they do know how close these bids were to their own value. 77

Conclusion 2.3: Bidding over time is qualitatively consistent with expectations regarding differences between the 11-3 and 11-10 treatments. For SC there is minimal bid shaving under 11-3 compared to substantial bid shaving for 11-10 sessions (with the notable exception of session 6 in which low-value bidders were bidding substantially above their values). Further, for 11-3 mid-value bid constants for both SC and DI regressions lie within the range of an LEFEU, as value bidding is the upper bound of this interval. For 11-10 under DI there is crisscrossing between mid and high-value bids in later rounds as bidders compete for the second advertising slot at favorable prices. Further, the constant for the mid-value bids lies within the range of an LEFEU, but above the VCG-like equilibrium. The sharp price cutting under DI with 11-10, is a nice example of Bertrand competition with incomplete information.70

2.5.3. Nash Equilibrium Bidding and Best Responding

Although bidding is not consistent with a VCG-like equilibrium, the question addressed here is how often was a Nash equilibria (NE) achieved, along with the costs associated with failing to achieve one. Table 2.5 shows the frequency of bid profiles in the NE in each treatment.71

The frequency with which bid profiles are a NE is substantially higher under 11-3 compared to 11-10, with the absolute frequency quite high (low) for 11-3 (11-10). These results are not surprising given the contrasting equilibrium properties of the two

70 There are no experimental studies of Bertrand pricing with incomplete information that we are aware of. The closest approximation to Bertrand competition over time, with incomplete information, are posted-price double auctions markets, where buyers are usually simulated (see Chapter 4 in Davis and Holt, 1993, for a summary of this literature). These too converge slowly, over time, to the competitive equilibrium. 71 NEs with dominated strategies are included here. They range between 30-39% for both SC and DI auctions, with low-value bidders responsible for the largest share of these above value bids, with the exception of 11-3 SC where high-value bidders bid above value most often. 78

treatments: Under 11-3, value bidding constitutes a Nash equilibrium and serves as a focal point. In contrast, value bidding is not a NE under 11-10 and, since both ad-slots have almost the same value, there is a coordination issue in achieving a NE. The net result is that the set of NE is much smaller under 11-10.

Table 2.5. Frequency of bid profile in Nash equilibria CTRs Information Frequency of NE # of observation 11-3 SC 76.1% 180 DI (All periods) 76.3% 1440 DI (last period) 73.6% 144 11-10 SC 15.0% 180 DI (All periods) 13.8% 1440 DI (last period) 18.0% 144

A more interesting question is whether bid profiles evolve to a NE with repetition, particularly in the DI treatment. Figures 2.6 reports this for SC and DI auctions. Since under 11-10 the NE set is quite narrow, in both cases the frequency with which bid profiles are close to the NE is also reported (close as measured by within a 5 ECU or a 10

ECU radius of the NE).

For both SC and DI there is not much of an increase in NE over time for 11-3 as NE are relatively high to begin with. For 11-10 the frequency of NE increases from the first couple of SC auctions to later auctions, with some dips along the way, it being hard for inexperienced bidders to coordinate to achieve a NE equilibrium, even in a one-shot SC game. However, with experience bidders learn the equilibrium structure of the 11-10 auctions, resulting in a noticeable increase when accounting for either 5 or 10ECU miscalculations. For 11-10 DI auctions the frequency of NE increases monotonically over

rounds within an auction, as bidders utilize information based on prior rounds bids. This increase is particularly striking for both the 5 ECU and 10 ECU bands, to the point that in the last round of bidding, the 10 ECU band achieves a NE around 60% of the time.

Figure 2.6. Frequency of Nash Equilibria over Time: SC (top panel) and DI (bottom panel)

*11-10 auctions also report outcome within a 5 ECU or a 10 ECU radius of the NE.

Table 2.6 shows profits of bidders relative to best responding. Under 11-3, bid profiles mostly constitute a NE, so that losses relative to best responding are quite small - $0.05 for SC and $0.13 in the last round with DI. Losses relative to best responding are higher with 11-10 as a consequence of the coordination issues inherent in that treatment,

averaging $0.20 with SC and $0.50 in the last round with DI.72 However, given the large standard errors, the difference in percentage losses between SC and the last round of DI are not significant for both CTRs.

Table 2.6. Average Losses Relative to Best Responding CTR 11-3 CTR 11-10 Losses: Losses: Profit if Percent Profit if Percent Round dollar dollar BR loss BR loss amount amount $0.17 $1.88 $0.70 $2.57 DI 1 9.0% 27.3% (0.12) (0.97) (0.35) (1.42) $0.11 $1.82 $0.54 $2.26 2 6.2% 24.0% (0.11) (1.00) (0.25) (1.32) $0.10 $1.80 $0.45 $2.34 3 5.3% 19.2% (0.03) (1.05) (0.21) (1.39) $0.19 $1.93 $0.53 $2.36 4 9.6% 22.4% (0.08) (1.17) (0.32) (1.49) $0.13 $1.90 $0.52 $2.34 5 7.0% 22.4% (0.05) (1.16) (0.30) (1.45) $0.17 $1.98 $0.52 $2.37 6 8.5% 21.9% (0.07) (1.26) (0.30) (1.51) $0.10 $2.05 $0.45 $2.41 7 4.8% 18.6% (0.03) (1.26) (0.29) (1.56) $0.13 $1.94 $0.49 $2.42 8 6.8% 20.1% (0.06) (1.15) (0.29) (1.57) $0.13 $2.02 $0.41 $2.35 9 6.4% 17.6% (0.05) (1.20) (0.21) (1.51) $0.13 $1.99 $0.50 $2.33 10 6.5% 21.4% (0.04) (1.15) (0.35) (1.48) $0.05 $0.73 $0.20 $0.79 SC 7.3% 24.8% (0.03) (0.37) (0.08) (0.40) (standard errors in parentheses)

Much of the literature on GSP auctions focuses on the VCG-like equilibrium for SC auctions although in practice GSP auctions involve dynamic incomplete information.

72 Note that absolute costs and profits are higher for DI compared to SC since the conversion rate from ECUs to dollars was tripled under DI to compensate that payment was only for 1 out of 10 rounds of bidding. 81

The idea being interactions between competitors will converge to a VCG-like equilibrium. Table 2.2, reported earlier, showed that there were significant deviations from VCG-like with respect to revenue under 11-10 for both SC and DI auctions, as well as for 11-3 auctions with DI. What follows extends this investigation of VCG-like outcomes, in terms of the behavior of mid-value bidders.

Table 2.7 shows the extent to which median bids of mid-value bidders’ deviate from a

VCG-like equilibrium along with Kolmogorov-Smirnov tests for whether the distribution of outcomes differs between SC outcomes and each round of DI bids. For 11-3, median deviations from the VCG-like outcome are not significantly different under SC compared to any round of DI. This is consistent with the argument for focusing on SC auctions, but outcomes under SC are significantly higher (25.9%) than the VCG-like equilibrium. No doubt the similarity between DI rounds compared to SC happens as value bidding serves as a focal point, and under the normalization employed here, is the same for DI and SC.

In contrast, for 11-10 the VCG-like equilibrium entails bidding close to v3 along with serious price competition between high and mid-value bidders. So in this case the large percentage deviations from value bidding under the VCG-like equilibrium (3.81 compared to 42) are no doubt inhibited by the persistent bidding above value on the part of low-value bidders. However, here too the median percentage deviation between the last round of DI bidding and SC auctions, is consistent with using SC auctions as a model for DI outcomes.73

73 For 11-10 the difference is significant at better than the 5% level, but not terribly different in terms of the economic significance. 82

Table 2.7. Median percentage deviation of mid-value bidders from VCG prediction CTR 11-3 11-10 Rounds Median Std. err. K-S testa Median Std. err. K-S testa % (p-value) % (p-value) 1 27.7 0.01 0.71 165.4 0.01 0.16 2 26.8 0.01 0.62 163.8 0.02 0.32 3 28.3 0.01 0.47 160.9 0.03 0.35 4 28.1 0.01 0.52 155.0 0.04 0.04** Dynamic 5 27.3 0.01 0.68 159.3 0.04 0.13 complete 6 27.2 0.01 0.57 155.9 0.07 0.03** information 7 26.8 0.01 0.81 156.5 0.04 0.01** 8 28.6 0.01 0.11 157.3 0.03 0.05** 9 28.5 0.01 0.52 160.1 0.04 0.05** 10 28.5 0.01 0.22 152.9 0.05 0.04** Static complete 25.9 0.04 - 163.6 0.02 - information a Kolmogorov-Smirnov test for equal distribution of outcomes between each DI round and SC. ** Significantly different at the 5% level. The bootstrap standard errors for median are reported.

Conclusion 2.4: The frequency of Nash equilibrium outcomes with 11-3 is relatively high much lower under 11-10 (around 75% versus15%). In addition, average losses relative to best responding are substantially higher under 11-10 (around 25% versus 7% under 11-3). Both of these differences can largely be accounted for by the fact that value bidding constitutes an NE under 11-3, but entails sharp price cutting, along with coordination issues, under 11-10. Median percentage deviations from the VCG- like equilibrium are substantially lower with 11-3. This can be accounted for by the sharp price discounting under 11-10 in conjunction low-value bids commonly exceeding valuations. However, for both CTRs, these percentage deviations from the VCG-like equilibrium are essentially the same between SC and DI auctions, consistent with the theorists focus on SC auctions as a stand in for DI auctions, which are closer in structure the GSP auctions in field settings. But these results also cast doubt on the relevance of the VCG-like equilibrium as a reference point for auction outcomes.

2.6. Summary and Conclusions

This paper experimentally investigates outcomes under GSP auctions for selling on line ad-slots. There are two main innovations in the paper: (i) using a cross-over design to compare static complete information (SC) auctions to dynamic incomplete information

(DI) auctions so that outcomes are observed for the same set of subjects, and (ii) restricting the relationship between random valuations across auctions designed to maintain the contrasting predictions between the two contrasting click through rates

(CTRs) studied. Consistently high efficiency levels are observed for both CTRs under both SC, and in later rounds of DI auctions. The primary comparative static predictions of the theory are shown to be correct in that there is sharp, Bertrand like, competition between bidders when the value of the CTRs are close in value, but minimal competition when they are not. In addition, in comparing SC and DI auctions, SC outcomes are shown to be quite similar to DI outcomes under both sets of CTRs. This provides empirical support for the common theoretical practice of using behavior in SC auctions to model outcomes in DI auctions, where the latter are closer in structure to GSP auctions outside the lab.

GSP auctions have a large number of Nash equilibria (NE). Around 75% of auction outcomes correspond to a NE with the 11-3 set of CTRs, although there is a fairly high frequency of bidding above value on the part of the high value bidder. Strict NE are observed only about 15% of the time under the 11-10 CTRs. Strict NE are inherently much less likely to be observed under 11-10 due to the strong competition for the second ad-slot at a favorable price. However, this rate is substantially higher and increasing with 84

experience after accounting for modest trembles around the NE. One equilibrium that theorists focus on is the VCG-like equilibrium under an auction mechanism that does not support truthful bidding. However, the experiment shows that VCG-like outcomes are not consistently observed. Part of the reason for this is the tendency for low value bidders to bid above their values, a common outcome in single unit second-price auctions conducted the laboratory, and in the one field experiment using experienced bidders.74 The experimental results also call for new theoretical approaches that incorporate behavioral elements studying GSP auctions.

In the GSP auction model studied here (two ad-slots and three bidders), bidding behavior is determined by the CTR ratio between the top and the bottom slot (c2/c1). With more ad-slots and more bidders, outcomes will be determined by the more complicated interactions between the different CTRs. For example, when four bidders compete for three ad-slots, the two ratios, c2/c1 and c3/c2, will both affect bidding. If the CTRs are all close, for example 11-10-9, so that both ratios, c2/c1 and c3/c2, are high, there will be strong price cutting behavior between all three, similar to what was observed with 11-10.

However, suppose the top two slots are close in value, with the third much lower, say 11-

10-3, so that c2/c1 is high, but c3/c2 is low. The two top bidders will be involved in price cutting to get the second position, while the others will bid close to their valuations.

Whether these predictions will hold, remains to be investigated, along with more complicated relations; e.g., where one bidder (say Amazon) has such name brand

74 See footnote 43 for references. 85

recognition that their CTR is inherently high, with ad placement near the top of little marginal value.

Chapter 3. A Pro-Competitive Effect of Joint Bidding in Multi-Unit Uniform Price

Auction with Asymmetric Bidders

3.1. Introduction

Studies on anti- and pro-competitive effects of joint bidding have been a subject for public policy debates over the past few decades. Ever since Markham (1970), researchers have been concerned about the anti-competitive effects of joint bidding, as it reduces the number of bidders, thus reducing competition. However, pro-competitive arguments of joint bidding have been reported as well. First, joint bidding serves as a vehicle for circumventing capital constraints, thus encouraging entry by small bidders (Hendricks and Porter, 1992). Second, when auctioned objects are risky, joint bidding allows bidders to share the risk, leading to more aggressive bidding (Millsaps and Ott, 1985). Third, in common value auctions, joint bidding allows bidders to pool their information to have more precise estimates of the value of the item, thus mitigating the winner’s curse, leading to more aggressive bids (DeBrock and Smith, 1983; Krishna and Morgan, 1997).

In this paper, I demonstrate a pro-competitive effect of joint bidding based on mitigating demand reduction rather than capital constraint, risk, and information pooling.

Previous studies showing pro-competitive effects of joint bidding focus on single-unit auctions. In contrast, the current paper reports a pro-competitive effect of joint bidding in

multi-unit uniform price auctions where bidders have private values and demand different numbers of units. I consider a uniform price auction where three identical items are sold, and a big bidder demands two units and two small bidders each demand one unit. In equilibrium, the small bidders submit bids equal to their values and the big bidder reduces demand for his second unit to get the first unit at a cheaper price. Now suppose the two small bidders merge and bid jointly75. The merger has two effects: first, the merged bidders now also reduce demand for their second unit, which adversely affects the seller’s revenue. Second, their reduction in demand for the second unit encourages the big bidder to bid more for his 1So that they behave as a single bidder demanding up to two units. second unit, as he now has a chance of winning the second unit at an advantageous price. When the second effect dominates the first effect, it increases the clearing price and the seller’s revenue.

The asymmetry among bidders in the model plays an important role as the pro-competitive effect of joint bidding is attributed to the fact that the joint bidding alters an asymmetric competition structure into a symmetric one. Thus, joint bidding among symmetric bidders would not have such a pro-competitive effect. Levin (2004) analyzes the competitive effect of joint bidding of symmetric bidders who originally demand a single unit then, after they are randomly paired up, demand up to two units with joint bidding. In his model, no demand reduction exists without joint bidding so joint bidding only increases monopsony power and strengthens anti-competitive behavior.

75 So that they behave as a single bidder demanding up to two units. 88

3.2. The Model

Suppose a seller auctions off three identical items in a uniform price auction and there are three bidders 1,2 and 3. Bidder 1 demands up to two units (big bidder) and the other bidders each demand a single unit (small bidder), so the total demand is four units. All valuations for the items are independently drawn from a CDF, F(v), whose support is [0,

1]. Bidder 1’s valuations are {푣1̂ , 푣1} where 푣1̂ ≥ 푣1 (note that bidder 1’s valuations are two independent random draws from 퐹(푣)) and bidder 2 and 3’s valuations are v2 and v3, respectively.

3.2.1. Without Joint Bidding

̂ 퐼 퐼 ̂ 퐼 퐼 In the auction, bidder 1 submits bids { 푏1, 푏1} where 푏1 ≥푏1 and bidder 2 and 3

퐼 퐼 76 respectively submit 푏2 and 푏3. Without joint bidding, the three highest bids win the items and the lowest bid sets the price. Bidders 2 and 3 have a (weakly) dominant

퐼 퐼 strategy of bidding their values (푏2(푣2) = 푣2, 푏3(푣3) = 푣3). Bidder 1 is guaranteed to get his first unit, so the question is how much he bids on the second unit. Thus, the

퐼 equilibrium is characterized by the optimal bid function 푏1, given the other bidders

̂ 퐼 submit bids equal to their values. Since bidder 1 wins two units when 푏1 > 푚𝑖푛(푣2, 푣3) and one unit otherwise, it is convenient to define 푣푙 = 푚𝑖푛(푣2, 푣3). Since 푣2 and 푣3 are

76 Superscript I means bids without joint bidding (Individually bidding)

2 independent, 푣푙 is a random draw from a CDF, 퐻(푣) = 1 − (1 − 퐹(푣)) . The expected

퐼 payoff of bidder 1 who bids 푏1 for his second unit is

퐼 푏1 퐼 퐼 퐼 훱(푏1; 푣̂1, 푣1) = 푣̂1 + ∫ (푣1 − 2푣푙)ℎ(푣푙)푑푣푙 + (−푏1)(1 − 퐻(푏1)) 0

퐼 By differentiating the expected utility with respect to 푏1, I get the following necessary

퐼 condition for the optimal bid 푏1(푣1):

푑훱 퐼 퐼 퐼 퐼 = (푣1 − 푏1)ℎ(푏1) − [1 − 퐻(푏1)] ≤ 0 (1) 푑푏1

퐼 with strict inequality only if the optimal bid is the corner solution 푏1(푣1) = 0.

From the necessary condition, it is clear that bidding the value of the second unit is never

푑훱 optimal ( 퐼 |푏퐼=푣 < 0), so bidder 1 must exercise demand reduction for his second unit. 푑푏1 1 1

3.2.2. With Joint Bidding

Now suppose bidders 2 and 3 merge into a bid consortium and bid jointly. The bid consortium demands up to two units and the valuations for the two units follow the original valuations of the two bidders {푣2, 푣3}. For notational convenience, I denote the higher one of the two values 푣̂푐 and the lower one 푣푐, so the valuations of the bid consortium are 푣̂푐, 푣푐, where 푣̂푐 ≥ 푣푐. Note that 푣푐 is a random draw from a

2 CDF, 퐻(푣) = 1 − (1 − 퐹(푣)) since it is the lower of two independent draws {푣2, 푣3} from 퐹(푣).

̂ 퐽 ̂ 퐽 ̂ 퐽 ̂ 퐽 In the auction, bidder 1 submits bids {푏1 , 푏1 } where 푏1 ≥ 푏1 and the bid consortium bids

̂ 퐽 ̂ 퐽 ̂ 퐽 ̂ 퐽 77 푏푐 , 푏푐 where 푏푐 ≥ 푏푐 . The three highest bids win the items and the lowest bid sets the price. Note that now there are two symmetric bidders (bidder 1 and bid consortium) who both demand two units, with each bidder guaranteed to get his first unit. Therefore, a symmetric Bayesian Nash equilibrium is obtained by characterizing a symmetric bid

퐽 퐽 function for the second unit, 푏 (푣). The expected payoff of bidder 1 who bids 푏 (푣1) for his second unit is:

퐽 −1 퐽 (푏 ) (푏1) 퐽 퐽 퐽 퐽 −1 퐽 훱(푏1 , 푣̂1, 푣1) = 푣̂1 + ∫ (푣1 − 2푏 (푣푐))ℎ(푣푐)푑푣푐 + (−푏1 )[1 − 퐻((푏 ) (푏1 ))] 0

퐽 By differentiating the expected payoff with respect to 푏1 and using that it is optimal to bid 푏퐽(푣) at an equilibrium, I get the following necessary condition for an equilibrium bid function.78

퐽 퐽 (푣1 − 푏 (푣_1))ℎ(푣1) − (푏 )′(푣1)(1 − 퐻(푣1)) = 0

Solving this ODE gives the equilibrium bid function, which is a special case of Theorem

6 of Engelbrecht-Wiggans and Kahn (2002)

푣1 ( ) 퐽 1 − 퐻 푣1 푏 (푣1) = 푣1 − ∫ 푑푥 (2) 0 1 − 퐻(푥)

From the bidding function (2), it is clear that bidder 1 and the bid consortium exercise demand reduction. Without joint bidding, only bidder 1 reduces his demand, but with the joint bidding, all bidders reduce their demands. So does this mean that the joint bidding

77 Superscript J denotes bids with Joint bidding 78 Note that 푏퐽(푣) = 0 never occurs with the joint bidding. If 푏퐽(푣) = 0 for any interval, then bidding slightly more than 0 will break the tie and ensures the second unit, breaking the equilibrium 91

reduce the seller’s revenue? The answer is not necessarily. Demand reduction of the merged bidders mitigate big bidder’s incentive to submit a low bid for the second unit as the big bidder now have a chance of winning the second unit at a cheaper price. I will show an example where the joint bidding mitigates the demand reduction of bidders 1 (i.e

퐼 퐽 푏1(푣1) < 푏 (푣1)) and the seller’s revenue increases.

3.2.3. An Example of the Pro-Competitive Effect of Joint Bidding

Example 3.1. When 퐹(푣) = 1 − √1 − 푣, so 퐻(푣) = 푣, bidder 1’s optimal bid without

퐼 퐽 joint bidding is 푏1(푣1) = 0 and the optimal bid with the joint bidding is 푏 (푣1) = 푣1 −

1 (1 − 푣1)푙푛 . Therefore, the seller’s revenue is greater with the joint bidding. 1−푣1

퐼 To see this, note that the optimal bid of bidder 1 without the joint bidding, 푏1(푣1), is characterized by (1):

푑훱 퐼 퐼 퐼 퐼 퐼 퐼 = (푣1 − 푏1)ℎ(푏1) − [1 − 퐻(푏1)] = (푣1 − 푏1) − (1 − 푏1) = 푣1 − 1 < 0, 푑푏1

for all 푣1 ∈ [0,1)

퐼 퐽 1 Thus, 푏1(푣1) = 0. On the other hand, 푏 (푣1) = 푣1 − (1 − 푣1)푙푛 > 0 is obtained by 1−푣1

(2).

Without the bid consortium, bidder 1 completely reduces his demand for his second unit, setting the clearing price at 0, hence the seller earns nothing. Since the two small bidders submit bids equal to their valuations and their valuations are likely to be high (퐹(푣) has 92

higher probability density at higher values), it is more profitable for bidder 1 to avoid the competition for his second unit with the small bidders and set a lower price for his first item. However, with joint bidding, bidder 1 finds it worth submitting a positive bid for his second unit. This is because the demand reduction of the bid consortium for their second unit gives a chance for a bidder 1 to get his second unit at a favorable price.

Example 3.2. Suppose 퐹(푣) = 푣, so 퐻(푣) = 1 − (1 − 푣)2 and ℎ(푥) = 2(1 − 푥).

퐼 퐼 퐽 2 푏1(푣1) = 0 when 푣1 < 0.5 and 푏1(푣1) = 2푣1 − 1 when 푣1 ≥ 0.5 while 푏1 (푣1) = 푣1 .

퐼 퐽 Note that 푏1(푣1) < 푏1 (푣1) as in figure 3.1, which means the bidder 1 submit higher bid after bidder 2 and 3 merged. The seller’s expected revenue increases from 0.15 to 0.2 with the merger.

Figure 3.1 Bid functions

Proof. See the Appendix C.1 93

퐼 Proposition 3.1. If 퐻(푣) ≤ 푣 (graphically, 퐻(푣) is below 45-degree line), then 푏1(푣1) =

0 for all 푣 .

Proof. See the Appendix C.2

Intuitively, 퐻(푣) < 푣 implies that, in expectation, a random draw from 퐻(푣) is likely to

1 79 be high as 퐸(푣) = ∫0 [1 − 퐻(푣)]푑푣. Thus, bidder 1 is likely to face rivals with high values. In the non-merger case, bidder 2 and 3 bid their values. Therefore, bidder 1 can better off by giving up competing for the second unit and set the price at 0 regardless of the realized value of bidder 1.

Corollary 3.2. If 퐻(푏) ≤ 푏, the seller’s revenue under non-merger is 0. Thus, the seller’s revenue unambiguously increases with the merger.

푎 퐼 Example 3.3. When 퐻(푣) = 푣 and 푎 ≥ 1, 푏1(푣1) = 0. On the contrary, when 퐻(푣) =

푎 퐼 푣 and 푎 < 1, 푏1(푣1) > 0 for all 푣 > 0.

Proof. By proposition 3.1, the former statement is true. For the latter one, see the appendix C.3

79 For non-negative random variable, integration above a CDF is the expected value. 94

3.3. Efficiency, Incentive to Merge, Generalizability

In this model, the efficiency of the auction unambiguously increases with the joint bidding of the small bidders. Without joint bidding, the big bidder avoids competition for his second unit, occasionally losing the second unit even though his value for that unit is greater than the values of bidder 2 or 3. With joint bidding, the bidder with a higher value for the second unit always wins that unit, resulting in increased efficiency. The increased efficiency in this model implies that a properly designed merger that recovers symmetry among bidders can increase social welfare as well as the seller’s revenue.

A limitation of this model is that the smaller bidders may not have an incentive to bid jointly as it may reduce their profits. For instance, in example 1, the interim expected profits of smaller bidders given the valuations are 푣̂푐 and 푣푐, while the expected joint

1 profit of bidding jointly is 푣̂ + 푣2.80 However, if participating in the auction is costly, 푐 2 푐 the smaller bidders can have an incentive to bid jointly. Suppose the participating cost is

푣푐. Then without joint bidding, both small bidders enter the auction and the expected profits are 푣̂푐 − 푣푐 and 0. With joint bidding, on the contrary, the bid consortium’s

1 expected joint profit is 푣̂ + 푣2 − 푣 , which is greater than the sum of profits in 푐 2 푐 푐 individual bidding.

Future research can investigate under what conditions that the pro-competitive effect holds; comparing expected revenue of the seller with and without joint bidding is a

80 Recall that the higher of {푣2, 푣3} is 푣̂푐 and the lower is 푣푐 95

challenging task due to the asymmetry of the bidders and the implicit form of the bid

퐼 function 푏1(푣1)

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Appendix A. Chapter 1 Appendix

A.1. Characterization of an equilibrium in UPA

Let 휷푈푃퐴(푣, 푤) = (훽1(푣, 푤), 훽2(푣, 푤), … , 훽푚(푣, 푤)) denote a symmetric equilibrium of

1 푚 1 푚 1 푚 1 푚 UPA. Let 퐜−퐢 = (훽1 , … , 훽1 , … , 훽푖−1, … , 훽푖−1, 훽푖+1, … , 훽푖+1, … , 훽푛, … , 훽푛 ) be the competing bids facing bidder i and the distribution of the random variable 푪−퐢 has a

푘 density given by ℎ(풄). Then, let 푐−푖 be kth highest element among 풄−푖 . An equilibrium in UPA is acquired by solving the following equation.

푈푃퐴 1 휷 (푣푖, 푤) = argmax ∫ 푚(푣푖 − 푐푖 )ℎ(풄)풅풄 1 푚 1 푚 푚 푘 1 푚 (푏 ,…,푏 )∈{(푏 ,…,푏 )|∑푘=1 푏푖 ≤푤} {풄−푖 : 푐−푖<푏푖 }

2 푚 + ∫ (푚 − 1)(푣푖 − 푚푎푥{ 푐−푖, 푏푖 })ℎ(풄)풅풄 2 푚−1 1 푚 {풄−푖 : 푐−푖<푏푖 푎푛푑 푐−푖> 푏푖 }

+ … ….

푚 2 + ∫ (푣푖 − 푚푎푥{ 푐−푖, 푏푖 })ℎ(풄)풅풄 푚 1 푚−1 2 {풄−푖 : 푐−푖<푏푖 푎푛푑 푐−푖 > 푏푖 }

Similarly, an equilibrium of UPA-M is characterized by substituting (푏1, … , 푏푚) ∈

{(푏1, … , 푏푚)|∀푘 ≤ 푀, 푏푘 ≤ 푤/푘} to the budget constraint restrictions.

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A.2. An example of UPA equilibrium.

A seller sells two equal shares and there are two bidders (1,2). Each bidder has a private per-unit value and the per-unit value is the same for the two shares. Bidder i's private value vi is independently drawn from a uniform distribution [0,1]. Each bidder can submit up to two bids and the third highest bid sets the market price. The two bidders have a common budget constraint w.

1 The following strategy constitutes an equilibrium for any w: bidder i submits bi = min

2 (vi, w) for the first share and submit bi = 0 for the second share. Both bidders each win one share and pay 0. Both bidders do not have an incentive to deviate from this strategy assuming the other bidder sticks to this strategy. For the first bid, it is dominated to bid more than vi and it is not possible to bid more than w if constrained. There is also no incentive to reduce the first bid since reducing it only decreases the chance of getting the first share and the saved budget by reducing the first bid will not be used for the second bid as the second bid is 0. For the second bid, increasing the second bid is not profitable.

If bidder 1 increases his second bid, this surely increases the market price he pays, which directly reduce his payoff. But bidder 1 wins the second share only if he increases his bid

1 second bid enough to beat b2 (the first bid of the second bidder). It can be easily shown that the first effect is greater than the second effect (as in Ausubel et al., 2014). Thus, the strategy profile is an equilibrium.

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A.3. Efficiency and revenue predictions with n=5 and n=10

Table A.1. Benchmark predictions (n=5) FPA PSA UPA UPA-M Efficiency 72.2% 86.2% 72.2% 84.0% w=20 Revenue 20.0 43.2 31.0 44.0 Efficiency 90.3% 86.4% 91.4% 95.8% w=50 Revenue 49.4 43.6 61.7 66.0

Table A.2. Benchmark predictions (n=10) FPA PSA UPA UPA-M Efficiency 65.9% 87.4% 81.0% 84.2% w=20 Revenue 20.0 58.4 50.1 53.7 Efficiency 82.6% 87.4% 94.0% 95.9% w=50 Revenue 50.0 58.5 75.8 79.7

Under n=5, when bidders are strongly budget constrained (w=20), PSA and UPA-M are predicted perform better than FPA and UPA. However, when bidders are weakly budget constrained, both UPA and UPA-M are predicted to perform the best, and FPA is predicted to perform better than PSA. Under n=10, the comparative statics is in general similar to n=5 case except PSA is predicted to achieve higher efficiency and revenue than

FPA even when bidders are weakly budget constrained.

When the number of bidders increases, efficiency and revenue of all share auctions (PSA,

UPA, UPA-M) are predicted to increase. However, efficiency and revenue of FPA are not benefitted by increased number of bidders.

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A.4. Dominated bids in PSA

Proposition 1.2. In PSA, bidding more than 1/4 vi is dominated by bidding 1/4 vi, regardless of n and F(v).

Proof. Let 풃−푖 = (b1, b2, … bi−1, 푏푖, … 푏푛) be bids submitted by bidders except bidder i.

Bidder i’s payoff of bidding 푏푖 given 푣푖 is

푏푖 Π(푏푖, 풃−푖, 푣푖) = 푣푖 − 푏푖 푏푖 + ∑−푖 푏−푖

The marginal payoff is

∂Π(푏푖, 풃−푖, 푣푖) ∑−푖 푏−푖 = 2 푣푖 − 1 ∂푏푖 (푏푖 + ∑−푖 푏−푖)

Since others’ bids affect bidder i’s payoff only through the summation, let

∑−푖 푏−푖 = 푥 for notational convenience. I will show that if 푏푖 > 1/4푣푖, the marginal payoff is always negative regardless of 푥. This means that payoff of bidding any bid 푏푖 >

1/4푣푖 is smaller than bidding 푏푖 = 1/4푣푖

푥 푏푖 1 2 푣푖 − 1 ≤ 2 푣푖 − 1 = 푣푖 − 1 < 0 (푏푖 + 푥) (푏푖 + 푏푖) 4푏푖

The first inequality is holds since the first term is maximized when 푥 = 푏푖. The last inequality holds since 푏푖 > 1/4푣푖.

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A.5. Dominated bids in UPA and UPA-M

Proposition 1.3. In UPA and UPA-M, bidding higher than vi/k for each unit is dominated by bidding vi/k. In UPA-M it is a dominant strategy to bid vi/k for the first unit, while it is no longer a dominant strategy in UPA.

Proof. Bidding higher than vi/k gives more winning units than bidding vi/k, only when the market price is above vi/k, in which case bidder i does not want to get more units. In

UPA-M, bidding less than vi/k for the first unit only reduces the chance of winning the unit but never lower the marker price that bidder i has to pay (if bidders i received any quantity of units, his first bid is never the market price). In UPA, however, it could be profitable to save budget for the first unit by bidding less than vi/k and bids more for later units.

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A.6. Regression analysis for efficiency and revenue

Table A.3. Efficiency and revenue with auction format dummy variables VARIABLES Efficiency Efficiency Revenue Revenue (w=20) (w=50) (w=20) (w=50)

Period -0.001 0.003* 0.030 0.047 (0.002) (0.002) (0.087) (0.210) PSA 0.054*** -0.065*** 9.909*** -8.840*** (0.008) (0.006) (0.493) (0.304) UPA -0.060*** -0.154*** 1.411*** -3.801*** (0.010) (0.012) (0.295) (1.213) UPA-M -0.013 -0.130*** 7.890*** -3.422 (0.008) (0.006) (1.908) (2.331) Constant 0.818*** 0.943*** 19.588*** 46.333*** (0.010) (0.010) (0.487) (1.190)

Observations 770 770 770 770 R-squared 0.057 0.208 0.162 0.029 Robust standard errors in parentheses *** p<0.01, ** p<0.05, * p<0.1

This table shows regression results of efficiency and revenue with dummy variables for auction formats, clustering errors at session level. The base treatment is FPA. While testing differences between dummy variables I used F-test. Negligible size of coefficients on Period variable show that the auction outcomes were stable across different auction periods.

Hypothesis 1.1 predicts clear efficiency and revenue reversal between FPA and PSA under w=20 and w=50 treatments. PSA achieved 5.4% higher efficiency than FPA under w=20 (p<0.01) and 6.5% lower efficiency than FPA under w=50 (p<0.01). When it comes to revenue, PSA achieved 9.9 higher revenue than FPA under w=20 (p<0.01) and

8.84 lower revenue than FPA under w=50 (p<0.01)

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Hypothesis 1.2 predicts PSA performs better than UPA in terms of efficiency and revenue under w=20, but under w=50, UPA achieves similar efficiency and raises higher revenue.

Under w=20, PSA achieved 11.4% higher efficiency (p<0.01) and 8.498 higher revenue than UPA (p<0.01), being consistent with the predictions. Under w=50, PSA achieved

8.9% higher efficiency than UPA (p<0.01), which is against the prediction, and UPA achieved 5.039 higher revenue than PSA (p<0.01), consistent with the prediction.

Hypothesis 1.3 predicts UPA-M uniformly improves the performance of UPA under both w=20 and w=50. Under w=20, UPA-M achieved 4.7% higher efficiency and 6.479 higher revenue. Both differences were statistically significant (p<0.01). Under w=50,

UPA-M achieved 2.4% higher efficiency and 0.379 higher revenue. However, both differences are slight and not significant at 5 percent level (p=0.06, p=0.89). This implies that the improvement of UPA-M due to its relaxed budget constraint enforcement rule is only effective when the budget constraint is stringent.

Hypothesis 1.4 predicts UPA-M achieves similar performances with PSA under w=20 but higher efficiency and revenue under w=50. Against the predictions, PSA achieves significantly higher efficiencies than UPA-M under both w=20 and w=50, the differences being 6.7% under w=20 (p<0.01) and 6.5% under w=50 (p<0.01). When it comes to revenue, PSA raised 2.019 higher revenue than UPA-M under w=20, but the difference was not statistically significant (p=0.326). Under w=50, UPA-M raised 5.418 higher revenue than PSA and the differences were significant at 5 percent level (p=0.040). The last two observations were consistent with the predictions of Hypothesis 1.4.

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Appendix B. Chapter 2 Appendix

B.1. Ratio Values Required to Maintain Contrasting Predictions Between Treatments.

While VCG-like equilibrium predicts bid shaving regardless of realized click values, the strongly contrasting predictions between the 11-10 and 11-3 treatments depend, in part,

10 푣1−푣2 on realized click values. For example, if v2 v3 are so close that ≤ , value bidding 11 푣1−푣3 constitutes a NE under the 11-10 treatment (see below) contrary to the strong bid shaving generally predicted under 11-10. On the other hand, if v2 is relatively high compared to

3 푣1−푣2 than v3 so that ≥ , under 11-3 the high-value bidder has a strong incentive to 11 푣1−푣3 get the second ad-slot at a cheaper price. This incentive is likely to trigger price cutting between the high and mid-value bidder, resulting in considerable bid shaving similar to what is generally expected under the 11-10 treatment. Thus, although a VCG-like equilibrium predicts more bid shaving in 11-10 than 11-3, it is possible to observe the

10 푣1−푣2 3 푣1−푣2 opposite outcome if the realized values are ≤ in 11-10 treatment or ≥ 11 푣1−푣3 11 푣1−푣3

3 푣1−푣2 10 in 11-3 treatment. As such we restrict click values to be ≤ ≤ . 11 푣1−푣3 11

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B.2. Value bidding as a Nash Equilibrium

Proposition 2.1. Given valuations (v1,v2,v3), value bidding constitutes a Nash equilibrium

1 2 푐2 푣 −푣 in GSP auctions iff ≤ 1 3. 푐1 푣 −푣

1 2 푐2 푣 −푣 Proof. (←) Suppose ≤ 1 3 . We claim that value bidding is a NE. When all bidders 푐1 푣 −푣 submit their own values, no bidder has an incentive to bid for the higher ad-slots since doing so results in losses. Now we show that bidders also do not have incentive to underbid to get a lower position. Since all winners earn positive payoff, no winner wants to underbid to be ranked 3rd and get no ad-slot. Thus, the only potentially profitable deviation is that the bidder with the top ad-slot deviates to get the second ad-slot.

However, this deviation is not profitable by the assumption.

1 2 2 3 (푣 − 푣 )푐1 ≥ (푣 − 푣 )푐2

1 2 푐2 푣 −푣 Thus, when ≤ 1 3 , value bidding is a NE. 푐1 푣 −푣

1 2 푐2 푣 −푣 (→) (Proof by contrapositive) Suppose > 1 3 . When all bidders submit bids equal 푐1 푣 −푣 to own value, the bidder with the top ad-slot has an incentive to deviate to get the second ad-slot. Thus, value bidding is not a NE.

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Appendix C. Chapter 3 Appendix

C.1. Proof of Example 3.2.

The necessary condition for optimal bids in non-merger case is (푣 − 푏)ℎ(푏) −

[1 − 퐻(푏)] = 2(푣 − 푏)(1 − 푏) − (1 − b)2 ≤ 0. Since (1−b) > 0, the condition can be re-written 2(푣 − 푏) − (1 − 푏) = 2푣 − 푏 − 1 ≤ 0. For 푣 < 0.5, the inequality holds

퐼 strictly, thus, 푏1(푣1) = 0. For 푣 ≥ 0.5, the condition holds equality when 푏 = 2푣 − 1,

퐼 thus, 푏1(푣) = 2푣1 − 1 (Note that the second derivate is −2(푣 − 푏) < 0, so it is indeed an optimal bid). Let 푅퐼 denotes the revenue under individual bidding.

1 1 1 2푣 −1 2 1 푅 = ∫ ∫ 0ℎ(푣 )ℎ(푣 )푑푣 푑푣 + ∫ ∫ 3푣 ℎ(푣 )ℎ(푣 )푑푣 푑푣 퐼 푙 1 푙 1 1 푙 푙 1 푙 1 0 0 0 2

1 1 + ∫ ∫ 3(2푣 − 1)ℎ(푣 )ℎ(푣 )푑푣 푑푣 = 0 + 0.075 + 0.075 = 0.15 1 1 푙 1 푙 1 2푣 −1 2 1

퐽 2 Let 푅퐽 denotes the revenue under the joint bidding. bid function 푏 (푣1) = 푣1 by (2). Let

푣̃ denotes the lowest value among all bidders.푣̃ is a random variable drawn from 퐻̃(푣̃) =

4 1 − (1 − 퐹(푣̃))

1 2 푅퐽 = ∫ 3(푣̃ )ℎ(푣̃)푑푣̃ = 0.2 0

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C.2. Proof of Proposition 3.1.

From (1), the marginal benefit of increase bidding at 푏 is (푣 − 푏)ℎ(푏) − [1 −

푘 퐻(푏)] ≤ 0. I will show ∫0 (푣 − 푏)ℎ(푏) − [1 − 퐻(푏)]푑푏 < 0 for any positive k such that 0 < k ≤ v, hence bidding 0 is optimal.

claim.

푘 푘 ∫ (푣 − 푏)ℎ(푏) − [1 − 퐻(푏)]푑푏 ≤ ∫ (푣 − 푏) − [1 − 푏]푑푏 < 0 0 0

The second inequality is trivial and −[1 − H(b)] ≤ −[1 − b] by assumption. Now I will

푘( ) ( ) 푘 show ∫0 푣 − 푏 ℎ 푏 푑푏 ≤ ∫0 (푣 − 푏)푑푏.

Begin with the LHS

푘 푘 ∫ (푣 − 푏)ℎ(푏)푑푏 = 푣[퐻(푘) − 퐻(0)] − ∫ 푏ℎ(푏)푑푏 0 0

푘 푘 = 푣퐻(푘) − [푏퐻(푏)]0 + ∫ 퐻(푏)푑푏 0

푘 푘 푘2 = (푣 − 푘)퐻(푘) + ∫ 퐻(푏)푑푏 ≤ (푣 − 푘)푘 + ∫ 푏푑푏 ≤ (푣 − 푘)푘 + 0 0 2

푘2 푘 = 푣푘 − = ∫ (푣 − 푏) 2 0

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C.3. Proof of Example 3.3

푎 퐼 When 퐻(푣) = 푣 and 푎 < 1, the necessary condition for 푏1(푣1) to be non-zero is

푎−1 푎 푎−1 (푣 − 푏)푏 − [1 − 푏 ] = 0 for some 푏 ∈ (0,1]. Since 푙𝑖푚푏→0+푏 = ∞,

푎−1 푎 푎−1 푎 푙𝑖푚푏→0+[(푣 − 푏)푏 − [1 − 푏 ]] = ∞. Plus, 푙𝑖푚푏→푣−[(푣 − 푏)푏 − [1 − 푏 ]] < 0.

Therefore, there must be 푏∗ ∈ (0, 푣) that satisfies [(푣 − 푏∗)(푏∗)푎−1 − [1 − (푏∗)푎]] = 0,

퐼 which implies 푏1(푣1) > 0

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