Efficiency in Auctions and Public Goods Mechanisms

The Pennsylvania State University

The Graduate School

Department of Economics

EFFICIENCY IN AUCTIONS AND PUBLIC GOODS MECHANISMS

A Thesis in

Economics

Oleksii Birulin

c 2004 Oleksii Birulin °

Submitted in Partial Fulﬁllment of the Requirements for the Degree of

Doctor of Philosophy

December 2004 The thesis of Oleksii Birulin has been reviewed and approved* by the following:

Vijay Krishna Professor of Economics Thesis Adviser Chair of Committee

Tomas Sjöström Professor of Economics

Kalyan Chatterjee Distinguished Professor of Economics and Management Science

Susan H. Xu Professor of Management Science and Supply Chain Management

Robert C. Marshall Professor of Economics Head of the Department of Economics

*Signatures are on ﬁle in the Graduate School. Abstract

Efficient allocation of scarce resources is one of the central themes in economics. The essays in this thesis approach two aspects of this problem. In the first essay I consider the problem of the efficient provision of a public good with congestion in a setting with asymmetric information. Mailath and Postlewaite (1990) and Rob (1989) show that in such a setting an efficient, budget balanced and individually rational mechanism for provision of pure public goods does not exist. Many public goods are not pure public goods, that is, they are excludable and to some extent congested. I show that when congestion is taken into account, in a wide class of economies it is possible to construct a mechanism that produces the public good efficiently, balances the budget ex-post and satisfies individual rationality constraints. It is known, see Krishna and Perry (2000), that a desired mechanism exists if and only if the Vickrey-Clarke-Groves (VCG) mechanism results in an expected budget surplus. In the VCG mechanism each consumer is levied a tax equal to the externality that she exerts on the rest of the society. When a good is pure public a consumer pays only if she is pivotal. When a public good is congested even a consumer that is not pivotal exerts an externality on the others. This externality is stronger when congestion is higher, hence the payment from such a consumer "increases with the level of congestion." On the other hand, when congestion is higher, fewer consumers are given access to the good, hence fewer consumers pay atall.Duetothistradeoff,itistoosimplistictosaythattheVCGmechanism runs an expected budget surplus if the good is congested enough. There are simple examples where the VCG mechanism runs an expected budget surplus with less congested good and a deficit with more congested good. I show that it is most likely that the VCG mechanism runs a surplus if the level of congestion is neither too high nor too low. I present a simple and intuitive condition on the parameters of the economy that guarantees that the VCG mechanism results in an expected budget surplus. As a result, in these economies, there also exists an efficient, incentive compatible, ex post budget balanced and individually rational mechanism. The second and third essays deal with the problem of efficient allocation of private goods. In the second essay, written jointly with Serguei Izmalkov, we study efficiency properties of a single-object irrevocable exit English auction in the setting with interdependent and asymmetric values. Maskin (1992) shows that the

iii pairwise single-crossing condition is both necessary and sufficient for efficiency of the English auction with two bidders. Our paper extends both Maskin’s result and the single-crossing condition to the English auction with N -bidders. The pairwise single-crossing imposes the following: if starting from a signal profile where the values of two bidders are equal and maximal we slightly increase the signal of one of the bidders, her value becomes the highest. We introduce the generalized single- crossing condition (GSC) a fairly intuitive extension of the pairwise single-crossing. GSC requires the following: if starting from a signal profile where the values of a group of bidders are equal and maximal we slightly increase the signals of a subset of the group, no bidder outside of the subset can attain the value higher than the maximal value attained among the bidders from the subset. Generalized single- crossing both implies the pairwise single-crossing and reduces to it in the case of two bidders. We show that GSC is both necessary andemphsufficient condition for existence of an efficient equilibrium in the English auction with N -bidders. In the third essay I deal with the setting of interdependent and asymmetric values and consider the class of auctions that possess an efficient ex-post equilibrium. An equilibrium is ex-post if none of the bidders wants to change his strategy after all the information that was private before the auction is revealed to her. Intuitively, this seems to be a significant restriction on the strategies, so that, hopefully, auctions that have an efficient ex-post equilibrium have few other ex-post equilibria. This intuition is, in fact, incorrect. I show that every auction that implements efficient allocation in ex-post equilibrium also has a continuum of inefficient ex-post equilibria in undominated strategies.

iv Contents

List of Figures vii

List of Tables viii

Acknowledgments ix

1Introduction 1 1.1Publicgoodswithcongestion:amechanismdesignapproach.... 3 1.1.1 Relationtotheliterature...... 5 1.2 On efficiency of the N -bidderEnglishauction...... 7 1.3 Inefficient ex-post equilibria in efficientauctions...... 9

2 Public goods with congestion: a mechanism design appoach 11 2.1 Preliminaries ...... 11 2.1.1 Mechanismandobjectives...... 12 2.2 Main results ...... 14 2.3TheVickrey-Clarke-Grovesmechanism...... 15 2.3.1 Pivotal consumers and budget deﬁcit...... 16 2.3.2 Congestedversuspurepublicgoods...... 18 2.3.3 Onmonotonicityofthebudgetsurplus...... 19 2.4Stepcongestionschedules...... 22 2.4.1 Asymptoticresults...... 27 2.5Congestionschedulesofthegeneralform...... 28 2.6Extensions...... 30

3Oneﬃciency of the N -bidder English auction 32 3.1 Preliminaries ...... 32 3.1.1 Pairwisesingle-crossing...... 33 3.1.2 TheEnglishauction...... 33 3.2Generalizedsinglecrossing...... 35 3.2.1 Results...... 36 3.2.2 Examples ...... 37 3.3 Suﬃciency...... 39

v 3.4 Necessity ...... 41 3.4.1 Anillustration...... 41 3.4.2 ProofofProposition3.2...... 42

4Inefficient ex-post equilibria in efficient auctions 46 4.1 Preliminaries ...... 46 4.2 English auction ...... 46 4.3 Ex-post efficientauctions...... 50

AOneﬃciency of the N -bidder English auction 53 A.1 Equivalence Lemma...... 53 A.2 Suﬃciency...... 56 A.3 Necessity ...... 57 A.3.1Supportingresultsforvaluefunctions...... 61

Bibliography 63

vi List of Figures

2.1Upperboundsonthepercapitacost...... 25

4.1 Discontinuous ex-post equilibrium in the English auction ...... 48

vii List of Tables

2.1Budgetsurpluswithstepcongestionschedules...... 21

viii Acknowledgments

I am very grateful to Vijay Krishna for his guidance, countless invaluable comments and insights, and enormous support and encouragement in preparing this thesis. I am indebted to Serguei Izmalkov, who co-authors the second essay and has an incredible positive impact on all my research. I wish to thank Tomas Sjöström, Joris Pinkse and Andrei Karavaev for many helpful discussions, Motty Perry for his encouragement at the early stage of this work, all of my committee members for their patience and support, Peter Norman for his detailed comments on the ﬁrst essay, and all participants of the conferences and seminars at which parts of this thesis were presented for their comments and suggestions.

ix Chapter 1

Introduction

Efficient allocation of scarce resources is one of the central themes in economics. The essays in this thesis approach two aspects of this problem. The first essay concentrates on provision and allocation of an excludable public good with congestion. The presence of congestion implies that consumers may be worse off if “too many”of them are using the good. Efficiency in this context implies, first, that the public good should be produced only if this increases the welfare of the society, and second, that the rights to use the good should be allocated such that that welfare is maximized. At least since Samuelson (1954) it has been recognized that the problem is with incentives, it is hard to induce the consumers to reveal how much they value the good. One possible instrument to generate correct incentives is to tax for the right to use the public good. These taxes have to be paid in the some other good, for example, private, which creates a new problem. The total tax collected should not exceed the cost of the good, this would contradict welfare maximization, that is desired efficiency. What is collected cannot be redistributed back in the society without distorting the incentives. Hence we need to produce the good and simultaneously guarantee that the budget is balanced–the amount of taxes collected equals the cost of the good, whenever it is produced. The problem becomes even more challenging if we need to guarantee voluntary participation, one should have the right not to consume the good and not to pay the tax. Mailath and Postlewaite (1990), Rob (1989) have shown that it is impossible to produce the pure public good efficiently, balance the budget and satisfy voluntary participation. A public good is called pure if it is both non-excludable and non-congested. Of course, this concept is an idealization. Many public goods are excludable, and in many cases consumers do care about how many others consume the good. These goods are rival to some extent, or in our terminology–congested. Examples of such goods are abundant. A swimming pool that charges membership fees is an excludable public good, and when it becomes crowded, customers, obviously, adversely affect each other’s welfare.

1 The first essay is devoted to the problem of financing the provision of excludable public goods with congestion. I show, in particular, that when congestion is taken into account, in a wide class of economies it is possible to produce the good efficiently, balance the budget and satisfy voluntary participation. This possibility result holds even if the information about the efficient level of the public good is dispersed among the potential consumers, and every consumer just knows her private valuation for the good and the distribution of others’ valuations. In such setting an efficient, incentive compatible, BB and IR mechanism for provision of a pure public good does not exist. Essays two and three deal with efficiency in the private good environment. The good is already produced, only one person can consume it; all we need to do is to allocate it efficiently–to the consumer with the highest value. Auctions have been used since antiquity to sell a wide variety of goods. Efficiency–an issue of whether at the end of an auction the object is sold to the buyer who values it themost–wasalmostexclusivelyatheorist’sconcern.Nowadaystheemphasisis changing. Large-scale privatization of state-owned assets in Russia, China, United Stated, United Kingdom, Australia, and many other countries is probably the most quintessential area feeding an interest in the search of efficient auctions. When no buyer’s private information entirely determines how she values the asset–the private values setting, many auction forms are efficient: second-price sealed bid or open ascending price (English) auctions are among them. When size- able in value assets are being offered for sale, the interdependent values setting– a buyer’s private information can be essential in how others are evaluating the assets–seems to be a more plausible description of the environment. For instance, Porter (1995) reports that in U.S. offshore oil and gas lease auctions, “firms are permitted to gather seismic information prior to the sale . . . On-site drilling is not permitted, but firms owning adjacent tracts can conduct off-site drilling, which may be informative.” Clearly, other firms might be interested in the results of such private tests. In the interdependent values setting, achieving efficiency is a highly non-trivial problem. Second-price auction ceases to be efficient if the values are interdependent. The English auction is efficient under quite restrictive conditions. Known mechanisms that are efficient under most general conditions are complex and almost prohibitively unsuitable for actual implementation. The second essay, written jointly with my colleague, Serguei Izmalkov, examines the question of when the standard single-object English auction has an efficient equilibrium. Thethirdessaystudiestheentireclassofauctionsthehaveanefficient ex-post equilibrium. An equilibrium is ex-post if no bidder wants to change his strategy after all the information that was private prior to the auction were revealed to her. This restriction on the strategies turned out to be quite popular; all the recently proposed efficient auctions: Dasgupta and Maskin (2000), Perry and Reny (2001), Perry and Reny (2002), Krishna (2003) allocate efficiently in ex-post equilibrium.

2 Since ex-post equilibrium is quite a stringent restriction it may also seem that these auctions have few ex-post equilibria. In the third essay I show that this intuition is false, every auction that has an eﬃcient ex-post equilibrium, also has a continuum of ineﬃcient ex-post equilibria.

1.1 Public goods with congestion: a mechanism design approach

The problem of achieving a Pareto efficient allocation in economies with public goods is an old one. At least since Samuelson (1954), it has been recognized that the main difficulty is with incentives–the agents need to be induced to truthfully reveal how much they value the good. Rational consumers would misrepresent their preferences in an attempt to “free ride,” thereby leading to the underprovision of public goods. Since then our understanding of the issue has been greatly extended. In particular, Clark (1971) and Groves (1973) showed that perhaps Samuelson’s conclusions were too pessimistic, since the use of their “pivotal” mechanisms led to truthful revelation of preferences. As a counter to this optimistic finding, however, it was shown that no such dominant strategy mechanism could balance the budget (Green and Laffont (1977)). Therefore, a search for weaker mechanisms was undertaken and indeed efficient, budget balanced and Bayesian incentive compatible mechanisms were constructed (Arrow (1979), d’Aspremont and Gérard-Varet (1979)). A final blow to this program came when it was shown that with pure public goods, no such mechanism could guarantee voluntary participation (Mailath and Postlewaite (1990), Rob (1989)). A public good is called pure if it is both non-excludable and non-rival. This definition implies, firstly, that nobody can be excluded from consumption of the good, and secondly, no consumer cares about how many others she shares the good with. Of course, the concept of a pure public good is an idealization. Many public goods are excludable, and in many cases consumers do care about how many others consume the good. These goods are rival to some extent, or in our terminology–congested. Examples of such goods are abundant. A swimming pool that charges membership fees is an excludable public good, and when it becomes crowded, customers, obviously, adversely affect each other’s welfare. A toll road is an excludable public good, but traffic jams happen even there. A university library is restricted to the students and the faculty, but a desired book may be checked out. This paper concentrates on the problem of financing the provision of excludable public goods with congestion. I show, in particular, that when congestion is taken into account, in a wide class of economies it is possible to construct an incentive compatible mechanism that always produces the good (ex-post) efficiently, balances the budget ex-post (satisfies BB) and satisfies voluntary participation or

3 individual rationality constraints (satisfies IR). This possibility result holds even if the information about the efficient level of the public good is dispersed among the potential consumers, and every consumer just knows her private valuation for the good and the distribution of others’ valuations. In such setting an efficient, incentive compatible, BB and IR mechanism for provision of a pure public good does not exist. It turns out that the existence of a suitable mechanism hinges on whether or not the well-known Vickrey-Clarke-Groves (VCG) mechanism runs an expected budget surplus.1 Specifically, the following result, due to Krishna and Perry (2000), is very useful in what follows: There exists an efficient, Bayesian incentive compatible, BB and IR mechanism if and only if the VCG mechanism runs an expected budget surplus.2 It is useful to begin by considering two polar cases–that of a private good and a pure public good. A private good is such that consumption or use by one person precludes its consumption or use by any other person–in the framework of this paper, the congestion externality is maximal. In this case, the VCG mechanism is the same as the second-price auction with the reserve price equal to the cost of the good. Clearly, in this case the VCG mechanism runs a budget surplus, regardless of the cost of the good. By the result above, in the case of a private good there exists an efficient, incentive compatible, BB and IR mechanism. At the other extreme, consider a pure public good. A pure public good can be consumed or used by all members of society simultaneously–in the framework of this paper, there is no congestion externality. In this case, it is known that, no matter what the cost, the VCG mechanism runs an expected budget deficit, and so there does not exist a budget balanced mechanism with the desired properties. What can be said concerning the goods with intermediate levels of congestion? Recall that in the VCG mechanism each individual is levied a tax equal to the externality she exerts on the other members of society. This suggests that the greater is the level of congestion–and so also the greater is the externality exerted on others–the greater is the taxes collected and the budget surplus. Hence, one can expect that, holding everything else fixed, the expected budget surplus in the VCG mechanism increases as the level of congestion increases. This conjecture implies a simple policy: smaller, “more exclusive” public goods have better chances of being self-financing. The conjecture, however, turns out to be incorrect. There are examples in which an increase in the level of congestion–modelled in quite a natural manner–

1 The term budget surplus denotes the difference between the budget revenue and the cost. The statement that amechanismrunsabudgetsurplusimplies that the budget surplus is non- negative. Budget deficit is a negative budget surplus. 2 The “only if” part of this result also follows from Williams (1999). Williams shows that every efficient and Bayesian incentive compatible mechanism is (interim) payoff equivalent to a Groves mechanism.

4 leads to a fall in the expected budget surplus (see Example 1). Specifically, there are situations in which goods with intermediate levels of congestion lead to an expected budget surplus, but goods with higher levels of congestion lead to an expected budget deficit. Thus the intuition offered above–that increased levels of congestion/externality lead to higher budget surpluses–is oversimlified. With a congested good even a consumer who is not pivotal is taxed since she exerts an externality on the others. If at a given realization some consumer is pivotal, the VCG mechanism runs a budget deficit (see Lemma 1). Thus, if the VCG mechanism runs an expected budget surplus, the contribution of the non- pivotal consumers to the budget has to dominate. This contribution by itself need not be “monotonic in congestion.” When the good becomes more congested, those who still consume it indeed exert a higher externality and pay more. On the other hand, with a more congested good, fewer consumers are given access, hence fewer payatall.Duetothistradeoff, other things being fixed, it is likely that the VCG mechanism runs an expected budget surplus when the level of congestion is intermediate. The main contribution of this paper is, therefore, as follows.

Proposition (Informal Statement) For a wide class of distributions and a wide range of costs, if the level of congestion is neither too high nor too low, the VCG mechanism runs an expected budget surplus. As a result, in these circum- stances, there also exists an eﬃcient, Bayesian incentive compatible, ex-post budget balanced and individually rational mechanism.3

1.1.1 Relation to the literature We start with the literature on the pure public goods. If the information about the impact of the public project on the society’s welfare is publicly available, the problem of achieving efficiency via BB and IR mechanism has a solution, see i.e. Hurwics (1979). Complete information assumption can be relaxed quite substantially. Jackson and Moulin (1992) offer a remarkable mechanism that satisfies BB and IR and implements the public project efficiently whenever every consumer knows her private value of the project, and at least two consumers know the sum of all the private values. If the information that each consumer possess is restricted to his private value and the distribution of others’ values, satisfying both BB and IR becomes a problem. Mailath and Postlewaite (1990) and Rob (1989) show that no efficient, Bayesian incentive compatible mechanism can simultaneously balance the budget and satisfy individual rationality constraints.4

3 Two features determine consumer preferences in my model: a) values drawn from distribution F and b) “congestion schedule” that speciﬁes how consumers aﬀect each others’ utility while sharing the public good. 4 Independence of “types“ with an assumption that there is a continuum of possible private valuations is important here.

5 Moreover, they show that the probability of ever undertaking the project goes to zero when the number of consumers goes to infinity and the cost of the project is increasing with the size of the economy. Hellwig (2003), in contrast, demonstrates that when the cost of the project is fixed, and the number of the consumers increases, the allocation of the balanced budget mechanism converges in distribution to the efficient one. I show that when a public good is congested it may be possible to balance the budget in the efficient mechanism when the economy grows, and the total cost of the project grows proportionally to the size of the economy. Several authors consider the setting where the public good, while still completely non-rival, is excludable. Dearden (1997) and Ledyard and Palfrey (1999), in more general environment, provide a characterization of an ex-ante efficient, incentive compatible, BB and IR mechanism.5 Norman (2003) shows that an ex- ante efficient, incentive compatible, BB and IR mechanism can be asymptotically approximated by a simple fixed entry fee mechanism. In my formulation the planner can exclude consumers, but can do so only on the grounds of efficiency; if due to the congestion it is beneficial to provide the good to exactly m consumers, exactly m consumers are given access. In the literature on excludable, non-rival public goods the planner is also free to exclude consumers, even though, after such a good is produced, it is inefficient to exclude anybody. If some of the consumers are not excluded, however, the mechanism runs a budget deficit, and the project is not undertaken. Hence, at most second-best level of efficiency can be achieved, versus first-best level in my paper. The (quasilinear setting) literature on pure and excludable public goods often approaches the problem by combining the BB, IR and incentive compatibility into one integral constraint. Then the planner’s objective is stated as a program of constraint optimization. My method is based on the analysis of the VCG mechanism. This allows for simpler proofs, even for already known results, and, in my view, leads to more intuitive understanding of the problem. There is extensive (complete information) literature on excludable public goods with congestion (also called club goods), starting from Buchanan (1965). For a nice survey see Cornes and Sandler (1986). Very little is done in incomplete information setup. This paper is the only one, to my best knowledge, that treats public goods with congestion in the setting with quasilinear utility. Jackson and Nicolò (2003) study the strategy-proof provision of club goods. Consumers in their setting have single-peaked preferences over the public good level and also care about how many other consumers have access to the good. Jackson and Nicolò (2003) show that strategy-proofness and efficiency are in general incompatible with individual stability, a notion that requires that consumers, who are given access to the good, weakly prefer to consume, and those who are

5 Cornelli (1996) studies the problem of the provision of an excludable public good by proﬁt maximizing monopoly and obtains very similar characterization.

6 excluded, weakly prefer not to consume.6 In contrast to Jackson and Nicolò’s my setting allows for transfers. As a result the findings are more positive. I show that an efficient and strategy-proof mechanism (the VCG mechanism) satisfies individual stability and balances the budget (in expectation) under quite permissive conditions. Moulin (1994) and further Dearden (1998) and Olszewski (1999) examine the properties of an appealingly simple and robust serial cost-sharing mechanism for provision of excludable, non-rival public goods. They show that serial mechanism Pareto dominates any BB, IR, coalition strategy-proof mechanism that satisfies anonymity.7 Deb and Razzolini (1999a) further demonstrate that serial mechanism Pareto dominates any strategy-proof, BB, anonymous mechanism that satisfies individual stability in the sense of Jackson and Nicolò (2003). Serial mechanism excludes consumers and therefore achieves only second-best efficiency.

1.2 On eﬃciency of the N -bidder English auction

How to sell a good efficiently–to the buyer who values it the most–is one of the main questions of the theory of auctions. The task becomes harder as the informational environment gets more complex. When the valuations of the buyers are asymmetric and depend on the private information of the others the set of efficient mechanisms is quite limited. Among these is the open ascending price, or English, auction. It is typically modeled as the irrevocable exit clock auction, and this model is known to possess an efficient equilibrium when value functions satisfy certain conditions. What is the minimal (necessary and sufficient) condition for efficiency of the English auction is a long-standing problem. This paper provides a solution. In a classic paper, Milgrom and Weber (1982) introduce the irrevocable exit model of the English auction. They show that in the setting with symmetric interdependent values the English auction has an efficient equilibrium, and, if signals are affiliated, it generates higher revenues to the seller than other common auction forms. Maskin (1992) indicates that the pairwise single-crossing condition is necessary for efficiency of the asymmetric English auction, and shows that it isalsoasufficient condition when the number of bidders is two. Perry and Reny (2001) provide an example with three bidders where the pairwise single-crossing is satisfied and no efficient equilibrium exists, and presents a pair of sufficient conditions for efficiency of the N-bidder English auction–an average-crossing and a cyclical-crossing conditions.8

6 Strategy-proofness and eﬃciency can also be incompatible with IR. 7 Serial mechanism by construction is budget balanced. 8 The average-crossing condition requires that, if starting from a signal proﬁle where the values

7 We introduce the generalized single-crossing condition which is a natural extension of the pairwise single-crossing to the case of N bidders. The pairwise single-crossing imposes the following: if starting from a signal profile where the values of two bidders are equal and maximal we slightly increase the signal of one of the bidders, her value becomes the highest. This implies that the private in- formationheldbyabidderaffects her valuation more than the valuations of her competitors. Our condition requires the following: if starting from a signal profile where the values of a group of bidders are equal and maximal we slightly increase the signals of a subset of the group, no bidder outside of the subset can attain the value higher than the maximal value attained among the bidders from the subset. The generalized single-crossing both implies the pairwise single-crossing and reduces to it in the case of two bidders. Two main results of this paper are the necessity: if the generalized single- crossing condition is violated at some interior signal profile, then no efficient equilibrium in the N-bidder English auction exists; and the sufficiency: if value functions satisfy the generalized single-crossing condition both in the interior and on the boundary of the signals’ domain, then there exists an efficient ex post equilibrium in the N-bidder English auction. If the generalized single-crossing is violated only on the boundary an efficient equilibrium may or may not exist, see Section 3.2.2. Given that the gap between the necessity and sufficiency statements is the set of measure zero we simply refer to the generalized single-crossing as to the necessary and sufficient condition.9 The English auction is not the only efficient mechanism in the interdependent values setting, and the generalized single-crossing is not the weakest condition for efficiency. What makes the English auction so special, aside from its widespread use, is the strategic simplicity, transparent set of rules, and ease of conducting. In the English auction, even if the values are interdependent, the strategy in the efficient equilibrium is nothing but “...drop out when the price reaches what you believe your value is.” The “contingent bid” mechanism of Dasgupta and Maskin (2000) requires each buyer to submit a price she is willing to pay given the realized values of the others–a (N 1)-variable function. This auction is efficient if the pairwise single-crossing holds.− Utilizing the fact that two-bidder sealed bid and ascending price auctions are efficient Perry and Reny (2002) and Perry and Reny (2001) design two elegant mechanisms that incorporate a concept of “directed bids”–every buyer bids against every other buyer, thus managing N 1 bids simultaneously. These auctions require the strong form of the pairwise single-crossing− of several bidders are equal and maximal, the signal of one of them is increased, the corresponding increments to the values of the others are lower than the average increment. The cyclical-crossing requires that the increments to the values are ranked in the prespecified cyclical order–the effect on the own value is the largest and decreases for each subsequent bidder in the cycle. 9 In fact, even in the two-bidder case, the pairwise single-crossing is necessary up to the boundary, see Example 3.1 in Section 3.1. In this sense, our results are the exact extenstion of Maskin’s.

8 for efficiency.10 Theabovemechanismsareremarkableconstructions,designedto allocate multiple units efficiently. In their single unit version, however, they are significantly more complex than the English auction.11 Izmalkov (2003) proposes an alternative model of the English auction. In this model the bidders are allowed to reenter–become active again after they dropped out. Izmalkov shows that the English auction with reentry is efficient under the conditions that are weaker than the generalized single-crossing.12 At the same time the possibility of reentry substantially enriches the strategy space and provides opportunities to exchange messages, which, potentially, may allow bidders to coordinate on a collusive outcome. In contrast, the irrevocable-exit English auction is robust to collusion within the auction. The fact that the exits are irrevocable implies that the bidders cannot coordinate their actions: the only way a bidder can send a message is by exiting, which makes winning impossible.

1.3 Ineﬃcient ex-post equilibria in eﬃcient auctions

An “efficient auction” is one with an equilibrium in which the allocation maximizes the social surplus. The efficiency of an auction depends, of course, on the informational environment. For instance, with private values, a first-price sealed- bid auction is efficient only when bidders are symmetric, whereas a second-price sealed-bid auction is always efficient. Recently, there has been renewed interest in finding efficient auctions in situations where values are interdependent–bidders’ values depend on the private information held by others.13 In this context, three classes of auctions/mechanisms have been studied.

1. English and other open ascending price auctions (Milgrom and Weber (1982), Ausubel (1997))

2. Generalized Vickrey-Clarke-Groves mechanisms (Crémer and McLean (1985), Maskin (1992))

10The pairwise single-crossing has to be satisfied for any pair of bidders with equal values, not only when their values are maximal. Thus, the strong pairwise single-crossing and the generalized single-crossing are not comparable. 11The generalized Vickrey auction is simple and efficient if the pairwise single-crossing holds, but, being a direct revelation mechanism, it requires the auctioneer to know everything that the bidders know about each other, which is “utterly unworkable in practice ” (see Maskin (2003) for further discussion). 12The generalized single-crossing would imply that no reentry happens in the efficient equilibrium. 13In the literature this case is sometimes refered to as the case of “common values,” which may cause a confusion with (pure) common values case. For example, Milgrom and Weber (1982) by common values mean the case where the value of the good is equal for all bidders.

9 3. Contingent bid (Dasgupta and Maskin (2000)) and directed bid (Perry and Reny (2001) and (2002)) mechanisms.

In each of these, research has focused on finding conditions under which there is an ex-post equilibrium which allocates efficiently. An ex-post equilibrium is a Bayesian equilibrium with no regret–after the good is allocated, no bidder would wish to change his action even if all the information were made available to him.14 This, of course, is a strong requirement and it is remarkable that the mechanisms have such an equilibrium. One may conjecture that with such a strong requirement, the set of equilibria would be small. The purpose of this essay is to show that this is, in fact, false: all of the efficient auctions/mechanisms mentioned above have a continuum of inefficient undominated ex-post equilibria. One might also expect a tight link between the “no ex-post regret” criterion and ex-post efficiency. This essay demonstrates that such a link does not exist: an equilibrium may be ex-post, even undominated,but inefficient. The multiplicity of equilibria in English and second-price auctions is well known. Milgrom (1981) and Bikhchandani and Riley (1991) provide examples of multiple equilibria in the pure common value model. Bikhchandani and Riley’s construction results in only perfect Bayesian equilibria. Milgrom’s equilibria are ex-post, but assumption of pure common values is crucial for his construction–there is no natural extension of such equilibria if values are not common; besides, with pure common values the efficiency question is moot. Bikhchandani, Haile and Riley (2002) consider the model of Milgrom and Weber (1982) and present a continuum of efficient equilibria; however, these equilibria are not ex-post. This essay presents the methodology of constructing the undominated ex-post equilibria, which extends to all the efficient auctions proposed so far. I first present the construction for the the English auction, which is probably the most illustrative case. In this auction the proposed inefficient ex-post equilibrium involves discontinuous strategies, by varying the size and the number of the “jumps ”one can construct a continuum of equilibria. Then I show that every auction that implements efficient allocation in ex-post equilibrium necessarily also has a continuum of inefficient ex-post equilibria that do not involve dominated strategies.

14Crémer and McLean (1985) seem to be the first to use the notion of ex-post equilibrium in the auction context. For the formal definition see Definition 4.2.

10 Chapter 2

Public goods with congestion: a mechanism design appoach

2.1 Preliminaries

The economy consists of a set of n potential consumers, N, and a social planner. Each consumer has an endowment of a numeraire good, which is convenient to think of as money. The social planner is endowed with a technology that can transform the numeraire into the public good. The technology can produce one unit of the public good and requires an amount C of the numeraire as an input. The public good is excludable and, to some extent, rival–any consumer can be excluded from the use of the public good, and the consumer’s utility from the use of the good may depend on how many others have access to it. Such a good is called congested good.

Each consumer i draws a real valued value vi, distributed according to a smooth distribution function F with the density f(v) > 0 on the interval [0, 1].1 The draws are independent. The distribution F is common knowledge, but vi is consumer i’s private information. This vi is the value that consumer i attaches to the public good if she consumes it alone as a private good. As usual, v denotes the “state of the world”–the proﬁle of the realized values (v1,v2, ..., vn) . Where necessary, we write (vi, v i) , where v i stands for the values of all the consumers except i. The set of all possible− realizations− of the values is denoted V. Consumers are risk-neutral and have quasilinear utilities. The utility that consumer i derives from consumption of the public good depends on the private value vi and (adversely) on the number of the consumers who are given access to the public good. The extent to which the consumers aﬀect each others utility is determined by the congestion schedule associated with the particular public good.

1 This assumption is made for the purposes of tractability. See Section 2.6 for the generaliza- tions.

11 Deﬁnition 2.1. A congestion schedule is a mapping α : N [0, 1] satisfying → α1 =1and αk αk+1 for all k. ≥ Thecongestionschedulecorrespondingtothecaseofapure public good is denoted by ρ,whereρk =1for all k. The congestion schedule for a private good is α1 =1and αk =0for k>1. Thus our model incorporates both the private and the pure public good as special cases, and so formalizes the idea that pure public and private goods constitute opposite ends of the entire spectrum of goods. If m consumers are given access to the public good, then the utility that any one of them, say consumer i, derives from consumption, is αmvi. Thefactthata congestion schedule is decreasing implies that whenever an extra consumer gets access to the public good, the utilities that all the former users derive from the good decrease. The congestion schedule is assumed to be commonly known among the consumers and the social planner. The cost of production of the public good is denoted by C.Itisnaturalto suppose that if α and β with α 5 β are the congestion schedules associated with two diﬀerent public goods, then associated costs of production, denoted Cα and Cβ satisfy Cα Cβ. ≤ 2.1.1 Mechanism and objectives The prerogative of the social planner is to choose a pair (M,t).First,theplanner determines the allocation M, that is, whether to produce the public good or not and if it is produced, which consumers are given access to the good. Formally, the set M N denotes the set of the consumers who are given access to the public ⊆ good. If the good is not produced, M = ∅. Second, the planner determines the payments t, that is, how much a consumer has to contribute (or receive), under the provision that the sum of the contributions covers C–the amount of the input necessary, ti C. As a result of≥ the planner’s decision, consumer i experiences a gain in utility P

α αmvi ti if i M, Ui (M,t,vi)= − ∈ ti if i/M, ½ − ∈ where m =#M and ti stands for the contribution that consumer i is asked to make. The social welfare in the economy is deﬁned as

αm vj C if M = ∅, α j M − 6 SW (M,v) ∈ ≡ ( P0 if M = ∅. The objective of the social planner is maximization of the social welfare in the economy. Since the consumers may create congestion and adversely aﬀect each others’ utility from consumption, the social planner has the power to exclude

12 consumers from the use of the good. The planner, however, excludes a consumer if and only if this exclusion increases social welfare. The efficient outcome may depend on congestion schedule α and the realization v which constitutes consumers’ private information.2 The planner may ask the consumers to reveal their values and then make a decision based on the reports. For this purpose, the planner designs a set of rules or a mechanism (M,t) that maps the reports into the allocation and the payment and announces the mechanism to the consumers.3 Denote the set of all possible allocations by and the set of all possible payments by .Amechanism(M,t):V M is (ex-post) efficient if at every v V theT allocation M maximizes social→ welfareM×TSWα(M,v). A mechanism (M,t):∈V is (ex-post) budget balanced if at every v V → M×T ∈ such that M = ∅ the payment tα satisfies 6 α ti = C, (BB)

α X and ti =0 otherwise. To achieve efficiency the planner needs to design a game, where each of the consumers has an incentive to, first, participate and second, tell the truthP about his value. The decision on whether to participate and whether to tell the truth is made at the interim stage, that is, when the consumer has already drawn his private valuation, but before the outcome is determined. De- α α fine Ui (vi) Ev i [Ui (M(v), t(v),vi)].Amechanism(M,t):V is ≡ − → M×T (Bayesian) incentive compatible if for all i N, at every vi, for every vi, ∈ α α Ui (vi) Ev i Ui (M(vi, v i), t(vi, v i),vi) . ≥ − − − e £ ¤ That is, “truth telling” constitutes a Bayesian-Nashe e equilibrium of the resulting game. A mechanism (M,t):V is (interim) individually rational if for → M×T all i N, every vi, ∈ α U (vi) 0, (IR) i ≥ where 0 stands for consumer i’s individual rationality level.4 The assumption that IR is satisfied at the interim stage (and not ex-post) implies that the social planner is endowed with some coercive power, so that ex- post she can force the consumers to make the payments that they have agreed to pay at the interim stage. Alternatively, one can think of a situation where

2 This dependence is made explicit where necessary. 3 I allow for the same notation for the outcome of the mechanism and for the mechanism itself since this creates no confusion. As usual, without loss of generality, I restrict the attention to direct revelation mechanisms. 4 The individual rationality level represents consumer i’s outside option. In public goods literature it is typically assumed to be equal to zero. This is not without loss of generality, but I follow the convention as soon as my main focus is on the role of congestion. The possible extensions of the model to the case of non-zero individual rationality level are discussed in Section 2.6.

13 consumers are first asked to make their payments, and only after that the allocation is announced. Both the notation and the requirements on the mechanism given above are standard in the public goods literature. In my setting, however, it is convenient to make a slight departure from the standard notation. By efficiency, if some consumers are excluded from the use of the public good, those are the consumers with the lowest values. Therefore, it is convenient to rank the values in descending order. From now on vi:n denotes the i-th highest value among n draws. Since the number of draws remains fixed in most of the paper, index n is suppressed until Subsection 2.4.1.

2.2 Main results

Below I establish my main result. Proposition 2.1, provides a weak sufficient condition for the efficient allocation of public goods of a particular sort–fixed capacity public goods. With such goods if no more than m consumers are given access to the good, then there is no congestion; on the other hand, if more than m consumers are allowed to use the good, then the congestion is extreme. As an example, consider a roller coaster with m seats. The first m consumers create no congestion, but if the m +1’st consumer is included, then the ride simply cannot start since everyone must be seated and buckled up. The congestion schedule associated with a public good of fixed capacity (say, m) is particularly simple: it consists of m 1’s followed by 0’s. I refer to such a schedule as a step congestion schedule

σm (1,...,1, 0,...,0). ≡ m For ﬁxed capacity public goods, I| show{z } in Section 2.4:

Proposition 2.1. For step congestion schedule σm with associated cost C,an eﬃcient, Bayesian incentive compatible, BB and IR mechanism exists if

C E [vm+1] . ≥ m The sufficient condition in the proposition should be viewed as being rather permissive. Suppose that the distribution of values F is such that for all v, F (v) 1 ≤ v. Further suppose that the per-capita cost C/m does not exceed 2 .Inthatcase when n is approximately 2m the sufficient condition is satisfied. Roughly, this suggests that for a wide class of value distributions, fixed capacity public goods with moderate congestion–one half of the population does not suffer from it–can be efficiently provided via budget balanced mechanism.

14 In general, that is, for any distribution F and any per-capita cost C/m < 1, when the size of the population n is large enough, the sufficient condition is also satisfied.5 Another set of results highlights the difference between the provision of a pure and that of a congested public good in a large economy. Consider a sequence of economies n (n, σm,C(σm,n)) with n such that the “capacity” of the public goodE increases≡ with the size of the→∞ economy, m =[µn] , where µ (0, 1) and [µn] is an integer part of µn. In the setting with the pure public good∈µ =1, and the results of Mailath and Postlewaite (1990), Rob (1989) and Hellwig (2003) suggest that in the limit an efficient, Bayesian incentive compatible, BB and IR mechanism exists if and only if the total cost of the good stays bounded and the C per capita cost of the good n 0 as n . With the congested public good such mechanism may exist even if→ the total→∞ cost of the good grows with the size of the economy and the corresponding per capita cost does not go to zero. Proposition 2.1 implies that:

Corollary 2.1. Consider a sequence of economies n (n, σm,C(σm,n))n∞=1 such C σ ,n E ≡ that m =[µn] for µ (0, 1) , and lim ( m ) = c [0, 1). In the limit an eﬃcient, ∈ n m ∈ →∞ 1 6 Bayesian incentive compatible, BB and IR mechanism exists if c

2.3 The Vickrey-Clarke-Groves mechanism

The existence of an eﬃcient, budget balanced mechanism that satisﬁes individual rationality constraints is known to be problematic in many settings (for instance with pure public goods). The following Theorem provides a decisive answer to this existence question.

Theorem 2.1 (Krishna and Perry (2000)). An eﬃcient, Bayesian incentive compatible, BB and IR mechanism exists if and only if the VCG mechanism runs an expected budget surplus.7

5 The large population result, in fact, extends beyond step congestion schedules. Suppose for a congestion schedule α there exists m such that αk =0for all k>m.Then there exists N such that for all n N in the economy with n consumers an efficient, Bayesian incentive compatible, BB and IR mechanism≥ exists. See Section 2.5 for more detail. 6 The characterization of Corollary 2.1 is quite sharp. I also show that such mechanism does 1 not exist if c>F− (1 µ) , see Subsection 2.4.1. 7 Notice the trade-off−that this Theorem offers. The VCG mechanism is efficient, dominant strategy incentive compatible and satisfies IR ex-post. If it runs an expected budget surplus,

15 In the VCG mechanism every consumer is asked to report her value to the social planner, who then determines the allocation of the public good and the payment from each consumer.

Deﬁnition 2.2. Kα(v) is an allocation rule of the VCG mechanism if at every v V ∈ Kα(v) arg max SWα(M,v). ⊆ M We need some more notation. Similarly to SWα(M,v),deﬁne

αm vj C if M = ∅, α i=j M − 6 SW i(M,v) 6 ∈ − ≡ ( P0 if M = ∅.

α α Introduce v i (0, v i) . Let SW i(M,v i) and K (v i) denote the social welfare − ≡ − − − − andanallocationintheeconomywherevi is set at 0. b b b Definition 2.3. tα(v) is the payment rule of the VCG mechanism if for every i N, at every v ∈ α α α ti (v)=SW i(K (v i), v i) SW i(K (v), v). − − − − − It is well known that truth-telling constitutes a weakly dominant strategy in b b the VCG mechanism. The VCG mechanism is efficient and satisfies IR ex-post.

2.3.1 Pivotal consumers and budget deﬁcit Following the literature on pure public goods introduce:

Deﬁnition 2.4. Consumer i is pivotal under schedule α at realization v, if Kα(v) = α 6 ∅, while K (v i)=∅. − Clearly, more than one pivotal consumer may exist. Suppose v >v and b i j consumer j is pivotal. Then consumer i is obviously pivotal also. As the following lemma demonstrates, the presence of pivotal consumers has important implications for the budget surplus in the VCG mechanism.

Lemma 2.1. Suppose under schedule α at realization v it is efficient to produce the good for at least two consumers, #Kα(v) 2. Suppose also that at v there exists a pivotal consumer. Then the VCG mechanism≥ runs a budget deficit at v. there exists an efficient, Bayesian incentive compatible mechanism that satisfies IR interim but balances the budget ex-post. Makowski and Mezzetti (1994) show that an efficient, Bayesian incentive compatible, BB and IR mechanism exists if and only if there exists an efficient, dominant strategy incentive compatible, IR mechanism that runs an expected budget surplus.

16 Proof. Suppose the allocation is Kα(v) with #Kα(v) 2 and consider the pay- α α α ≥ α ment of consumer i K (v),ti (v)=SW i(K (v i), v i) SW i(K (v), v). The payoﬀ that consumer∈ i derives in the VCG− mechanism,− − − −

α α b bα Ui (K(v), t(v),vi)=SW(K (v), v) SW i(K (v i), v i). − − − − That is, i’s payoﬀ constitutes his direct marginal contribution to the social welfare. b b By construction

n n α α Ui (K(v), t(v),vi)=αk vi ti (v). i Kα(v) − i=1 ∈ i=1 X P X α α α Notice that Uj (K(v), t(v),vj)=0and tj (v)=0for consumer j/K (v). Then n ∈ α ti (v) C is equivalent to i=1 ≥

P n α αk vi C Ui (K(v), t(v),vi), or i Kα(v) − ≥ ∈ i=1 P X n α α α SW(K (v), v) n SW(K (v), v) SW i(K (v i), v i) (2.1) − − − ≥ · − i=1 X Inequality (2.1) is a familiar “feasibility” requirement; the marginalb contributionsb α of the consumers (their payoffs Ui (K(v), t(v),vi)) should add up to no more than the total social welfare. α Now suppose there is a pivotal consumer. When i is pivotal she pays ti (v)= α α α SW i(K (v), v). Hence Ui (K(v), t(v),vi)=SW(K (v), v), pivotal consumer’s payo− ff−alone constitutes SW(Kα(v), v).Ifconsumeri uses the good as a private good, the budget in the VCG mechanism can be balanced. If the good is shared with other consumers, they obtain a positive (IR is satisfied ex-post) payoff.Thus, whenever the good is consumed not as a private good and a pivotal consumer exists, the VCG mechanism runs a budget deficit.

Requiring the VCG mechanism to run an expected budget surplus corresponds to requiring (2.1) to hold in expectation. An eﬃcient, incentive compatible, BB and IR mechanism exists if and only if

(n 1) α (n) α nE − [SW(K (v), v)] (n 1) E [SW(K (v), v)] ,(2.2) ≥ − (n 1) where E − [ ] denotesanexpectationinthesamplewithn 1 draws. I will use this condition· later. − Corollary 2.2. With a pure public good the VCG mechanism runs an expected budget deﬁcit.

17 Proof. Indeed, if at v the pure public good is not produced, the budget is exactly balanced. If at v the good is produced and there is a pivotal consumer, by Lemma 2.1 the VCG mechanism runs a budget deﬁcit. If at v the good is produced and no consumer is pivotal, the total payment is 0 and the budget deﬁcit is C. −

Corollary 2.2 together with Theorem 2.1 re-establishes the impossibility result for pure public goods, obtained by Mailath and Postlewaite (1990) and Rob (1989). In the following two subsections I perform some “comparative statics” exercises. Namely, suppose there are two goods with diﬀerent congestion schedules. One is “more congested” than the other. I investigate the relationship between the level of congestion and the budget surplus in the VCG mechanism. To concentrate on the role of congestion, I assume the cost C to be independent of the congestion schedule.

2.3.2 Congested versus pure public goods This subsection compares a congested public good with a pure one and demonstrates that the VCG mechanism runs higher budget surplus when the good is congested. The cost C is assumed to be the same for both goods. Recall that ρ denotes the congestion schedule for the pure public good.

Lemma 2.2. Suppose Kα(v) = ∅ for a good with congestion schedule α.Then at v the budget surplus in the VCG6 mechanism with the congested good exceeds the corresponding surplus with a pure public good.

Proof. Clearly if it is eﬃcient to produce the good under some congestion, it is also eﬃcient to produce the good as a pure public good. Thus, Kα(v) Kρ(v) and we just need to show that ⊆

tα(v) tρ(v). i ≥ i X X Choose an arbitrary consumer i Kα(v). Obviously i Kρ(v). There are two possibilities–either consumer i is pivotal∈ under ρ orsheisnotpivotal.∈ Suppose i is pivotal under ρ at v. Obviously, if i is pivotal under ρ, she is pivotal under α as well. Then,

α α ρ ρ ti (v)= SW i(K (v), v) SW i(K (v), v)=ti (v) − − ≥− − as ρ α SW i(K (v), v) SW i(K (v), v). − ≥ − ρ Suppose i is not pivotal under ρ at v. Then ti (v)=0, and

α α α ρ ti (v)=SW i(K (v i), v i) SW i(K (v), v) 0=ti (v), − − − − − ≥ b b 18 where the inequality follows from efficiency of the VCG mechanism at the profile v i under the schedule α. − Thus tα(v) tρ(v) for any i Kα(v). i ≥ i ∈ b Notice finally that any consumer j/Kα(v) cannot be pivotal under ρ by the α ∈ α ρ supposition that K (v) = ∅ and hence tj (v)=tj (v) for all such j. 6 Remark 2.1. Whenever Kα(v)=∅, that is, a realization is such that the congested good is not produced, the VCG mechanism with a pure public good cannot run a budget surplus (see the proof of Corollary 2.2). Thus a congested good is “better” than the pure public good in terms of the budget surplus. Lemma 2.2 does not provide a definite answer to the question of whether the VCG mechanism in an economy with a congested good runs a budget surplus or a budget deficit, however, it indicates a direction of the search. When public good is congested every consumer, who is given access to the good, exerts a negative externality on the rest of the society, hence every such consumer pays a positive amount. A consumer obviously pays when she is pivotal, but by Lemma 2.1 this only leads to a budget deficit. At the realizations where no consumer is pivotal, the VCG mechanism may run a budget deficit, but it may also run a budget surplus. When the latter realizations “contribute enough” to the budget, the VCG mechanism runs an expected budget surplus.

2.3.3 On monotonicity of the budget surplus The previous section compared congested and non-congested goods in terms of the budget surplus in the VCG mechanism. This section compares goods with diﬀerent “degrees of congestion.” It starts with the analysis of a two-agent economy. Sup- pose the economy consists of two consumers whose values are v1 and v2.Consider two congestion schedules α =(1,a) and β =(1,b) with b>a,so that schedule α is more congested than schedule β. Again assume that cost C is the same for both goods. It turns out that in the two-agent economy the budget surplus with the more congested good is higher. More precisely: Lemma 2.3. Suppose n =2and consider two congestion schedules α and β such that α 5 β. In every realization v the budget surplus in the VCG mechanism with schedule α exceeds the corresponding surplus with schedule β.

α β α β Proof. Ishowthat ti (v) ti (v) when K (v) = ∅ and ti (v) C when ≥ 6 ≤ Kα(v)=∅ and Kβ(v) = ∅. Suppose first thatPK6α(v)=P1, 2 . Then by efficiency Kβ(vP)= 1, 2 . By the definition of the payments in the{ VCG} mechanism { }

α α α ti (v)= SW i(K (v i), v i) SW(K (v), v), and − − − β β − β ti (v)= SW i(K (v i), v i) SW(K (v), v). P P − − − − b b P P b19 b β α Notice that SW i(K (v i), v i)=SW i(K (v i), v i) for every i since the social welfare in the economy− without− − one of− the two consumers− − cannot possibly depend β on the second coefficientb ofb the congestion schedule.b b Clearly, SW(K (v), v) α α β ≥ SW(K (v), v) since α is more congested than β, and therefore ti (v) ti (v). α ≥α Next suppose K (v)= 1 . Then the good is sold as a private good, t2 (v)=0 α { } α β P Pβ and t1 (v)=max v2,C . Since K (v)= 1 either K (v)= 1 or K (v)= { } β {β } { } 1, 2 . In the former case t2 (v)=0and t1 (v)=max v2,C . In the latter it { } β α β { } follows that t (v) t (v). Indeed, suppose K (v)= 1, 2 , that is b(v1 +v2) i ≤ 1 { } ≥ v1, and C>v2. Consumer 1 is then pivotal at v, and by Lemma 2.1 the VCG P β α mechanism runs a budget deficit with schedule β. Hence ti (v) C = t1 (v) β α ≤ in this case. Now suppose K (v)= 1, 2 and C v2. Then t1 (v)=v2 and β { } ≤ P ti (v)=(1 b)v i. None of the consumers is pivotal when the schedule is β and they both have− to− “compensate” each other for the congestion that they cause. Then β β α t (v)+t (v)=(1 b)(v1 + v2) v2 = t (v), 1 2 − ≤ 1 where the inequality follows from the fact that b(v1 + v2) v1. α ≥ β Suppose finally that K (v)=∅. Then v1

A private good corresponds to the congested good with the schedule α =(1, 0). Lemma 2.3 then implies that the budget surplus with a private good exceeds the budget surplus with a congested public good. Therefore, a new impossibility result follows.

Corollary 2.3. Suppose n =2and the cost C is such that the good is ineﬃcient to produce as a private good. Then the VCG mechanism runs a budget deﬁcit with thegoodwithanycongestionschedule.

Example 2.1 below demonstrates that the two-agent economy is, in fact, a special case and Corollary 2.3 is incorrect if n>2. The example that follows deals with step congestion schedules. The next section treats such congestion schedules in detail. Here I concentrate on the question of whether the expected budget surplus in the VCG mechanism is “monotonic in congestion”. Fix the size of the population. Now consider goods with diﬀerent step congestion schedules. Notice that it is eﬃcient to produce the good with, say, step schedule σ2 if and only if v1 + v2 C and only the two consumers with the ≥ highest values are given access to the good. With schedule σ3 the three consumers

20 with the highest values are given access and so on. Thus the good with schedule σ2 is more congested than the good with schedule σ3 in a very natural sense.

Example 2.1. The are 6 potential consumers, n =6. The values are drawn from 4 the uniform distribution on [0, 1]. The cost of the good is ﬁxed at C = 3 .The VCG mechanism runs an expected budget surplus with schedule σ3 and an expected 8 budget deﬁcit with schedule σ2.

The upper row in the table below indexes the congestion schedule. The column with m =1corresponds to the private good, m =2–to schedule σ2, ..., m =6– to the pure public good. The middle and the lower rows represent, E [Sσm (v)]– the expected budget surplus in the VCG mechanism under congestion schedules 9 σ1,..., σ6, (from numerical simulations) and its sign.

m 1 2 3 4 5 6 E [Sσm (v)] 0 0.01 0.02 0.15 0.6 1.3 − − − − signE [Sσm (v)] + − − − − Table 2.1: Budget surplus with step congestion schedules

Since C>1, the good is not produced as a private good. It is not too hard to explain why the expected budget surplus may be non-monotonic. The good with schedule σ2 is produced iﬀ v1 + v2 C while with schedule σ3 it is produced iﬀ ≥ v1 + v2 + v3 C. Consumers 1 and 2 use the good with either schedule. It follows σ2 ≥ σ3 that ti (v) ti (v) for i =1, 2. That is, every consumer who does consume the more congested≥ good exerts higher externality and pays more when the good is more congested. But on the other hand, when the good is more congested, fewer consumers are given access to it, hence fewer consumers pay at all.

It is harder to explain the expected budget deficit with schedule σ2 and expected budget surplus with schedule σ3. It can be verified that the probability that a pivotal consumer exists is higher when the schedule is σ2. By Lemma 2.1 the VCG mechanism then runs a budget deficit “more often” with σ2.Inaddi- tion, when no consumer is pivotal, the total payment in the VCG mechanism with schedule σ2 is 2v3, while with schedule σ3 it is 3v4 (the next section is more explicit 8 9 on this). In the example E [2v3]= 7 and E [3v4]= 7 , hence non-pivotal consumers contribute more to the budget when the schedule is σ3.

8 4 The fact that C is constant throughout the example is not crucial. If C(n)= 3 √n,the distribution is quadratic on [0, 1] ,n=10. The VCG mechanism runs an expected budget deﬁcit with schedule σ2 and an expected budget surplus with schedules σ3, σ4, ..., σ7. 9 σm m σm m σm For given m, S (v)= t (v) C if vi C, and S (v)=0otherwise. i=1 i − i=1 ≥ P P 21 2.4 Step congestion schedules

This section contains a proof of Proposition 2.1 for fixed capacity public goods. Examples of such goods are numerous. Section 1.1 mentions a roller coaster. Another example is a parking garage. It has fixed capacity and the first m cars create no congestion. If the m +1’stcarcanonlybeparkedintheentrance, allowing the m +1’st consumer to park creates quite severe congestion. Notice that both a private good and a pure public good can be viewed as public goods with fixed capacity, in the former case the capacity is equal to 1, in the latter it is equal to n. Recall that a public good with fixed capacity m is associated with step congestion schedule σm. Clearly, such good is efficient to provide either for the m consumers with the highest values or for nobody.10 Preceding sections compare public goods with different congestion schedules. The cost of the good, somewhat artificially, was assumed to be the same for the goods with different congestion schedules. In this section the cost of the good, C (σm,n), may depend on the congestion schedule and on the size of the economy, n. This formulation is quite similar to the model of local public goods that is used in public economics literature, see i.e. Section 2.2 in Wildasin (1986). Suppose the objective is to provide a certain level of a public good to m consumers, certain quality of education in a public school, for example. If the number of the consumers grows to m0, the additional consumers create congestion and the quality of education goes down for everyone. To maintain the desired level of education additional investment is required. After suitable normalization one can think of the cost of the public good, C (σm,n) as of the cost of providing the good such that m consumers can use it without suffering from congestion. Thus C (σm0 ,n) >C(σm,n) if m0 >m.On the other hand, in the economy with n potential consumers, when only m of them areallowedtousethegood,C (σm,n) should reflect the expenses on keeping the rest of the population out of the good. Then C (σm,n0) C (σm,n) if n0 >n. Further I use a shortcut notation for the cost, C, bearing in≥ mind the fact that it may be differentforthegoodswithdifferent congestion schedules. The rest of this section proves Proposition 2.1. I derive a lower bound on the expected budget surplus in the VCG mechanism under step schedule σm.Ifthe lower bound is non-negative, the VCG mechanism runs an expected budget surplus and by Theorem 2.1 an efficient, incentive compatible, BB and IR mechanism exists.11

10Deb and Razzolini (1999b) auction-like mechanism in a similar setting allocates the good C C to m consumers with the highest values if vm m for the payment of max vm+1, m . Their mechanism generates an expected budget surplus≥ and satisfies IR, however, it is inefficient. 11Computing the exact value of the expected budget surplus turns out to be© quite hard;ª unlike in the auction for a private object, where just the winner pays, when the congested good is provided for m consumers they all pay and since some of them are pivotal and some are not they may pay different amounts. Even in case of the uniform on [0, 1] distribution the resulting

22 If the cost C>m, the budget of the VCG mechanism is trivially balanced since it is never eﬃcient to produce the good. Further it is assumed that C m. The following lemma introduces the lower bound on the ex-post budget revenue≤ of 12 the VCG mechanism under step congestion schedule σm.

Lemma 2.4. For 1 m n 1 suppose Kσ(v) = ∅.Then, ≤ ≤ − 6 n σ ti (v) mvm+1 i=1 ≥ P and the lower bound is strict.

Proof. Pick an arbitrary i Kσ(v). ∈ σ σ Suppose i is not pivotal at v.Thenbyeﬃciency, #K (v i)=#K (v)=m. Then, −

m+1 mb σ ti (v)= vj vj = vj vj = vm+1. i=j=1 i=j=1 σ − σ − j K (v i) i=j K (v) 6 6 ∈ X − 6 ∈X X X e σ m Now suppose i is pivotal at v. Then ti (v)=C i=j=1 vj. Suppose that m σ − 6 C i=j=1 vj

Thus, the total budget revenue at the realization, where the good is eﬃcient to produce, is bounded from below by mvm+1, hence the budget surplus in the VCG mechanism is bounded from below by mvm+1 C.Atrealizationv,such − that Kσ(v)=∅, the good is not produced and the budget is exactly balanced. The expected budget surplus in the VCG mechanism, under the schedule σm is bounded from below as follows:

σm E [S (v)] EF [ mvm+1 C I v1 + ... + vm C ] ,(2.3) ≥ { − }· { ≥ } where I v1 + ... + vm C is an indicator function that takes a value of 1 when { ≥ } v1 + ... + vm C and 0 otherwise. The notation EF stands for the expectation with the joint≥ density of the n order statistics of n independent draws from the distribution F . Further, where this creates no confusion index F is omitted. Now I use the derived lower bound to characterize the economies where the congested public good is ﬁnanced via budget balanced mechanisms. polynomial is very complicated. 12In what follows I use explicit notations for the relevant social welfares and omit the index m on the schedule σ where it can be inferred from the context.

23 Lemma 2.5. The VCG mechanism runs an expected budget surplus under step congestion schedule σm with associated cost C if C E [vm+1] . ≥ m Proof. It suﬃces to show that the right-hand side of (2.3) is non-negative. I have to introduce some deﬁnitions here. Random variables (v1,v2, ..., vn) are associated if E [γ (v) φ (v)] E [γ (v)] E [φ (v)] (2.4) · ≥ · for every pair of increasing functions γ and φ such that the corresponding expectations exist.13 By Theorem 5.2.2 in Tong (1980) increasing functions of associated random variables are associated random variables. Consider the family of functions that maps the vector of random draws into its order statistics. Obviously, each function from this family is increasing. If n draws are made independently from thesamedistributionF ,then draws are associated. Hence their order statistics (v1,v2,...,vn) are associated. It is easy to see that mvm+1 C and I v1 + ... + vm C are increasing functions, therefore − { ≥ }

E [ mvm+1 C I v1 + ... + vm C ] E [mvm+1 C] E [I v1 + ... + vm C ] . { − }· { ≥ } ≥ − · { ≥ }

Clearly, E [I v1 + ... + vm C ]=Pr v1 + ... + vm C . Thus, the expected budget surplus in{ the VCG mechanism≥ } is bounded{ from≥ below} by

[mE [vm+1] C] Pr v1 + ... + vm C . − · { ≥ }

The second multiplier is obviously non-negative, in fact, it is positive when E [vm+1] C ≥ m . (Notice that this condition implies C

Proposition 2.1 in Section 2.2 now follows from Theorem 2.1. The sufficient condition in Lemma 2.5 is rather intuitive. Each of the m consumers who has access to the good creates an externality of at least vm+1 and hence pays at least vm+1. If in expectation this payment covers the per capita cost, the budget in the VCG mechanism can be balanced.14 13Association implies correlation and is implied by affiliation. The latter property is often used in auction theory and requires the condition (2.4) to be satisfied for the expectation conditional on any sub-lattice in V. For two vectors v0 and v notation v0 v implies v0 vi for every i =1, 2, ..., n. A real-valued = i ≥ function φ(v) is increasing if φ(v) φ(v0) for every v and v0 = v. 14For a corresponding result with≤ complete information see Chapter 11, pp. 186-187 in Cornes and Sandler (1986).

24 Remark 2.2. The assumption that values are drawn independently can be relaxed for Lemma 2.5. Indeed, what was shown is that the VCG mechanism runs an C expected budget surplus if E [vm+1] m when the joint distribution of the random draws is associated. Theorem 2.1, however,≥ holds only if the draws are independent. Lemma 2.5 implies that the VCG mechanism runs an expected budget surplus in the economy with n =3, step congestion schedule σ2 and the uniform 1 on [0, 1] distribution if C 2 . This characterization of the “admissible” cost is based on the lower bound≤ on the expected budget surplus. Therefore for C such 1 that 2

F G C m U

m 01n Figure 2.1: Upper bounds on the per capita cost

m Introduce n ,theshare of the population that can consume the public good without suﬀering from congestion. Figure 1 schematically represents the implications of Lemma 2.5. The size of the population is ﬁxed throughout the picture.

25 Different curves correspond to different distributions. If distribution F stochasti- cally dominates distribution G, the “admissible” per capita cost is higher for the m m distribution F for all shares n ,forfixed n.Verysmalln with m close 1 corresponds to the case of the “almost private” good and the VCG mechanism runs an expected budget surplus even if the per capita cost of the good is close to 1. When m thegoodispurepublic, n =1,andC>0 the VCG mechanism runs an expected budget deficit regardless of the distribution.15 When, the good is “almost public,” m n is close to one, only very small per capita cost can be covered so that the VCG mechanism runs an expected budget surplus. A public good with fixed capacity is an abstraction. Nevertheless, the results of this section (and the following subsection), in my view, apply to quite realistic situations. Consider a highway. Until a certain number of cars is on the highway at a given time, the highway is not congested, the speed limit is what determines the speed. After the number of cars reaches certain capacity, say m, the highway becomes too crowded and the congestion is binding on the speed. Thus, the first m users consume the public good without any congestion, as a pure public good. An appropriate congestion schedule includes m 1’s followed by a decreasing part, which represents more and more serious congestion when the number of consumers goes beyond the capacity. The cost of such a project is primarily determined by the value of m,onecan think of it as of the cost of overcoming congestion on the highway. It is reasonable to assume that more expensive highway, other things equal, will be less congested. With a higher investment, the resulting m is higher. m Suppose α1 = ... = αm =1and αm+j m+j for j 1.Thenfromeffi- ciency perspective if the highway is constructed≤ at all, access≥ is provided to the m consumers with the highest values. Indeed, the inequality

αm+j (m + j) vm mvm ≤ implies αm+j (v1 + ... + vm + vm+1 + ... + vm+j) v1 + ... + vm. ≤ Thus, the characterization of Lemma 2.5 extends to such congestion schedules.

Lemma 2.6. Suppose congestion schedule α satisfies α1 = ... = αm =1and m αm+j m+j for any j 1. Then the VCG mechanism runs an expected budget ≤ C ≥ surplus if E [vm+1] . ≥ m Theliteratureontraffic networks frequently uses the following model of conges- m tion, see Sheffi (1985). The trafficdelay∆τ = β n m , where m is the actual traffic flow, n isthe“capacityofthenetwork”–thevolumeoftra− ffic that would congest the network entirely and β is a calibration constant. This specification implies that the impact of congestion increases with the number of users very rapidly. Thus, simple characterization of Lemma 2.5 is quite useful. 15This cannot be deduced from the lower bound argument, but follows from Corollary 2.2.

26 2.4.1 Asymptotic results For a pure public good Mailath and Postlewaite (1990) and Rob (1989) show that when the cost of the good grows with the size of the economy, the probability of the efficient provision via budget balanced mechanism converges to zero. Hellwig (2003) shows that if the cost stays bounded while the economy grows, the allocation of the budget balanced mechanism converges in probability to the efficient one. For a congested public good (with step congestion schedule) I show that to balance the budget in the efficient mechanism the cost need not be bounded. The VCG mechanism may run an expected budget surplus even when the economy grows, and the number of consumers who can use the good without suffering from congestion, m, and the total cost of the project, C, grow proportionally. It is important here that some share of the population is not given access to the good and a condition on the cost similar to the one in Lemma 2.5 is satisfied. For µ (0, 1) let m =[µn] , where [µn] is the integer part of µn. Then the ∈ m +1-st order statistics out of n, vm+1:n, represents the µ-th sample quantile 16 of distribution F. Consider a sequence of economies n (n, σm,C(σm,n))n∞=1 such that m =[µn] . For every n agivenshare of the populationE ≡ does not suffer from 1 congestion. It can be shown that E [vm+1:n] F − (1 µ) as n . Suppose → − C(σm,n) →∞ the cost of the good increases such that per capita cost m c [0, 1) as n . In the limit for the efficient mechanism to balance the budget,→ ∈ the per →∞ 1 capita cost need not go to zero. When n and m =[µn], F − (1 µ) c provides a transparent (and sharp, see Lemma→∞ 2.7 below) upper bound on− the≥ per capita cost that can be “covered” by the payments in the VCG mechanism.

Lemma 2.7. Consider a sequence of economies n (n, σm,C(σm,n))n∞=1 such C σ ,n E ≡ that m =[µn] for µ (0, 1) , and lim ( m ) = c [0, 1). In the limit the VCG ∈ n m ∈ →∞ 1 17 mechanism runs an expected budget deﬁcit if c>F− (1 µ). − Proof. Notice that c<1 implies that the good is produced with positive probability. The distribution of the sample quantile is asymptotically normal,

1 1 (vm+1:n F − (1 µ)) d √n f F − (1 µ) − − N (0, 1) , · − µ (1 µ) −→ ¡ ¢ − see, for example, Theorem 8.5.1 in Arnold,p et. al. (1992). This implies, in partic- 1 1 ular, that plim vm+1:n = F − (1 µ) and E [vm+1:n] F − (1 µ). n − → − By Theorem→∞ 4 in Al-Najjar and Smorodinsky (2000) the probability that a 1 given consumer is pivotal goes to 0 at the rate √n as n . Since m grows at thesamerateasn, the proportion of pivotal consumers among→∞ those m who pay

16Since the size of the sample in this subsection is not ﬁxed n appears in the index. 17By allowing c [0, 1) I accomodate both the case where m and C grow proportionally and thecasewherem grows∈ faster than C.

27 for the good also goes to 0 as n grows. The payment of any consumer is bounded above by her own value, hence in the limit the impact of pivotal consumers on the expected revenue goes to 0. Since a non-pivotal consumer with probability 1 pays less than per capita cost c, the VCG mechanism runs an expected budget deﬁcit 1 if c>F− (1 µ). −

Remark 2.3. For the sequence n (n, σm,C(σm,n))n∞=1 with m =[µn] for µ C σ ,n E ≡ ∈ (0, 1) and lim ( m ) = c [0, 1) Lemma 2.5 implies that the VCG mechanism n m ∈ →∞ 1 runs an expected budget surplus in the limit if F − (1 µ) >c.Whenthelast inequality is weak to show that the VCG mechanism− runs an expected budget C(σm,n) surplus one has to add a requirement that E [vm+1:n] for the elements of ≥ m the sequence (n, σm,C(σm,n))n that correspond to large n. It should be stressed that the findings for the congested public goods are quite concordant with those for the pure public goods. In my setting the per capita cost that can be covered with a budget surplus in the VCG mechanism depends both on m, the number of consumers who do not suffer from congestion, and on the size of the economy n. Inthecaseofapurepublicgoodm = n. My argument does not apply to this case, but for the sake of comparison it suffices to consider m = n 1. C(σn −1,n) Then the VCG mechanism runs an expected budget surplus if E [vn:n] n −1 . ≥ − As n ,E[vn:n] 0. Theaboveconditiononthecostmaystillbesatisfied for →∞ → C(σn 1,n) large n if the per capita cost n −1 goes to 0 as n , for example if the cost of the good stays bounded as the economy− grows, and→∞ it cannot be satisfied if the per capita cost converges to c>0. This is clearly consistent with the possibility result for large economies with bounded cost due to Hellwig (2003) and the impossibility result for the economies with bounded from 0 per capita cost due to Mailath and Postlewaite (1990) and Rob (1989). Another asymptotic result arises when σm and the associated C stay fixed as n . This result is a particular case of Lemma 2.9 in the next section. →∞ 2.5 Congestion schedules of the general form

The main advantage of step congestion schedules is their analytical tractability. However, the message of the paper–that congested public goods can be produced in efficient, budget balanced mechanisms–is not restricted to step congestion schedules. This section illustrates how some of the results extend to the schedules of quite general shape. Fix congestion schedule α and the associated cost of the project C.Further assume that for schedule α there exists a cut-off number of users, L, such that αL > 0 and αm =0for any m>L.Such an assumption seems reasonable for a project with the fixed cost, the highway of a given size cannot be “stretched” to accommodate the number of cars without bounds. Now we increase the size of the

28 population (α and C stay ﬁxed) and demonstrate that when the economy is large enough the VCG mechanism runs an expected budget surplus. First, Lemma 2.4 is extended to the congestion schedules of the general form.

Lemma 2.8. Suppose Kα(v)= 1, 2, ..., m . Then { } n α ti (v) mαmvm+1. i=1 ≥ P Proof. Consider consumer i Kα(v). I further show that for such i, tα(v) ∈ i ≥ αmvm+1. By deﬁnition of the payments in the VCG mechanism,

α α α ti (v)=SW i(K (v i), v i) SW i(K (v), v), − − − − − m m+1 α α where SW i(K (v), v)=αm vjb C. bThen αm vj C = SW i(K (v), v)+ − i=j=1 − i=j=1 − − α 6 6 α αmvm+1,thatisSW i(K (v), Pv)+αmvm+1 can be expressedP as SW(K (v), v i), − − the social welfare that the economy with vi =0can attain if the good is still produced for the m consumers with the highest values. By eﬃciency of the VCGb α α α mechanism, SW i(K (v i), v i) SW(K (v), v i), hence ti (v) αmvm+1 for every i Kα(v)−. For consumer− − j/≥ Kα(v) follows −tα(v)=0. ≥ ∈ ∈ j Lemma 2.9. For a congestionb b schedule α withb the cut-oﬀ number of users L, there exists N such that for all n N in the economy with n consumers the VCG mechanism runs an expected budget≥ surplus.

Proof. Given C, the good is provided with positive probability for, say, m consumers only if mαm >C.If there is m such that mαm = C, the good is provided for m consumers with probability 0.Choosek arg min mαm .Notice ≡ m : mαm>C { } that a strict inequality is required. Since mαm =0for every m>Lthe search for k is over the ﬁnite set. If C mαm for every m L, the good is produced with probability zero and the budget≥ in the VCG mechanism≤ is trivially balanced. Fur- ther it is assumed that α and the associated C are such that the good is produced with positive probability and therefore k introduced above exists. By Lemma 2.8 in every realization where the good is provided with positive probability, the revenue is bounded below by kαkvL+1, where L is the cut-oﬀ number for schedule α. Therefore, the expected revenue in the VCG mechanism is bounded from below by

E kαkvL+1 I max αm (v1 + ... + vm) C , · m { } ≥ h n oi where I max αm (v1 + ... + vm) C takes a value of 1 if max αm (v1 + ... + vm) m { } ≥ m { } ≥ n o C and 0 otherwise. It is easy to verify that I max αm (v1 + ... + vm) >C is m { } n o 29 an increasing function (see Footnote 16 for deﬁnitions). Therefore, by the same argument as in Lemma 2.5, the above expression is bounded from below by

kαkE [vL+1] Pr max αm (v1 + ... + vm) >C , · m { } n o where Pr max αm (v1 + ... + vm) >C is, in fact, the probability that the good m { } with congestionn schedule α and cost C iso eﬃcient to provide. The expected budget surplusisthenboundedfrombelowby

[kαkE [vL+1] C] Pr max αm (v1 + ... + vm) >C . − · m { } n o For any smooth distribution F on [0, 1] with positive density, for any m, 18 lim E [vm]=1. By construction kαk >Cand clearly Pr max αm (v1 + ... + vm) >C n m { } ≥ 0→∞. n o

The limit result of Lemma 2.9 is less satisfactory than the characterization of Lemma 2.5. Indeed one may say that to obtain an expected budget surplus in the VCG mechanism one has to increase the size of the population to inﬁnity, restrict- ing the good for almost private consumption. This is clearly the consequence of the generality of the result. If the distribution is very skewed towards zero, the value that each and every consumer derives from the good is very low and a very large number of potential consumers is needed to cover the cost of the project.

2.6 Extensions

I have made several assumptions to simplify the exposition. Here I show how some of these can be relaxed. More specifically, I examine how the sufficient condition of Proposition 2.1 is affected. First, suppose consumer i draws his value from Fi on [ai,bi] with ai 0 and ≥ IRi = IRi (vi),whereIRi is a non-decreasing function. Draws are independent. The private values assumption is maintained. For consumer i introduce his base type vi (see Krishna and Perry (2000) for details)

α vi =arg min Ev i [SW (K (s, v i) , (s, v i))] IRi (s) . s [ai,bi] − − − − ∈ One can show that the payment of a consumer who is given access to the good is bounded from below by max v ,vm+1 IRi (v ).Consumerj who is not given { i } − i 18By the aforementioned Theorem 8.5.1 in Arnold, et. al. (1992) for µ>0 small enough lim E [vk+1:n]=1for k =[µn] . For any fixed m and any µ>0 there exists N such that for n →∞ any n N, [µn] >m,hence lim E [vm]=1. n ≥ →∞ 30 access to the good pays IRj v . Then condition (2.3) transforms into − j m ¡ ¢ n E max v ,vm+1 C I v1 + ... + vm C IRi (v ) 0. { i } − · { ≥ } − i ≥ ∙µi=1 ¶ ¸ i=1 P P m IRi (vi) Introduce C0 C + .Clearly, max vi,vm+1 is an in- ≡ Pr v1 + ... + vm C { } { P ≥ } i=1 creasing function; the sufficient condition of PropositionP 2.1 can now be written as m C0 E max v ,vm+1 . { i } ≥ m ∙i=1 ¸ NowP suppose the support of F is again [0, 1] and IRi =0for every i, but the values are interdependent and asymmetric. That is, consumer i draws a real- valued signal si [0, 1] and her value vi = vi (si, s i) , where vi (0)=0for every i; ∈ − vi is increasing in every signal and satisfies the pairwise single crossing condition. Suppose the realized signal profile is s andwiththisprofile Kσm (s)= 1, 2, ..., m . It can be shown that the payment of a consumer who is given access to{ the good} is bounded from below by vm+1 (θi(s i), s i) , where θi(s i) solves vi (θi(s i), s i)= 19 − − − − − vm+1 (θi(s i), s i). − − Under our assumptions vm+1 (θi(s i), s i) is an increasing function, and the − − m C condition of Proposition 2.1 transforms into E vm+1 (θi(s i), s i) . − − ≥ m ∙i=1 ¸ P

19 If vi (θi(s i), s i) >vm+1 (θi(s i), s i) for every θi(s i) [0, 1] set θi(s i)=1. − − − − − ∈ − 31 Chapter 3

On eﬃciency of the N -bidder English auction

3.1 Preliminaries

There is a single indivisible object to be auctioned among a set = 1, 2, ..., N of N { } bidders. Prior to the auction each bidder j receives a real valued signal sj [0, 1]. ∈ Signal sj is bidder j’s private information. Signals are distributed according to a joint density function f(s),whereprofile s =(s1,s2, ..., sN ) represents signals of all the bidders. It is assumed that f hasfullsupportandisstrictlypositiveonthe interior of it. If the realized signals are s, the value of the object to bidder j is Vj(s)–it depends potentially on the information obtained by the other bidders. The sale of an oil track is a typical example of such an environment–a firm’s estimate of the worth of the track may depend on the results of the “off-site” drilling conducted by a rival firm that owns an adjacent track, see Porter (1995). Value functions V =(V1,V2, ..., VN ) areassumedtohavethefollowingproper- ties. For any j,andanyi = j: Vj(0)=0; Vj(1) < ; Vj is twice-differentiable ∂Vj ∂Vj 6 ∞ in s; > 0;and 0. Value functions Vj for all j and distribution f(s) are ∂sj ∂si ≥ assumed to be commonly known among the bidders. We denote s =(sj)j –the signal profile of the bidders from a subset , and s –the signalA profi∈Ale of the bidders from .1 A ⊂ N −A N\A Definition 3.1. For a given profile of signals s the winners circle (s) is the set of bidders with the highest values imputed at s. Formally, I

i (s) Vi(s)=maxVj(s). (3.1) j ∈ I ⇐⇒ ∈N 1 We denote vectors and sets by bold and calligraphic letters correspondingly; a b (a b) À = denotes that ai >bi (ai bi) in every component. ≥ 32 Thus, the object is allocated eﬃciently at s, if the person it goes to–the winner–belongs to the winners circle (s). We require value functions to be regularI –at every s for any subset of bidders ∂Vi(s) (s) it is assumed that det DV =0,whereDV = ∂s is the matrix J ⊂ I J 6 J j i,j of partial derivatives (Jacobian). ³ ´ ∈J

3.1.1 Pairwise single-crossing Deﬁnition 3.2. The pairwise single-crossing (SC) condition is satisﬁed if at any s with # (s) 2, for any pair of bidders i, j (s), I ≥ ∈ I

∂Vi(sj, s j) ∂Vj(sj, s j) − − . (3.2) ∂sj ≤ ∂sj In words, take a group of bidders who have equal and maximal values. If the signal of one of the bidders from the group is increased, the corresponding impact on the value of that bidder is the highest among the group. We say that SC is violated at s if there exist bidders i, j (s) such that ∈ I

∂Vi(sj, s j) ∂Vj(sj, s j) − > − . ∂sj ∂sj

We say that SC is strictly satisﬁed if (3.2) holds with strict inequalities.

3.1.2 The English auction Following Milgrom and Weber (1982) we consider a model that became a standard model for the analysis of English auctions. Specifically, the price of the object rises continuously, and bidders indicate whether they are willing to buy the object at that price or not. A bidder who is willing to buy at the current price is said to be an active bidder. At a price of 0 all the bidders are active, and, as the price rises, bidders can choose to drop out of the auction. The decision to drop out is both public and irrevocable. Thus, if bidder j drops out at a price pj, both her identity and the exiting price pj are observed by all the bidders. Furthermore, once bidder j drops out she cannot “reenter” the auction at a higher price. The auction ends at the moment when at most one bidder remains active. The clock stops, the only remaining bidder is the winner. If no bidders remain active the winner is chosen at random among those who exited last. The winner is obliged to pay the price shownontheclock.2 2 We should complete the description of the gamebyspecifyingtheoutcomeinthecasewhere two or more bidders decide to remain active forever (do not drop out first), and an auction continues indefinitely. In this case we assign to every such bidder a payoff of . Alternatively, it suffices to set that the object is not allocated in this case. −∞

33 At price p all the bidders commonly know who was active at every preceding price. This public history H(p) can be effectively summarized as a sequence of prices at which bidders, inactive at p, have exited, H(p)=p ,where is −M M the set of active bidders just before p.Ifnobidderexitsatp [p0,p00),then ∈ H(p0)=H(p00).DenotewithH¯ (p) the public history H(p) together with all exits that happen at p. Therefore, if H¯ (p) = H(p), then there exists a bidder who exited at p. All the bidders are assumed to be6 active just before the clock starts at p =0, so H(0) = ∅. In the English auction a strategy of bidder j determines the price at which she would drop out given her signal and public history–given that no other bidder drops out earlier. Formally, following Krishna (2003), a bidding strategy for bidder N M j is a collection of functions βjM :[0, 1] R+− R+,where is the ∈ M × −→ M set of active bidders just before p, M =# > 1.FunctionβM determines M j the price βjM(sj; H(p)) at which bidder j will drop out when the set of active bidders is , j’s own signal is sj, and the bidders in have dropped out at M N\M prices H(p)=p = pj j . The rules of the English auction require that −M { } ∈N\M βjM(sj; p ) > max pj : j . If active bidders are able to extract true signals s −M of inactive{ bidders∈ N\M} from their exit prices p , the strategies can be −M −M equivalently written as βjM(sj; s ). The equilibrium concept we use−M throughout this paper is a Bayesian-Nash equilibrium. The equilibrium we present in Section 3.3 is also ex-post and efficient. Definition 3.3. An ex-post equilibrium is a Bayesian-Nash equilibrium β with the property that β remains a Nash equilibrium even if the signals (s1,s2, ..., sN ) are commonly known. This notion is equivalent to the notion of the robust equilibrium (see Dasgupta and Maskin (2000)), which requires that the strategies remain optimal under any initial distribution of signals. Definition 3.4. An equilibrium in the English auction is efficient if the object is allocated to the bidder with the highest value in every realization of signals (s1,s2, ..., sN ). Maskin (1992) establishes that: Claim 3.1 (Maskin, sufficiency). The pairwise single-crossing is sufficient for the existence of an efficient ex-post equilibrium in the English auction with two bidders. Claim 3.2 (Maskin, necessity). Suppose the pairwise single-crossing is violated at some interior signal profile. Then the English auction with N 2 bidders does not possess an efficient equilibrium.3 ≥

3 This claim is indicated in Maskin (1992). The proof is straightforward.

34 The following example illustrates that eﬃcient equilibria may exist even when SC is violated on the boundary of the signals’ domain. Example 3.1. Consider the English auction with two bidders with value functions of the form

2 1 V1 = 3 s1 + 3 s2, V2 = s1 + s2. There exists an eﬃcient ex post equilibrium.

At the point s1 = s2 =0, V1 = V2, the pairwise single-crossing is violated, while at any other s it is vacuously satisfied. Strategies β1(s1)=s1, β2(s2)= (bidder 2 never drops out first) form an ex post equilibrium, which is efficient. ∞

3.2 Generalized single crossing

For an arbitrary vector u consider u Vk(s)–the derivative of Vk in the direction ·∇ ∂Vk ∂Vk ∂Vk u,where Vk(s)= , ,..., is the gradient of Vk(s). ∇ ∂s1 ∂s2 ∂sN Definition 3.5.a (Directional³ formulation)´ The generalized single-crossing (GSC) condition is satisfied if at any s with # (s) 2,foranysubset of bidders (s), I ≥ A ⊂ I u Vk(s) max u Vj(s) , (3.3) j · ∇ ≤ ∈A { · ∇ } for any bidder k (s) and any direction u, such that ui > 0 for all i ∈ I \A ∈ A and uj =0for all j/ . ∈ A In words, select any group of bidders from (s)–bidders who have equal and maximal values. Increase theA signals of biddersI from only. Consider the corresponding increments to the values of bidders from (As). GSC in the directional formulation requires that the increments to the valuesI of bidders from (s) are at most as high as the highest increment among the bidders from .I Or,\A stated differently, at least one bidder from should be in the resulting winners’A circle. Note that in the case of = j ,AGSC reduces to the pairwise single-crossing. A { } Definition 3.5.b (Equal increments formulation) The generalized single-crossing (GSC) condition is satisfied if at any s with # (s) 2,foranysubset of bidders (s), I ≥ A ⊂ I uA Vk(s) 1 (3.4) · ∇ ≤ for any bidder k (s) , where vector uA =(uA, 0 ) is defined by ∈ I \A A −A uA Vj(s)=1, · ∇ for all j . ∈ A 35 Existence and uniqueness of vector uA follows from the fact that uA solves A vector system DV (s) uA = 1 (marginal increments to the values are equal), A 1 · A A thus uA =(DV (s))− 1 . By regularity assumption, det DV (s) =0.Wefurther A A · A A 6 refer to uA as to the equal increments vector corresponding to subset . A In words, select any group of bidders from (s). There exists the unique direction of the change of the signals of biddersA fromI such that along this direction the values of all the bidders from increase uniformly.A GSC in the equal increments formulation requires that alongA this direction the value to any bidder from (s) does not increase more rapidly. I \A Lemma 3.1. The formulations of GSC giveninDefinitions 3.5a and 3.5b are equivalent.

Proof. The proof is rather technical and is presented in Appendix A.1.

Thus the equal increments formulation of GSC is only seemingly less demand- ing than the directional formulation. In fact, both of them put the same restriction on the value functions. In the proofs that follow we use these formulations inter- changeably, whichever is more convenient for the speciﬁcargument. Now we state our main results.

3.2.1 Results Proposition 3.1 (Suﬃciency). Suppose value functions satisfy GSC. Then there exists an eﬃcient ex post equilibrium in the N-bidder English auction.

Deﬁnition 3.6. GSC condition (in the directional formulation) is violated at the signal proﬁle s for the proper subset (s) and bidder k (s) if there A ⊂ I ∈ I \A exists a vector u,withui > 0 for all i , uj =0for all j/ , such that ∈ A ∈ A

u Vk(s) > max u Vi(s) . (3.5) i · ∇ ∈A { · ∇ } Similarly a violation of GSC condition in the equal increments formulation can be defined. Hereafter whenever we say that GSC is violated it means that there exist some s, (s),andk (s) , such that (3.5) holds. A ⊂ I ∈ I \A Proposition 3.2 (Necessity). Suppose GSC condition is violated at some interior signal profile. Then no efficient equilibrium in the N-bidder English auction exists.

The proofs of Propositions 3.1 and 3.2 are presented in Sections 3.3 and 3.4 correspondingly.

36 3.2.2 Examples Here we present three examples to illustrate the link between the generalized single- crossing condition and efficiency. We start with the known example where the English auction fails to allocate efficiently and show that GSC is indeed violated there. In the other two examples we show that an efficient equilibrium may or may not exist if GSC is violated on the boundary of the signals’ domain and satisfied everywhere else.

Example 3.2 (Perry and Reny (2001)). Consider the English auction with three bidders with value functions of the form

V1 = s1 + s2s3, 1 V2 = 2 s1 + s2, V3 = s3.

There exists no eﬃcient equilibrium.

Perry and Reny (2001) contains the proof of the fact that the three-bidder English auction possesses no efficient equilibrium in this example. It is easy to see that GSC is violated here. Notice that at the signal profile s =(.3,.6,.75) all the values are tied. Now choose a subset A = 2, 3 and the direction u =(0, 1, 1). { } ∂V s ∂V s Then, u V (s)=u V (s)=1, while u V (s)= 1( ) + 1( ) =1.35 > 1. 2 3 1 ∂s2 ∂s3 The next· ∇ example· generalizes∇ the message· ∇ of Example 3.1 and illustrates that the English auction may possess an efficient equilibrium even when value functions violate GSC on the boundary of the signals’ domain. In Example 3.3, however, any bidder may have the highest value, hence the fact that an efficient equilibrium exists is not as trivial as it was in Example 3.1.

Example 3.3. Consider the English auction with three bidders with value functions of the form

2 V1 = s1 + 3 (s2 + s3), V2 = s2,

V3 = s3.

There exists an eﬃcient ex post equilibrium.

It is clear that GSC is violated at s =(0, 0, 0) for = 2, 3 , bidder 1 and vector (0, 1, 1). There is no other s at which values of allA three{ bidders} are equal. SC (or GSC for # =1)isclearlysatisfied everywhere. The following strategiesA form an efficient ex post equilibrium. When all the bidders are active, bidders 2 and 3 drop out when the price reaches their private values and bidder 1 never drops out first. After one of bidders 2 and 3,saybidder

37 2, drops out, bidder 3 stays active until the price reaches her private value. Bidder 2 1 drops out when the price p reaches s1 + 3 (p + s2),wheres2 is the revealed signal of bidder 2, who had dropped first. Note that if bidders 2 and 3 follow these strategies and drop out simultaneously, the value of the object to bidder 1 is always higher than the price that she has to pay. Thus the “waiting strategy“ is “safe” for bidder 1.Bidders2 and 3 use their dominant strategies. The next example illustrates that GSC being satisfied only in the interior of the signals’ domain is not sufficient for the existence of an efficient equilibrium.

Example 3.4. Consider the English auction with four bidders with the value functions of the form

3 V1 = s1 + 2 s2s3, 3 V2 = s2 + 4 s1, V3 = s3, 1 V4 = 2 (s1 + s2 + s3).

There exists no eﬃcient equilibrium.

3 At s = 0, GSC is violated for the subset = 1, 2, 3 , vector u = 1, 1, 1 4 and bidder 4. GSC and SC are satisfied(atleastweakly)ateveryinteriorsignalA { } © ª profile. There is no violation on subset 2, 3 and bidder 1 since whenever V2 = V3, 1 { 1 } 1 V4 >V3. Observe that 2 (s1 + s2 + s3)= 2 (s3 + 4 s1 + s3) >s3, where the equality 3 follows from s3 = s2 + 4 s1.ToverifySC observe that V12 V13 − − 2 4 in the interior profiles where V1 = V3. 2 Bidder 1 has the highest value whenever s2 = s3 > 3 hence she should not drop 1 1 5 out first. At the signal profile 5 , 4 , 16 ,x ,V2 is the highest. At the signal profile 1 1 5 1 5 10 , 4 , 16 ,x ,V4 is the highest. At the signal profile 0, 4 , 16 ,x ,V3 is the highest. Therefore neiter bidder 2, nor¡ bidders 3 or¢ 4 can drop out first. Thus, there is no ¡efficient equilibrium¢ in the English auction. ¡ ¢ The common feature of Examples 3.3 and 3.4 is that bidder 1 with s1 =0 has the highest value whenever V2 = V3 =0.Thedifference in predictions of the examples is due to the fact that GSC pertains6 only to the bidders with equal and maximal values, and additional restrictions on values that are not maximal determine whether an efficient equilibrium exists. If GSC is imposed on such values, as in Example 3,anefficient equilibrium exists, and if it is violated, as in Example 4,anefficient equilibrium may not exist. No such restrictions (on the values that are not the maximal) is needed at interior signal profiles. We view this “boundary problem” as a technicality, a simple and natural restriction on the value functions rules out the complication altogether.

38 Remark 3.1. Suppose that a bidder with the most pessimistic signal cannot have the highest valuation whenever some other bidder has a positive signal. Formally, suppose Vi(s) < maxj Vj(s) for every s = 0 with si =0.ThenGSC is both necessary and sufficient condition for efficiency6 of the English Auction: an efficient equilibrium exists if and only if GSC is satisfied.

3.3 Suﬃciency

In this section we show that GSC is sufficient for the existence of an efficient equilibrium in the N-bidder English auction. The proof is by construction.4 In this equilibrium, for a given public history H(p)=p ,activebidders from calculate σ(p, H(p))–profile of inverse bidding functions−M that are used to defiMne the strategies. For bidder j, to decide whether to be active at p or not 5 is sufficient to compare her true signal sj with σj(p). If σj(p) pj. Now we define the (equilibrium) strategies. Suppose there exists a profile of functions σ(p, H(p)),whichwecallinferences, such that:6

1. for an inactive bidder i , σi(p)=σi(pi,H(pi)),thatisσi(p) is ﬁxed ∈ N\M after bidder i exits at pi;

2. for any active bidder j , σj(p) [0, 1] solves Vj(σj(p), σ j(p)) = p,if ∈ M ∈ − such a solution exists with σj(p) 1, otherwise σj(p)=1with Vj(σj(p), σ j(p)) < p. ≤ −

Thus, for all active bidders, σ (p) are determined simultaneously, as a solution to M V (σ (p), σ (p)) 5 p1 , σ (p) 5 1 , M M −M M M M (3.6) j :(Vj(σ(p)) p)(σj(p) 1) = 0. ∀ − − For bidder j strategy βjM :(sj,H(p)) R+ is ∈ M −→

βjM(sj; p )=argmin σj(p) sj . (3.7) −M p { ≥ } 4 Milgrom and Weber (1982) propose the ideology of constructing efficient equilibria for the English auction with symmetric bidders; Maskin (1992) extends it to the case of two asymmetric bidders, and Krishna (2003) generalizes it to the case of N asymmetric bidders. 5 To shorten the notation we are omitting H(p) from the set of arguments, whenever the public history can be implied from the context. 6 Note that H(p) is a collection of exit prices, H(p)=p ,where is the set of active −M M bidders. Therefore, σ(p, H(p)) will be defined for all p maxi pi. ≥ 6∈M 39 Strategy βj can be interpreted as follows. If bidder j is active at p, given the public history H(p)=p of exits of inactive bidders ,bidderj is supposed to −M N\M exit the auction at pj = βjM(sj; p ), provided no other bidder exits before. If −M the current price p satisfies p<βjM(sj; p ),bidderj is suggested to maintain −M an active status; if p βjM(sj; p ) bidder j is suggested to exit at p. ≥ −M Once bidder j exits at pj, other bidders update the public history and, expecting bidder j to follow (3.7), infer s∗ = σj(pj).Ifσj( ) is non-decreasing the inferred s∗ is j · j unique and coincides with true signal sj. The strategies can then be reformulated as functions of the own and inferred signals of inactive bidders, βjM(sj; s )= −M βjM(sj; p ). To proceed−M with the sufficiency result we need the following: Lemma 3.2. Suppose GSC is satisfied. Then there exist inferences σ(p, H(p)), such that each σj( ,H(p)) is continuous and non-decreasing for any H(p),and · σj(p, H¯ (p)) = σj(p, H(p)) for all p such that H¯ (p) = H(p). For any active at 6 H(p) bidder j, j (σ(p)) if σj(p, H(p)) < 1. ∈ I Proof. Proof is presented in Appendix A.2. The following Lemma then proves Proposition 3.1. Lemma 3.3. Suppose value functions satisfy GSC.Thenβ defined by (3.7) constitute an efficient ex-post equilibrium in the N-bidder English auction. Proof. We first show that β are well-defined. For any bidder j, arbitrarily fixexit prices of other bidders, p j, possibly with pi = for some bidders. Then one can − ∞ obtain σj(p) defined for any p 0 as σj(p)=σj(p, H(p)), where H(p)= p

Vj(σj(p∗), s j)=maxVi(σj(p∗), s j)=p∗. (3.8) − i=j − 6

The pairwise single-crossing, σj(p∗) sj, and equation (3.8) imply that ≤

Vj(s) max Vi(s) p∗, (3.9) i=j ≥ 6 ≥ so bidder j is (one of) the bidder(s) with the highest value. Note that price p∗ that bidder j has to pay for the object does not depend on the signal of bidder j. Finally, we show that β form an ex-post equilibrium. Suppose every bidder other than bidder j follows the proposed strategy and bidder j deviates. The

40 payoff of bidder j can change only if the deviation affects whether bidder j obtains the object. If bidder j wins the object as a result of the deviation, she has to pay pj∗ =maxi=j Vi(σj(pj∗), s j).Ifbidderj were not the winner in the equilibrium, 6 − σj(pj∗) sj since σj(p) is non-decreasing, so Vj(s) pj∗ and the deviation is not profitable.≥ If as a result of the deviation bidder j is≤ not the winner while she is in the equilibrium, she is possibly forfeiting positive profits according to (3.9). Thus, no profitable deviation exists. Theaboveisvalidevenifsignalss are commonly known, hence the presented equilibrium is ex-post.

3.4 Necessity

In this section we establish that GSC is necessary for the existence of an efficient equilibrium in the N-bidder English auction. Proposition 3.2 (Necessity). Suppose GSC is violated at an interior signal profile. Then no efficient equilibrium in the N-bidder English auction exists. The proof is quite involved. There is a number of technical complications to be resolved. Before we proceed we would like to illustrate the main ideas behind the proof with a partial analysis of the three-bidder English auction.

3.4.1 An illustration Claim 3.3. Suppose there are three bidders in the auction, = 1, 2, 3 .Suppose N { } that GSC is violated for = 2, 3 and bidder 1 at the interior signal profile s0, A { } such that V1(s0)=V2(s0)=V3(s0). Suppose also that SC is strictly satisfied. Then, no efficient equilibrium exists with β2(s2; ∅) and β3(s3; ∅) continuous at s20 and s30 correspondingly. Proof. Suppose an efficient equilibrium exists. Step 1. Consider a stage in the auction when all three bidders are active, H(p)=∅. GSC is violated for = 2, 3 and bidder 1 at s0; that is, there exists A { } direction u =(0,u2,u3) with u2 > 0, u3 > 0, such that for every small enough ε>0, if the signals of the bidders 2 and 3 are increased along this direction, bidder 1 has the highest value,

V1(s0 + εu) > max V2(s0 + εu),V3(s0 + εu) . (3.10) { } Thus, eﬃciency prescribes that she must not be the ﬁrst to drop out,

β (s0 ) > min β (s0 + εu2),β (s0 + εu3) . (3.11) 1 1 { 2 2 3 3 }

Step 2.Weshowthatβ2(s20 )=β3(s30 ). Suppose not, without loss of generality consider β2(s20 ) <β3(s30 ). Bycontinuityofβ2(s2) and of β3(s3) at s20 and s30

41 correspondingly, for sufficiently small ε,wehaveβ2(s20 +εu2) <β3(s30 ) and β2(s20 + εu2) <β3(s30 + εu3). Together with (3.11), we get β1(s10 ) >β2(s2) for s2 >s20 close to s0 ,andsoβ (s2) < min β (s0 ),β (s0 ) . This contradicts efficiency since the 2 2 { 1 1 3 3 } value of bidder 2 is strictly the highest at (s10 ,s2,s30 ), so she must not be the first to drop out. Therefore, β (s0 )=β (s0 ) b. 2 2 3 3 ≡ Taking limits in (3.11) we obtain β (s0 ) b. 1 1 ≥ Step 3. Since bidder 2 has strictly the highest value at (s10 ,s2,s30 ) for s2 >s20 close to s0 ,wehaveβ (s2) > min β (s0 ),β (s0 ) . Similarly for bidder 3. Therefore, 2 2 { 1 1 3 3 } β2(s2) >b, β3(s3) >b, (3.12) for s2 >s20 and s3 >s30 close to s20 and s30 correspondingly. Step 4. Finally, by (3.10) and by continuity of value functions, for a given ε>0, there exists ε1 > 0, such that bidder 1 has the highest value at (s0 ε1,s0 +εu2,s0 + 1 − 2 3 εu3).Byefficiency, it must be that

β (s0 ε1) > min β (s0 + εu2),β (s0 + εu3) . 1 1 − { 2 2 3 3 } Together with (3.12) this implies

β (s0 ε1) >b= β (s0 )=β (s0 ). (3.13) 1 1 − 2 2 3 3 Thus, at (s10 ε1,s20 ,s30 ) bidders 2 and 3 drop out simultaneously, and bidder 1 wins. However,− since SC is strictly satisﬁed, bidder 1 has the lowest value. Inwhatfollowsasanintermediatestepwebasicallyshowthatifaneﬃcient equilibrium exists then β2(s2; ∅) and β3(s3; ∅) are almost everywhere continuous intheneighborhoodsofs20 and s30 correspondingly.

3.4.2 Proof of Proposition 3.2 We proceed from the contrary–we assume that an efficient equilibrium exists, while GSC is violated at some interior signal profile. Fix an efficient equilibrium

β, βi(si; H(p)) is the equilibrium strategy of bidder i with signal si. No restrictions on the strategies are imposed, they need not be monotonic and can be discontinuous everywhere. Our proof incorporates the following three main principles. Principle 1.Supposethatats exactly one bidder, say bidder j,hasthehighest value among all the bidders, # (s)=1. Then, in the equilibrium bidder j must win the object, so at any intermediateI history H(p) with being the set of active bidders, M

βj(sj; H(p)) > min βi(si; H(p)); (3.14) i ∈M that is, the bidder who has the highest value must not be the one to drop out ﬁrst. And, in particular, bidder j cannot be among the bidders who drop out simultaneously at the end, since then she will not obtain the object with certainty.

42 When two or more bidders have the highest value at some s,wedonotimpose any restrictions on who should be the winner among them. In particular, we do not require that each of them has to win the object with positive probability. Therefore, (3.14) must hold only for the eventual winner, not for any j (s). ∈ I Principle 2.IfGSC is violated at some interior signal profile, we can find a possibly different interior signal profile s,whereGSC is violated for bidder k and a minimal subset , that is the subset that contains the fewest possible number of bidders needed toA violate GSC. Indeed, for all interior profiles s and all pairs k and that exhibit a violation, the number of bidders in is an integer between 1 and AN 1,andsomin and arg min operators are well-deAfined. Then we can find s, minimal− subset , and bidder k =1(after relabeling), such that these are the only bidders in theA winners circle, +1 1 = (s). To separate bidders A ≡ A ∪ { } I = (s) +1 when = ∅, we can lower the values of all the bidders from in BsomeI manner,\A while keepingB 6 the values of all the bidders from +1 fixed andB the signals of all the others fixed. If the change is sufficiently small,A then by regularity and continuity of value functions and by continuity of their first derivatives the resulting signal profile s will be interior, (s)= +1,andGSC is violated for bidder 1 and minimal . I A A Our focus will always be on bidders +1, the signals (and so the strategies) of therestofthebiddersarefixed throughoutA the proof. Principle 3. This principle, or to be more exact, convention of how we use notation to make strong statements about bidding functions shortens and simplifies the proof by a lot. It is important then to describe it in detail. Suppose that at s, GSC is violated for bidder 1 and subset , is minimal, +1 A A (s)= ,ands (s) are fixed. By continuity, there exists an open neighborhood I A s −I of s (s), U (s), such that for any s0 =(s0 (s), s (s)), (s0) (s). In other words, if I I I −I I ⊆ I we slightly disturb the signals of the bidders from (s) only, all the bidders with thehighestvalueasaresultmustbelongto +1. I A +1 For any such s0,letp(s0) to be the first price at which a bidder from drops A out. Slightly abusing the notation, H(s0)=H(p(s0)) is the history of play just +1 prior to p(s0)–the sequence of exits of the bidders not from uptoamoment of the first drop-out of a bidder from +1. It is not necessaryA that all the bidders +1 A not from exit first, and we allow for a possibility that H(s0)=∅. A If it were the case that for any such s0 history H(s0) is the same (this would havebeenthecaseifallthebiddersnotfrom +1 exited before any of the bidders +1 A from for all s0, e.g.), then we would fixthishistory,H, and consider only A +1 parts of the strategies, β (sj; H) for all j , to derive results and reach a j ∈ A contradiction at the end. Unfortunately, for different s0, H(s0) may be different. +1 For example, suppose the first bidder from to drop out (at p(s0))isbidder A +1 j. Then if we change slightly sj,bidderj may no longer be the first from to A drop out, H(s0) may stay the same or lengthen depending on whether some other bidder from +1 or a bidder from +1 exits first instead. Even if bidder j is A N\A 43 still the first from +1 to drop out, the number of bidders who exit before j can decrease or increase.A To avoid dealing with potentially different histories, and, therefore, different parts of strategies, we propose the following. For any bidder j +1 for any ˆ ∈ A signal sj we calculate βj(sj)–the price level at which bidder j with sj would exit according to her equilibrium strategy if all other bidders from +1 remained active forever (or do not exit before her) and all the bidders +1Afollowed their N\A ˆ equilibrium strategies (their signals are fixed).Itispossiblethatβ (sj)= for j ∞ some bidder j with signal sj. In the proof we will be deriving results concerning these bidding functions. The fact that the bidder, say j, with strictly the highest value at s0 never drops out first implies that ˆ ˆ βj(sj0 ) > min βi(si0 ). i +1 ∈A +1 Indeed, in equilibrium, there must exist a bidder i ,who,ats0,dropsout +1 ∈ A ˆ the first among . But,thenthepriceatwhichshedoessoisequaltoβi(si). Bidder j (with theA highest value) must at least stay longer.

In what follows, to avoid excessive notation, we are writing simply βj(sj) in ˆ +1 place of βj(sj). We are also omitting the signals of the bidders from since these are fixed, so s denotes the profile of signals held by the biddersN\A from +1 only. We write s when referring to the full profile of signals. A N ProofofProposition3.2. Suppose that the minimal subset contains n bidders, and GSC is violated at some interior signal profile s for A and bidder 1,with +1 = (s ).Thefactthatn 2 follows from ClaimN 3.2. A A StepI1.N Consider trajectory≥s(t) that for each t solves

+1 Vj(s(t)) = V (s)+t, for all j . ∈ A Such a trajectory exists and is unique, since it can be found as a solution to the differential equation ds 1 =(DV (s))− 1. (3.15) dt · By continuity of value functions and their first derivatives, +1 = (s (t)),and GSC is violated at s (t) for and bidder 1 for all t in someA openI neighborhoodN 0 N A Ut of t =0. 0 Step 2.Considers0 = s(t) for an arbitrary t U .LemmaA.4inAppendix ∈ t A.3 shows that for any j ,thereexistbj(sj0 ) limsj s inf βj(sj),andthese ∈ A ≡ ↓ j0 limits are equal for the bidders from : for any j , bj(s0 ) b(s0) b(t) < . A ∈ A j ≡ ≡ ∞ In addition, for any j and sj >s0 sufficiently close to s0 , β (sj) b(t);and ∈ A j j j ≥ for bidder 1, β1(s10 ) >b(t). Step 3. Corollary A.2 in Appendix A.1 shows that either: (i) for any j , ∈ A sj(t0) >sj(t) while s1(t0) t;or(ii)foranyj , sj(t0) s1(t) for t0 >t. This, together with the results of Step 2,implies that b(t) is (weakly) monotonic in t. In Case (i) it is non-decreasing, in Case (ii) it is non-increasing. Step 4. Corollary A.3 in Appendix A.3 shows that if for some bidder j , ∈ A βj(sj(t)) = b(t),thent has to be a discontinuity point for b(t).Sinceb(t) is monotonic6 it has no more than a countable number of discontinuity points. Hence 0 for almost all t Ut , βj(sj(t)) = b(t) for every j . That is, when the signals of bidders from∈ belong to trajectory s(t), bidders∈ A from almost always exit simultaneously. A A Step 5. Consider two continuity points for b(t),tand t0, such that b(t0) b(t). ≥ In Case (i), t0 >t; in Case (ii), t0 b(t0) 1 ≥ b(t)=βj(sj(t)) for all j . ∈ A +1 Step 6. By construction, at t, (s1(t), s (t)) = . Since there exists a unique I A A solution to (3.15), and SC is satisfied for bidder 1,wehavethatats1 s1(t), ≡

∂V1(s1, s (t)) ∂Vj(s1, s (t)) A > max A . (3.16) ∂s j ∂s 1 ∈A 1

Therefore, we can ﬁnd t0 suﬃciently close to t such that: t0 is a continuity point for b(t), s1(t0)

45 Chapter 4

Ineﬃcient ex-post equilibria in eﬃcient auctions

4.1 Preliminaries

There is a single indivisible object to be auctioned among a set = 1, 2, ..., N of N { } bidders. Each bidder j receives a real valued signal sj [0, 1] prior to the auction. ∈ As usual, a signal proﬁle s =(s1,s2, ..., sN ) denotes the signals of all the bidders, s denotes the signals of the bidders from the set ,ands –the signals of the biddersA from the set . Signals are distributedA according−A to a joint density function f(s) with fullN\A support, and f is assumed to be strictly positive on the interior of its support. If the realized signals are s, the value of the object to bidder j is vj(s). Value functions vj are assumed to have the following properties: vj(0)=0;vj ∂vj ∂vj is diﬀerentiable in s; vj is increasing in the sense that > 0 and 0 for any ∂sj ∂si ≥ j and any i = j. In addition, we assume that value functions satisfy the pairwise single crossing6 condition.

Deﬁnition 4.1. Take any signal proﬁle s such that vj(s)=vi(s) for any i, j . Valuefunctionssatisfythepairwise single crossing condition if ∈ N

∂v (s) ∂v (s) j > i ∂sj ∂sj for any i, j and i = j. ∈ N 6 4.2 English auction

In this section we present an ineﬃcient equilibrium in discontinuous strictly increasing strategies, which is: i) ex-post, ii) does not involve dominated strategies.

46 We concentrate on the English auction with two bidders, however, this is not a restriction–the construction extends for the English auctions with arbitrary N bidders. The setup of the English auction repeats the one presented in Milgrom and Weber (1982) and extended for the asymmetric value functions in Maskin (1992) and Krishna (2001). The English auction is an open ascending price auction. Specifically, the price of the object rises, and bidders indicate whether they are willing to buy the object at that price or not. A bidder who is willing to buy at the current price is said to be an active bidder. At a price of 0 all bidders are active and, as the price rises, bidders can choose to drop out of the auction. The decision to drop out is both public and irrevocable. Thus, with two bidders, if say, bidder 2 drops out at a price p2, the auction stops, and the good is given to bidder 1 for the price p2. In the English auction with two bidders a strategy βj(sj) for bidder j determines the price at which he is going to drop out given his signal. With two bidders the English auction is strategically equivalent to the sealed-bid second price auction. Definition 4.2. An ex-post equilibrium in the English auction is a Bayesian- Nash equilibrium with the property that even if the signals (s1,s2) become commonly known, no bidder j can gain by changing his strategy given the strategy of the other bidder β j. − First consider an efficient equilibrium in the English auction with two bidders: the strategies that constitute this equilibrium will be further used in our construction.

Take a pair of strategies β1(s1),β2(s2) such that the inverse functions σ1(p),σ2(p) solve v1(σ1(p),σ2(p)) = v2(σ1(p),σ2(p)) = p (4.1) 1 for all p minj σ− (1). ≤ j Lemma 4.1. Maskin (1992). Suppose value functions are increasing and satisfy the pairwise single crossing condition. Then the inverse functions σ1(p),σ2(p) areincreasingandthecorrespondingstrategiesβ1(s1),β2(s2) constitute an ex-post equilibrium 1 Proof. Suppose that the solution to (4.1) exists and that β1(s1)=p1 >β2(s2)= p2. Since p2 = v1(σ1(p2),σ2(p2)), where σ2(p2)=s2 and s1 = σ1(p1) >σ1(p2), it must be that v1(s1,s2) >p2, because v1 is strictly increasing in s1. This implies that the winner (bidder 1) makes positive profit, cannot affect the price, and therefore cannot profitably deviate ex-post. Since p1 = v2(σ1(p1),σ2(p1)), where σ1(p1)=s1 and s2 = σ2(p2) <σ2(p1), it must be that v2(s1,s2)

47 An equilibrium is efficient if β1(s1) >β2(s2) if and only if v1(s1,s2) >v2(s1,s2). To see that β1(s1),β2(s2) is an efficient equilibrium notice that v1(σ1(p2),σ2(p2)) = v2(σ1(p2),σ2(p2)). Suppose β1(s1)=p1 >p2 = β2(s2). Since s1 = σ1(p1) >σ1(p2) and σ2(p2)=s2, the pairwise single crossing prescribes that v1(s1,s2) >v2(s1,s2). Thus, the presented equilibrium is efficient. We further refer to this equilibrium as to the regular equilibrium. Now we present a construction of the discontinuous inefficient ex-post equilibrium. Fix two arbitrary levels p0 and p00 >p0. Define the signals s0 σ1(p0),s0 1 ≡ 2 ≡ σ2(p0), where σ1(p0),σ2(p0) is the solution to (4.1) at p = p0. Similarly define the signals s00 σ1(p00),s00 σ2(p00). Functions σ1(p),σ2(p) are strictly increasing, 1 ≡ 2 ≡ hence sj00 >sj0 .

Discontinuous strategies β1(s1), β2(s2) areconstructedasfollows(seeFigure 1). b b 6 6 β10 β20 p00 v2( ,s00) · 2

v1(s0 , ) 1 · p0 v1(s10 ,s20 )=v2(s10 ,s20 )

- - s10 s100 s20 s200

Figure 4.1: Discontinuous ex-post equilibrium in the English auction

When the signal of bidder 1, s1 [0,s0 ] [s00, 1], and the signal of bidder ∈ 1 ∪ 1 2, s2 [0,s0 ] [s00, 1], the bidders use the “regular” strategies β (s1)=β (s1), ∈ 2 ∪ 2 1 1 β (s2)=β (s2), such that inverse functions σ1(p),σ2(p) solve (4.1) for all p p0 2 2 ≤ and p p00. b ≥ b When s1 (s0 ,s00), bidder 1 bids accordingly to β (s1)=v2(s1,s00). ∈ 1 1 1 2 When s2 (s0 ,s00), bidder 2 bids accordingly to β (s2)=v1(s0 ,s2). ∈ 2 2 2 1 Note that v1(s10 ,s20 )=v2(s10 ,s20 ). Value function bv2 is strictly increasing in s2, hence v2(s1,s00) >v2(s1,s0 ) v2(s0 ,s0 ) for any s1b >s0 .Thusthesuggested 2 2 ≥ 1 2 1 strategy of bidder 1 involves a jump at s10 . Similarly, s200 is a discontinuity point of the suggested strategy of bidder 2. By construction β1(s1) is continuous at s100 and strictly increasing everywhere. Similarly β2(s2) is continuous at s20 and strictly increasing. b b 48 Claim 4.1. Suppose value functions v1,v2 are increasing and satisfy the pairwise single crossing condition. Then strategies β1(s1), β2(s2) constitute an ex-post equilibrium. b b Proof. Discontinuous strategies β1(s1), β2(s2) are “built on” strategies β1(s1),β2(s2). It should be clear from the proof of Lemma 4.1 that, whenever say, bidder 1 uses the

“regular” bid β1(s1)=β1(s1) andb winsb against the “regular” bid β2(s2)=β2(s2), neither the winner, nor the loser can profitably deviate ex-post in all such (s1,s2). Consider bidderb 1 with the signal s1 [0,s0 ]. By construction,b bidder 1 uses ∈ 1 the bid p1 = β (s1), wins only against the bids β (s2) for s2 [0,σ2(p1)) and loses 1 2 ∈ against the bids β (s2) for s2 (σ2(p1),s0 ] [s00, 1], hence cannot profitably deviate 2 ∈ 2 ∪ 2 in any of these realizations. In addition, bidder 1 with the signal s1 [0,s0 ] loses ∈ 1 to all the bids β (s2)=v1(s0 ,s2) for s2 (s0 ,s00). Bidder 1’s value v1(s1,s2) 2 1 ∈ 2 2 ≤ β2(s2)=v1(s10 ,s2), hence 1 cannot gain positive profit by deviating and winning. Now considerb bidder 1 with s1 (s0 ,s00). She wins against β (s2) for s2 [0,s0 ] ∈ 1 1 2 ∈ 2 andb loses to β (s2) for s2 [s00, 1]. The value v1(s1,s2) is always higher than β (s2) 2 ∈ 2 2 for s2 [0,s0 ] and lower than β (s2) for s2 [s00, 1]. ∈ 2 2 ∈ 2 Notice that

β1(s1)=v2(s1,s200) >v2(s10 ,s200) >v1(s10 ,s200) >v1(s10 ,s2)=β2(s2) (4.2) for all sb1 (s10 ,s100) and s2 (s20 ,s200) (the second inequality followsb from the pairwise single crossing).∈ This implies∈ that in all such realizations bidder 1 wins and pays

β2(s2)=v1(s10 ,s2)

Proceeding the aboveb one can show that when bidder 2 uses the strategy β2(s2) against β1(s1), bidder 2 cannot deviate ex-post and increase her profit. b Thus the strategies β1(s1), β2(s2) constitute an ex-post equilibrium. b b b Recall that in our discontinuous equilibrium bidder 1 wins in all the realizations where s1 (s10 ,s100) and s2 (s20 ,s200). This is not the case in the regular equilibrium (4.1). Recall∈ also that the∈ regular equilibrium is always efficient. Thus in all the realizations where bidder 1 wins in the discontinuous equilibrium, but loses in the regular one, the discontinuous equilibrium allocates inefficiently.Thusthe discontinuous equilibrium allocates inefficiently in all the realizations s1 (s0 ,s00), ∈ 1 1 s2 (s0 ,s00) such that v1(s1,s2)

4.3 Ex-post eﬃcient auctions

In this section we show that inefficient ex-post equilibria are not a specificfeatureof the English auction. In fact, every auction that has an efficient ex-post equilibrium, in particular, the generalized VCG mechanism and Dasgupta and Maskin (2000) auction, also have a continuum of inefficient ex-post equilibria. The proof for an arbitrary efficient auction relies on the fact that the generalized VCG mechanism has inefficient ex-post equilibria, hence we start with this direct mechanism.5 The direct mechanism (the generalized VCG mechanism) was first mentioned in Crémer and McLean (1985). We describe it for the case of two bidders. The bidders submit their signals t1 and t2, then the auctioneer computes values v1(t1,t2) and v2(t1,t2) using these reports. The good is allocated to the bidder whose value is higher. If say v1(t1,t2) >v2(t1,t2), then bidder 1 receives the good, and the payment is equal to the value of bidder 1 with the signal t10 , such that the value of bidder 1 is just tied with the value of bidder 2, v1(t10 ,t2)=v2(t10 ,t2). In this mechanism truth-telling constitutes an efficient ex-post equilibrium when value functions are increasing and satisfy the pairwise single crossing. It can be shown that the generalized VCG mechanism possesses a continuum of inefficient ex-post equilibria, where bidders misrepresent their true signals.

2 ∂v1 ∂v2 Suppose there exists (s ,s ), such that v1(s ,s )=v2(s ,s ), and > 0, > 0 in 10 20 10 20 10 20 ∂s2 ∂s1 the neighborhood of (s10 ,s20 ) —sothatvaluevj is indeed sensitive to the signal s j.Takep0 = − v1(s10 ,s20 )=v2(s10 ,s20 ) and p00 = p0 + . If >0 is small enough — the resulting equilibrium β1(s1), β2(s2) is undominated. 3 Recall that we assumed that the signals s1 and s2 take the values on the square [0, 1] b [0, 1]. × b 4 See Milgrom (1981) for the analysis of equilibria in the English auction with pure common value. 5 The deﬁnition of an ex-post equilibrium for the English auction clearly translates into the deﬁnition of an ex-post equilibrium for any other allocating mechanism. All one needs to do is to replace the strategy space.

50 Fix some price range (p0,p00) and define the signal ranges (s10 ,s100) and (s20 ,s200) as was done for the discontinuous equilibrium in the English auction. Consider the following “non-truthful” strategies. When the signal sj [0,s0 ] [s00, 1] bidder j reports sj.Whenthesignal ∈ j ∪ j s1 (s0 ,s00), bidder 1 reports t1 σ1(p1), where p1 = v2(s1,s00) and the inverse ∈ 1 1 ≡ 2 functions σ1(p),σ2(p) solve (4.1). When the signal s2 (s0 ,s00), bidder 2 reports ∈ 2 2 t2 = σ2(p2), where p2 = v1(s10 ,s2). Similarly to how this was done for the English auction one can check that the “non-truthful” strategies constitute an ex-post equilibrium in the generalized VCG mechanism. When value functions exhibit interdependence, one can construct a continuum of inefficient undominated ex-post equilibria. Now consider any auction that has an efficient ex-post equilibrium.6 Efficiency determines the allocation rule or, equivalently, the winning bidder. If the equilibrium is ex-post, the price, that the winner has to pay, cannot depend on the winning bid and, more generally, it cannot depend on the signal of the winner. Thus, the only way the winner can affect the price ex-post is by changing the equilibrium allocation, making his bid not winning. If an equilibrium is ex-post, it must be that the winner’s value is never lower than the price. Then, clearly, any deviation, that changes the allocation, is not profitable for the winner. Similarly, if an equilibrium is ex-post, any deviation of the losing bidder, that changes the allocation, results in the price no lower than the value of that bidder. It turns out that requiring an auction to allocate efficiently in ex-post equilibrium imposes quite stringent restrictions on the auction design. Perry and Reny (2002) establish the following ex-post revenue equivalence result7 Lemma 4.2. Consider an auction that possess an ex-post equilibrium. In every realization of signals, for every bidder, thepaymentisdetermined,uptoaconstant, by the equilibrium allocation.8 Thus, two auctions that implement the same allocation rule, for example, both allocate efficiently, in ex-post equilibrium, generate the same revenue (up to a constant) in every realization of signals. Using this Lemma and the construction of inefficient ex-post equilibria for VCG mechanism we can show the following “counterpart” to the revelation principle for ex-post equilibria.

Lemma 4.3. Consider an auction A which has an eﬃcient ex-post equilibrium. If the generalized VCG mechanism has an ex-post equilibrium, where each bidder

6 By an auction, as usual, we mean a game form with well definedallocationandpayment rules. In this section we consider only deterministic allocations, but this is not a restriction on any of the results. 7 To the best of my knowledge this result appeared for the first time in the working paper version of Perry and Reny (2002), but did not make it to the published version. 8 The constant is a constant of integration. It can be specific to a particular bidder and an auction form, but not to a particular realization of signals.

51 misrepresents his true signal sj with feasible one tj,thenA has an ex-post equilibrium, where bidders with true signals sj act as they act in the efficient ex-post equilibrium with signals tj. Proof. Consider first the efficient ex-post equilibrium in the generalized VCG mechanism. Each bidder i reports signal ti truthfully. This completely determines the outcome in VCG mechanism, both the allocation–the winner’s identity j in every realization (tj, t j), and the payment by the winner–pj(t j). Now consider− a “non-truthful” equilibrium. In this equilibrium− in every realization (sj, s j) each bidder j reports a feasible tj instead of her true signal sj. Win- − ner’s identity j and her payment pj(t j) must be the same as in the efficient equi- − librium for signals (tj, t j). From the above discussion follows that if reporting tj − instead of sj is an ex-post equilibrium action, it must be that vj(sj, s j) pj(t j), − ≥ − for any s j and corresponding t j, (the value of the winner is never lower than the price); and− if any bidder i = j wanted− to deviate and win the good, the price that 6 he would have to pay pi(t i) vi(si, s i), for any s i and corresponding t i. − ≥ − − − Now take an auction A that has an efficient ex-post equilibrium. Suppose each bidder j has signal tj and uses the strategy that is prescribed for her in efficient equilibrium β.TheoutcomeofA must be the same allocation as in the generalized VCG mechanism and payments: pj(t j)+cj, by the winner, and ci by every other bidder i. Because the equilibrium is e−fficient and ex-post, by Lemma 4.2, the price pj(t j) must be the same as in the generalized VCG mechanism. The constants − cj cannot depend on the realization of signals. It implies that bidder j has to pay (or receive) cj regardless of whether she wins or loses. Now we present an inefficient ex-post equilibrium. Consider a profile of strategies β,suchthatat(sj, s j) all bidders act as they act in ex-post efficient equi- − librium β if their signals were (tj, t j). Thus the profile of actions for (sj, s j) is − − indistinguishableb from efficient equilibrium actions for true signals (tj, t j). Hence − the winner must be the same bidder j and his payment is the same pj(t j)+cj, − every other i pays ci. β is an ex-post equilibrium since vj(sj, s j) pj(t j) for all − ≥ − s j and corresponding t j, which follows from the fact that reporting tj instead of − − sj is an ex-post equilibriumb action in the generalized VCG mechanism. Similarly, no bidder i = j can profitably deviate since pi(t i) vi(si, s i) for any s i and 6 − ≥ − − corresponding t i. −

52 Appendix A

On eﬃciency of the N -bidder English auction

A.1 Equivalence Lemma

Lemma A.1. The formulations of GSC given in Deﬁnitions 3.5a and 3.5b are equivalent.

Proof. Fix s. Introduce µk(u) u Vk(s)–the derivative of Vk along the direction u. ≡ · ∇ 5a = 5b. It is enough to show that every component of the equal increments ⇒ vector uA is non-negative for all subsets (s). A A ⊂ I Step 1. Suppose inequalities in the directional formulation (3.3) are strict for all .ThenwecanshowthatuA 0 (every component is strictly positive). A A À This is done by induction on the number of bidders in .For# =1, 1 A A ∂V − uA = ∂sA > 0. Suppose for all (s) with # n 1, uB 0.Wewant A A B ⊂ I B ≤ − B À to show³ the´ same for an arbitrary subset (s) with # = n. A ⊂ I A Suppose, on the contrary, there exists (s) with # = n such that A ⊂ I A uA 0. We can partition = ,where , ,and are subsets of A 6À A BtCtD B C D bidders for which the corresponding components of uA are negative, equal to zero, A and positive correspondingly. By presumption, = ∅. Obviously, is also B ∪ C 6 D not empty. Note that uD 0 as # µj( u0). GSC in the directional formulation for the setB dictates that−µ (u) <µ−(u). D i j 53 Since uA = u ( u0),wehaveµ (uA) <µ(uA). We have a contradiction since − − i j µi(uA)=µj(uA)=1by deﬁnition of uA. Therefore, uA 0. A À Step 2. Suppose that weak inequalities in (3.3) are possible. Then we can slightly perturb the Jacobian of value functions at s, DV (s), in the following way: add ε>0 to every diagonal element, I

DV 0(s)=DV (s)+εI# . I I I All inequalities in (3.3) become strict after the perturbation–for any (s) and ∂Vj A ⊂ I u from Definition 3.5a we have µ0 (u)=ε +µj(u) >µj(u) for any bidder j , j ∂sj ∈ A while µ0 (u)=µ (u) for all i . i i 6∈ A If prior to the perturbation, there existed a subset (s) such that for A ⊂ I some i , uiA < 0, then, by continuity, for sufficiently small ε, uiA would still be negative∈ A after the perturbation which would contradict the result in Step 1. Therefore, uA 0 for all subsets (s). A À A ⊂ I 5b = 5a. Again we use the induction on the number of bidders in (s). For # ⇒=1the result is obvious. Fix the subset (s) with # A=⊂nIand supposeA that GSC is satisfied at s in the directionalA formulation⊂ I forA all subsets (s) with # maxj µj(u). Clearly, u = uA. ∈ I \A ∈A 6 Consider –the subset of bidders who have the highest increments to B ⊂ A their values in the direction u, i µi(u)=maxj µj(u).Sinceu = uA, ∈ B ⇔ ∈A 6 = . Consider vector w1(t)=u tuB.Notethatu 0 and, since # < # , B 6 A − B À B A by the induction hypothesis and the argument above, uB = 0. B At t =0,foranyj and i ,wehaveµ (w1(0)) <µ(w1(0)) < ∈ A\B ∈ B j i µk(w1(0)). Oncewestartincreasingt, that is, decreasing in a special direction the signals of all the bidders from only, all µ (w1(t)), for i , decrease uniformly B i ∈ B at rate t, while for any bidder l (s) (including k)theirµ (w1(t)) decrease at ∈ I \B l most at the same rate, because GSC is satisfied for .Introducet1–the minimal B value of t>0 such that: either µ (w1(t)) = µ (w1(t)) for some j and j i ∈ A\B every i ,orw1i(t)=0for some i . In the latter case, stop. If the former case applies,∈ B consider –a subset that∈ includesB and all the bidders j C B ∈ A\B such that µ (w1(t1)) = µ (w1(t1)).Define w2(t)=w1(t1) tuC. Find the smallest j i − t2 > 0 such that: either µ (w2(t2)) = µ (w2(t2)) for some bidder j and j i ∈ A\C every i ,orw2i(t2)=0for some i , in which case stop. Again, if the former case applies∈ C consider . Repeatthisprocedureuntilforsomebidder∈ C i , D ⊃ C ∈ A wmi(tm)=0. This will take at most # repetitions and may result in all bidders A i having wmi(tm)=0. ∈ A Note that by the induction hypothesis for bidder k (s) , µk(w1(t)) always decreased at a rate no higher than the corresponding∈ rateI for\A bidders from , , B C . . . . Thus, at any stage l m of the procedure, µk(wl(t)) > maxj µj(wl(t)).In ≤ ∈A 54 particular,

µk(wm(tm)) > max µj(wm(tm)). (A.1) j ∈A If for all j , wmj(tm)=0, then, by construction, wm(tm)=0,which ∈ A makes (A.1) impossible, otherwise deﬁne 0 to be the set of bidders j with A ∈ A wmj (tm) > 0.Since# 0

The following is an obvious corollary to Lemma 3.1.

Corollary A.1. GSC in the directional formulation is violated at s if and only if GSC in the equal increments formulation is violated at s.

It should be noted that for a violation to occur it is not necessary that the same subsets of bidders are involved under both deﬁnitions. The following Corollary is used in the proof of Proposition 3.2.

Corollary A.2. At a given s, consider an arbitrary (s) with # = n 2. Suppose GSC is satisfied at s for any subset A(s⊂) withI # 0 A ∈ I \A k j for all j or (ii): ukC > 0 and ujC < 0 for all j ,where k and uC is an equal∈ A increment vector for the subset . ∈ A C ≡ A ∪ { } C Proof. By the above conditions one needs at least n +1bidders to violate GSC. Therefore, as follows from the proof of Lemma 3.1, uA 0. A À (= ). Suppose first that ukC < 0, while uC 0. Consider vector u0 such that ⇒ A À uk0 =0, u0 k = uC k.SinceGSC is satisfied for = k , µk(u0) > maxi µi(u0). − − B { } ∈A Therefore, GSC is violated for k, ,andu0.NowsupposeukC > 0, while uC 0. A A ¿ Consider vector u0 such that uk0 =0, u0 k = uC k.ThenGSC is violated for k, − − − ,andu0. A ( =).IfuC =0,thenuC = uA. In this case GSC is satisfied with equality for ⇐ k bidder k and .IfukC > 0 or ukC < 0,wecansupposethatalltheinequalitiesin (3.4) are strictA since by disturbing the Jacobian, as we did in Step 2 of the proof of Lemma 3.1, we eliminate all equalities. If ε is small enough, whether GSC is violated for bidder k or not and the sign of ukC remain unchanged. As a result, for any i , ui =0. ∈ A 6 Step 1. We show that uC can have either 1 or n positive components. Partition C = (inequalities are strict), i (i )ifuC < 0 (uC > 0). Given that C BtD ∈ B ∈ D i i = ∅,supposethat# 1,n . Consider vector w1(t)=uC tuD.Wehave D 6 D 6∈ { } − uD 0, µj(uD) >µi(uD) for all j and i since # # 2 0. There exists the minimal t1 > 0 such that for some j , w1j(t)=0. Consider the subset of bidders l with ∈ D E ∈ D w1l(t) > 0,andvectorw2(t)=w1(t1) tuE . Increase t until for some bidder j , − ∈ E 55 w2j(t2)=0.Again,µ (w2(t2)) <µ(w2(t2)) for all j and i . Repeating this j i ∈ E ∈ B procedure we eventually obtain vector wm(tm) such that for all j , wmj(tm)=0. ∈ D Introduce vector w0 (tm) wm(tm). Note that for all i , w0 (tm)= uC > 0. m ≡− ∈ B mi − i Fix bidder j with wmj(0) > 0. Clearly µ (w0 (tm)) >µ(w0 (tm)) for all i . ∈ D j m i m ∈ B Therefore GSC is violated for bidder j, subset ,andvectorwm0 (tm),whichisa contradiction since # 0. We show that uC 0. Suppose the otherwise and A ¿ consider vector u0 such that uk0 =0and u0 k = uC k. Clearly, u0 = uA.Since − − 6 µ (uC)=µ (uC)=1for all i and GSC is satisfied for = k ,wehave k i ∈ A B { } µk(u0) mini µi(u0).SinceGSC is violated for and bidder k,thereexists ≤ ∈A A vector u,withu 0 and u = 0,suchthatµk(u) > maxi µi(u).Consider A À −A ∈A vector w(t)=u tu0. It follows that µk(w(t)) > maxi µi(w(t)) for t 0.Note that w (t)=0−for any t and w (0) 0. Since at least∈A one of the components≥ −C C A À of u0 = u is positive, there exist the smallest t0 > 0 such that for some i , A A ∈ A wi(t0)=0. Then, GSC is violated for bidder k,subset = i and vector B A\{} w(t0), which is a contradiction since #

A.2 Suﬃciency

Lemma A.3. Suppose GSC is satisﬁed. Then there exist inferences σ(p, H(p)), such that each σj( ,H(p)) is continuous and non-decreasing for any H(p),and · σj(p, H¯ (p)) = σj(p, H(p)) for all p such that H¯ (p) = H(p). For any active at 6 H(p) bidder j,ifσj(p, H(p)) < 1 then j (σ(p, H(p))). ∈ I

Proof. First, we construct inferences σ(p, H(p)),suchthateachσj( ,H(p)) is con- · tinuous at p for any H(p), and any bidder j active at p with if σj(p, H(p)) < 1 has thehighestvalueatσ(p).Define (σ(p, H(p))) to be the set of active bidders at A price p with σj(p, H(p)) < 1. Suppose at some p0 with H(p0)=H¯ (p0) thereexistsaprofile σ0(p0) that satisfies (3.6), and = (σ0(p0)) (σ0(p0)).Fixσ (p)=σ0 (p0) for p p0.ConsideraproA fileA of functions⊂ σI(p)=(σ (p), σ −(Ap)) such that−A σ (p) ≥ 0 A0 −A A satisfies (3.6) for every p [p ,p∗] for some p∗ >p. Finding a solution σ (p) to the system ∈ A V (σ (p), σ0 (p0)) = p1 (A.2) A A −A A 56 is equivalent to solving the system of differential equations

dσ 1 A =(DV )− 1 . (A.3) dp A A By the Caushy-Peano theorem, there exists a unique continuous solution σ (p) to the system (A.3) with initial condition σ (p0)=σ0 (p0), and this solution extendsA A A to all p p∗ ,wherep∗ is the lowest price at which σj(p∗ )=1for some bidder j . ≤ A A A ∈ A dσ Suppose GSC is satisfied. As long as (σ(p)), dpA = uA = 0 (this follows A ⊂ I A ∂Vi dσ from the proof of Lemma 3.1), and ∂σ dpA 1 for any i (σ(p)) by (3.4). 0 A ≤ dσ ∈ I \A 0 Since (σ(p )), it follows that (σ(p)) and dpA = 0 for all p [p ,p∗ ]. A ⊂ I A ⊂ I 0 ∈ A We have constructed σ(p, H(p)) for p [p ,p∗ ].Toextendσ(p, H(p)) beyond ∈ A p∗ ,wehavetosolveanewsystem(A.2)for 0 = (σ(p∗ ,H(p∗ ))) ( with A A A A A A initial condition σ 0 (p∗ )=σ (p∗ ). This is repeated until no bidder remains with A A A A σj(p) < 1,thereafterσ(p) is fixed. To provide σ(p, H(p)) for all prices and histories we need to specify for each 0 0 0 0 H(p) initial σ (p ),wherep =maxpj H(p) pj.Atp =0set σ (0, ∅)=0,then ∈ σ(p) are calculated as above with = ,forp [0,p∗ ]. At p0 such that H(p0) = H¯ (p0A),deNfine σ0(p∈0, H¯ (pN0)) = σ(p0,H(p0)).Obvi- ously, if (σ(p, H(p0))) 6 (σ0(p0)),then (σ0(p0, H¯ (p0))) (σ(p, H(p0))) and (σA0(p0)) (σ0(p⊂0))I. Then, we canA define σ(p, H¯ (p0))⊂.Notethatpro-A ceedingA this way⊂ allowsI us to maintain continuity of σ, or more formally, to link σ( ,H(p0)) and σ( , H¯ (p0)) at the price p0 where bidders exit the auction. · · A.3 Necessity

Throughout this section we follow our notational convention and assume that GSC is violated for and bidder 1, (s)= +1, is minimal, # = n 2,theset A I s A A A ≥ of signals considered is limited to U (s). All additional definitions and supporting results are located in Appendix A.3.1.I 0 Lemma A.4. Consider s0 = s(t) for an arbitrary t Ut , s(t) is the trajectory defined in Step 1 of the proof of Proposition 3.2. For∈ any j ,thereexist ∈ A bj(sj0 ) limsj s inf βj(sj), and these limits are equal, for any j , bj(sj0 ) b< ≡ ↓ j0 ∈ A ≡ . In addition, β (s0 ) >band for any j and sj >s0 sufficiently close to s0 , ∞ 1 1 ∈ A j j β (sj) b. j ≥ u u Proof. Consider trajectory sA(τ) s AA(τ) with s AA(0) = s0 as in Definition A.1 in Appendix A.3.1. Along this trajectory≡ the values of the bidders from are A increasing uniformly while s0 is fixed. Since GSC is violated for and 1, for any 1 A sufficiently small τ>0, V1(sA(τ)) > maxj Vj(sA(τ)), and therefore, ∈A

β1(s10 ) > min βj(sjA(τ)). (A.4) j ∈A 57 By continuity, for any s1 suﬃciently close to s10 , s1

β1(s1) > min βj(sjA(τ)). (A.5) j ∈A

Since uA 0, sjA(τ) is strictly increasing for any τ. A À Define as j if and only if j and bj(sj0 )=mini bi(si0 ),define B ∈ B ∈ A ∈A b mini bi(si0 ) and b mini bi(si0 ). Since we are not imposing any a prioriB ≡ restrictions∈A on the− biddingB ≡ functions,∈A\B we allow for b = and b = .1 Clearly, when # 0,wecanfind δτ > 0, such that for any τ (0,δτ ), ∈ j βi(siA(τ)) >b j ε for any i j and (A.4) holds. Consider s∗(r)=s0 + rv , − j − j ∈ A\{ } where vector v v (λ) is defined in Lemma A.5. Then we can find δλ (0, 1), ≡ ∈ such that for all λ with λ1 (δλ, 1), for all r (0,δj), where δj = sjA (δτ ) sj0 , ∈ j ∈ − for any i j such that vi > 0,wehaveβi(si∗(r)) >b j ε as well. The ∈ A\{ } − − existence of δλ follows from the fact that uA 0. A À Fix ε =(b j bj) /2, and consider an element sjm of the sequence sjm sj0 with − − ↓ β (sjm) bj(s0 ),suchthatβ (sjm)

β1(s10 ) >βj(sj∗(rm)) = min βj(sjA(τ m)), (A.6) j ∈A and, similarly to (A.5), there exist δ1 > 0 such that for any s1 (s0 δ1,s0 ), ∈ 1 − 1

β1(s1) >βj(sj(rm)). (A.7) By continuity and Lemma A.5, we can find sufficiently large m, and so sufficiently small rm > 0, such that (s∗(rm)) j 1 .Since is minimal, I ⊂ { } ∪ { } A 1 = (s∗(r)). BycontinuityandLemmaA.6,wheneverrm > 0 is sufficiently { } 6 I small, there exists δ0 > 0 such that for any s1 (s0 δ0 ,s0 ), 1 ∈ 1 − 1 1

j = (s1, s∗ (rm)). (A.8) { } I A

Pick rm (0,δj) and s1 such that both (A.7) and (A.8) hold. Then, even if we ∈ slightly increase the signals of all the bidders i with s∗(rm)=s0 ,bycontinuity, ∈ A i i 1 In what follows a statement a>btogether with b = implies a = . ∞ ∞ 58 j = (s), where s is the disturbed profile. Since, βi(si) >bj ε for any { } I − − si (si0 ,siA(δτ )), βj(sjm)=βj(sj)=mini +1 βi(si). We reached a contradiction; bidder∈ j has the highest value but drops out∈A the first. Step 2. Here we show that no matter what is, for any j ,forsj sufficiently B ∈ A j close to sj0 , βj(sj) b . Suppose otherwise. For any j pick trajectory s (r)= B j j ≥ j j ∈ A j s0 + rv (λ ),wherev (λ ) satisfies conditions of Lemma A.5 and λ (1 δλ, 1) 1 ∈ − for all j ,forδλ (0, 1) and arbitrarily close to 1. ∈ A ∈ For any given j ,thereexistsδrj,suchthatforanyr (0,δrj), j 1 j ∈ A j ∈ { }∪{ } ⊃ (s (r)) and j = (s1, s (r)) for any s1 sufficiently close to s10 (this may depend I { } I A on a particular r), s1 s10 .Pickδr > 0 with δr minj δrj andsuchthatforany ≤ j ≤ j i, j ,foranyr (0,δr), si (r) is such that whenever si (r) > 0,(A.4)holdsfor ∈ A j ∈ τ that solves si (r)=siA(τ). By our presumption, there exists bidder j , such that βj(sj) sj0 . We can always find bidder j with sj such that rj, the solution to j j sj = sj(r),satisfies rj (0,δr).Define ε = b βj(sj). Then, j 1 (s (r)) and by the procedure∈ similar to the one in StepB − 1, by slightly{ reducing}∪{ } ⊃ theI signal j of bidder 1 and slightly increasing signals of all i with si (r)=si0 ,weobtain profile s at which bidder j has strictly the highest value.∈ A There must be some other +1 bidder i with β (si) <β (sj). If it is always bidder 1, no matter how slight ∈ A i j is the decrease in her signal, then lims1 s sup β1(s1) b ε, which contradicts ↑ 10 ≤ B − (A.5). Otherwise, there exists bidder i with si >si0 , such that βi(si) 0, by construction and since GSC is satisfied for bidder 1 and , (sB(ρ)) 1 . By continuity and by Lemma A.6, for sufficiently small ρ>D I0, if we slightly⊂ B ∪ { decrease} the signal of bidder 1, all the bidders with the highest value as a result belong to . B Consider bj(ρ) bj(sjB(ρ)), clearly, limρ 0 bj(ρ)=bj(sj0 ) for all j . Then, ≡ → ∈ B for sufficiently small ρ, maxj bj(ρ) 0, and by induction (any

59 subset of has less than k elements), we have that bj(ρ)=b (ρ) for all j . B B ∈ B In general, as long as we stay suﬃciently close to s0,wecan,byslightlymoving in appropriate directions away from s0, possibly in several steps, separate bidders from in any given order. This implies that whenever two or more bidders from haveA equal and maximal values, the limits of their bids from the right have to Abe equal.

Similarly, by the argument as in Step 2, βj(sj) b (ρ) for any j and sj ≥ B ∈ A sufficiently close to sjB(ρ), sj >sjB(ρ). Therefore, since sjB(ρ) is strictly increasing for some j , b (ρ) is weakly increasing. ∈ B B Suppose that for any δρ > 0 we can find ρ (0,δρ) and a bidder j ,such ∈ ∈ B that βj(sjB(ρ)) >b(ρ). Then, fixing signal sjB(ρ) and bid βj(sjB(ρ)) for bidder B j, by induction and arguments similar to the above applied to 0 = j and B B\{ } 0 = j , we obtain a contradiction. If it is to the presumption that an efficient AequilibriumA\{ } exists or to # = k0 and any j , βj(sjB(ρ)) b (ρ). ∈ B ≤ B Combined with monotonicity of b (ρ) and the fact that for all j, βj(sj) b (ρ) B ≥ B for sj >sjB(ρ) (locally), we have that whenever βj(sjB(ρ))

βj(sjB(ρ)) = b (ρ). B Now, we add bidder 1 into the picture. First, consider the case, when for some j ,thereexistsδρ > 0, such that sB(ρ)=s0 for ρ (0,δρ). Then, ∈ B j j ∈ b (ρ)=b ,andsoforanyi ,withsiB(ρ) strictly increasing, βi(siB(ρ)) = b B B ∈ B B for all ρ (0,δρ).ConsidersA(τ),forasufficiently small τ>0 we can separate ∈ bids of the bidders from away from b and for each j , βj(sjA(τ)) b , A\B B ∈ B ≥ B while at least for some i , βi(siA(τ)) = b . Therefore, from (A.4), β1(s10 ) > ∈ B B b =minj βj(sjA(τ)).Let 0 be a subset of bidders i ,forwhomthebids B ∈A B ∈ B 0 βi(si)=b in the right neighborhood of si0 . Consider a trajectory s∗(r)=s0 +rvB . B Along this trajectory, the set of bidders with the highest value is a subset of 0 1 . By continuity and Lemma A.6, for a sufficiently small r>0,(A.5)holdsaswellB ∪{ } and, after slightly reducing the signal of bidder 1, all the bidders with the highest value as a result belong to 0. After slightly increasing the signal of each j B ∈ A with s∗(r)=s0 , we obtain signal profile s, at which all the bidders from 0 drop j j B out simultaneously at b =minj +1 βj(sj)–a contradiction. B ∈A In the remaining case, sB(ρ) are strictly increasing for all j , therefore β ( ) j ∈ B j · for all j are monotonic in the right neighborhood of s0 . For any small τ>0, ∈ B j since uA 0 and sjB(ρ) are strictly increasing, we can define ρj for each j , A À ∈ B such that sjB(ρj)=sjA(τ), let ρ0 minj ρj. For any ε>0 there exist δτ > 0,such ≡ ∈B that for any τ (0,δτ ), we have: (i) for any i , βi(siA(τ)) >b (s0) ε/2; ∈ ∈ A\B −B − (ii) for all j , ρj is such that b (ρ) b <ε/2 and the above results hold; ∈ B | B − B| that is, in particular, that the bidders from have the highest value at sB(ρ), B b (ρ) is monotonic, and for all j, βj(sjB(ρ)) b (ρ); (iii) starting from sB(ρ0) after B ≤ B 60 reducing s0 slightly all the bidders with the highest value as a result belong to . 1 B Pick any τ (0,δτ ) such that for all j , b (ρ) is continuous at ρ0. Then, ∈ ∈ B B consider i with ρ = ρ0.From(A.5)weobtainforanys1 sufficiently close to ∈ B i s10 , s1

β1(s1) > min βj(sjA(τ)) = min βj(sjA(τ)) = min βj(sjB(ρj)) = βi(siB(ρ0)) = b (ρ0). j j j B ∈A ∈B ∈B

Then, starting from sB(ρ0), reducing slightly s10 and increasing slightly sj for each j with sB(ρ0)=s0 we obtain signal profile s,atwhich (s) , but all the ∈ A j j I ⊂ B bidders from exit first simultaneously at b (ρ0)–a contradiction. Step 4.WehaveshownthatB # = n,andsoB = .Letb b . Since for all B B A ≡ A j , βj(sj) b(s0) for all sj close to sj0 , sj >sj0 ,from(A.4)wehaveβ1(s10 ) >b. ∈ItremainstobeshownthatA ≥ b< .Ifb = , then for each bidder j +1 ∞ ∞ ∈ A there exists a range of signals with βj(sj)= . As a result, the equilibrium payoff to each of the bidders is equal to ,whichcannothappeninanequilibrium∞ since each bidder can exit instead at−∞p =0and assure herself the payoff of 0.

Corollary A.3. If for some bidder j , βj(sj(t)) = b(t),thent is a discontinuity point for b(t). ∈ A 6

Proof. If for some j , β (sj(t)) >b(t), then by the argument similar to the one ∈ A j made in Step 3 of the proof of Lemma A.4, considering 0 = j ,wecanﬁnd A A\{ } aproﬁle s, at which all the bidders from 0 exit simultaneously prior to bidder 1 A and j, while the bidder or bidders with the highest value belong to 0. Monotonicity of b(t) is established in Step 3 of the proof of PropositionA 3.2.

From Lemma A.4 it follows that for all j , whenever sj(t0) >sj(t), βj(sj(t0)) ∈ A 0 ≥ b(t),fort and t0 from the considered neighborhood Ut . Therefore, if for some j , βj(sj(t))

A.3.1 Supporting results for value functions

Definition A.1. For a given s0, (s0), and vector x with x = 0,define B ⊂ I −B x y x V (s0)–the derivative of V along x.Define trajectory s B(τ) with xB ≡ · ∇ B x B s B(0) = s0 and s B (τ)=s0 as a solution to the system −B −B x V (s B(τ)) = (V + τ)y , B B x ds B(τ) where V =maxj Vj(s0).Clearly, Bdτ = x . ∈N τ=0 B ¯ Lemma A.5. For any proper subset (¯ there exists subset = ( ), B ¯ A D D B B ⊆ ( ,suchthatforanyk , uD Vk(s) < 1.Moreoverforanyε>0, D A ∈ A\D · ∇ there exist vector vB 0 such that vB uB <ε, vB =0for any k , = − k ∈ A\D vB Vi(s) < 1 for any i and vB Vj(s)=1for all j . · ∇ ∈ A\B ° · ∇ ° ∈ B ° ° 61 Proof. The proof is by induction on the number of bidders in . B Define as the set of bidders k such that uB Vk(s)=1.SinceGSC C ∈ A\B · ∇ is satisfied for , uB = uA and = .If = ∅, then set ,andvB uB. B 6 C 6 A\B C D ≡ B ≡ Whenever # =# 1=n 1 we have the result. Suppose that for all 0,with B A − − B # 0 > # the result holds. B B If = ∅ define 0 = ,thenuB Vk(s) < 1 for any k 0 .Pick C 6 B A\C · ∇ ∈ B \B ( 0).ConsidervB = λ1uB +(1 λ1) vB0 with λ1 (0, 1).Whenλ1 1, D ≡ D B − ∈ → vB uB. By induction, for all j 0, vB0 Vi(s) < 1. Therefore, for all → ∈ A\B · ∇ j , vB Vi(s) < 1 as long as λ1 (0, 1). ∈ A\B · ∇ ∈ Remark A.1. As follows from the proof of Lemma A.5 we can find a finite sequence ( 0 ( 00 ( ... ( ,suchthatvB can be represented as vB = λ1uB +λ2uB0 + B B B A λ3uB00 + ...,where λi =1, for any i, λi (0, 1). Lemma states that vB vB(λ) i ∈ ≡ with required properties can be found with λ1 being arbitrarily close to 1. P ∂V1 Lemma A.6. For any ,itiseithervB V1(s) < 1,or (s0) > ( ∂s1 ∂Vj B A · ∇ minj (s0). ∈B ∂s1

Proof. If the inequality in (3.4) is strict for and bidder 1, uB V1(s) < B · ∇ maxj uB Vj(s),andsovB V1(s) < 1. Otherwise, since # minj ∂s (s0). C 6 C C 6 C 1 ∈B 1

62 Bibliography

[1] Al-Najjar, N. and R. Smorodinsky (2000): “Pivotal Players and the Charac- terization of Influence,” Journal of Economic Theory, 92, pp. 318-342. [2] Arnold, B., N. Balakrishnan and H. Nagaraja (1992): “A First Course in Order Statistics,” John Wiley & Sons, Inc.: New-York, USA. [3] Arrow, K. J. (1979): “The Property Rights Doctrine and Demand Revelation under Incomplete Information,” in Economics and Human Welfare, ed. by M. Boskin, New York: Academic Press, pp. 23-39. [4] d’Aspremont, C. and L. A. Gérard-Varet (1979): “On Bayesian Incentive Compatible Mechanisms,” in Aggregation and Revelation of Preferences, ed. by J.-J. Laffont, Amsterdam: North-Holland Publishing Co., pp. 269-288. [5] Ausubel, L. (1997): “An Efficient Ascending Bid Auction for Multiple Ob- jects,” mimeo, University of Maryland, College Park, MD. [6] Bikhchandani, S. and J. Riley (1991): “Equilibria in Open Common Value Auctions,” Journal of Economic Theory, 53, pp.101-130. [7] Bikhchandani, S., P. Haile and J. Riley (2002): “Symmetric Separating Equi- libria in English Auctions,” Games and Economic Behavior, 38, pp. 19—27. [8] Buchanan, J. (1965), “An Economic Theory of Clubs,” Economica,32,pp. 1-14. [9] Chung, K.-S. and J. Ely (2000): “Efficient and Dominance Solvable Auctions with Interdependent Valuations,” mimeo, Northwestern University. [10] Clarke, E. (1971), “Multipart Pricing of Public Goods,” Public Choice,11, pp. 17-33. [11] Cornelli, F. (1996), “Optimal Selling Procedures with Fixed Costs,” Journal of Economic Theory,71,pp.1-30. [12] Cornes, R. and T. Sandler (1986), “The Theory of Externalities, Public Goods and Club Goods,” Cambridge University Press: Cambridge, UK.

63 [13] Crémer, J. and R. McLean (1985): “Optimal Selling Strategies under Uncer- tainty for a Discriminating Monopolist when Demands are Interdependent,” Econometrica, 53, pp. 345-362.

[14] Dasgupta, P. and E. Maskin (2000): “Eﬃcient Auctions,” Quarterly Journal of Economics, 115, pp. 341-388.

[15] Dearden, J. (1997), “Eﬃciency and Exclusion in Collective Action Alloca- tions,” Mathematical Social Sciences, 34, pp. 153-174.

[16] Dearden, J. (1998), “Serial Cost Sharing of Excludable Public Goods: General Cost Functions,” Economic Theory,12,pp.189-198.

[17] Deb, R. and L. Razzolini (1999a), “Voluntary Cost Sharing for an Excludable Public Project,” Mathematical Social Sciences, 37, pp. 123-138.

[18] Deb, R. and L. Razzolini (1999b), “Auction-Like Mechanisms for Pricing Ex- cludable Public Goods,” Journal of Economic Theory,88,pp.340-368.

[19] Green, J. and J.-J. Laﬀont (1977), “Characterization of Satisfactory Mecha- nisms for the Revelation of Preferences for Public Goods,” Econometrica,45, pp. 427-438.

[20]Groves,T.(1973),“IncentivesinTeams,”Econometrica, 41, pp. 617-663.

[21] Hellwig, M. (2003), “Public-Good Provision with Many Participants,” Review of Economic Studies,70,pp.589-614.

[22] Hurwicz, L. (1979), “Outcome Function Yielding Walrasian and Lindahl Al- locations at Nash Equilibrium Points,” Review of Economic Studies,46,pp. 217-225.

[23] Izmalkov, S. (2003): “English Auctions with Reentry,” mimeo,MIT.

[24] Jackson, M. and H. Moulin (1992), “Implementing a Public Project and Dis- tributing its Costs,” Journal of Economic Theory, 57, pp. 125-140.

[25] Jackson, M. and A. Nicolò (2003), “The Strategy-Proof Provision of Public Goods under Congestion and Crowding Preferences,” Journal of Economic Theory,forthcoming.

[26]Krishna,V.andM.Perry(2000),“Eﬃcient Mechanism Design,” mimeo,the Pennsylvania State University.

[27] Krishna, V. (2001), “Asymmetric English Auctions,” Journal of Economic Theory, 112, pp. 261-288.

64 [28] Ledyard, J. and T. Palfrey (1999), “A Characterization of Interim Eﬃciency with Public Goods,” Econometrica, 67, pp. 435-448.

[29] Mailath, G. and A. Postlewaite (1990), “Asymmetric Information Bargaining Problems with Many Agents,” Review of Economic Studies, 57, pp. 351-369.

[30] Makowski, L. and C. Mezzetti (1994), “Bayesian and Weakly Robust First Best Mechanisms: Characterization,” Journal of Economic Theory,64,pp. 500-519.

[31] Maskin, E. (1992): “Auctions and Privatization,” in Privatization: Sympo- sium in Honor of Herbert Giersch, ed. by H. Siebert, pp. 115-136. Institut für Weltwirtschaft an der Universität Kiel.

[32] Maskin, E. (2003): “Auctions and Eﬃciency,” in Advances in Economics and Econometrics: Theory and Applications, Eighth World Congress,ed.byM. Dewatripont, L. Hansen and S. Turnovsky, vol. 1. Cambridge Univerity Press, Cambridge, UK.

[33] Milgrom, P. (1981), “Rational Expectations, Information Acquisition, and Competitive Bidding,” Econometrica, 49, pp. 921-943.

[34] Milgrom, P. and R. Weber (1982): “A Theory of Auctions and Competitive Bidding,” Econometrica, 50, pp. 1089-1122.

[35] Moulin, H. (1994), “Serial Cost-Sharing of Excludable Public Goods,” Review of Economic Studies,61,pp.305-325.

[36] Norman, P. (2003), “Eﬃcient Mechanisms for Public Goods with Use Exclu- sions,” Review of Economic Studies,forthcoming.

[37] Olszewski, W. (1999), “Serial Mechanisms for Provision of Excludable Public Goods,” mimeo, Princeton University.

[38] Perry, M. and P. Reny (2002), “An Ex-Post Eﬃcient Auction,” Econometrica, 70, pp. 1199-1212.

[39] Perry, M. and P. Reny (2001), “An Eﬃcient Ascending Multi-Unit Auction,” mimeo, University of Chicago.

[40] Porter, R. (1995): “The Role of Information in U.S. Oﬀshore Oil and Gas Lease Auctions,” Econometrica, 63, pp. 1-27.

[41] Rob, R. (1989), “Pollution Claim Settlements with Private Information,” Journal of Economic Theory,47,pp.307-333.

65 [42] Samuelson, P. (1954), “The Pure Theory of Public Expenditure,” Review of Economics and Statistics, 36, pp. 387-389.

[43] Shaked, M. and J. Shanthikumar (1994), “Stochastic Orders and Their Ap- plications,” Academic Press: San-Diego, USA.

[44] Sheﬃ, Y. (1985), “Urban Transportation Networks, Equilibrium Analysis with Mathematical Programming Methods,” Prentice-Hall: Englewood Cliﬀs, USA.

[45] Tong, Y. (1980), “Probability Inequalities in Multivariate Distributions,” Aca- demic Press: New-York, USA.

[46] Vickrey, W. (1961), “Counterspeculation, Auctions and Competitive Sealed Tenders,” Journal of Finance,16,pp.8-37.

[47] Wildasin, D. (1986), “Urban Public Finance,” Harwood Academic Publishers: Chur, Switzerland.

[48] Williams, S. (1999), “A Characterization of Eﬃcient Bayesian Incentive Com- patible Mechanisms,” Economic Theory, 14., pp. 155-180.

66 Vita

Date & Place of Birth: 15 May 1973; Zaporozhye, Ukraine

Citizenship: Ukraine

Education: 2004, Ph.D. in Economics, Pennsylvania State University 1999, M.A. in Economics, New Economic School, Moscow, Russia 1996, Diplom in Physics (cum laude, M.S.), Moscow State University, Rus- sia ≈

Fields: Primary: Microeconomics, Game Theory Secondary: Monetary Theory, International Trade

Publications: “Ineﬃcient Ex-Post Equilibria in Eﬃcient Auctions,” 2003, Eco- nomic Theory, 22/3, pp. 675-683.

Presentations: “On Eﬃciency of the N-bidder English Auction,” presented at the 13th International Conference on Game Theory, SUNY Stony Brook, July 2002. “Public Goods with Congestion: a Mechanism Design Approach,” presented at the 14th International Conference on Game Theory, SUNY Stony Brook, July 2003; Mid-West Economic Theory Meetings, Indiana University, Bloom- ington, October 2003; seminars at University of Texas at Austin, University of Western Ontario, Australian National University, University of Sydney, University of Aarhus, New Economic School, the Pennsylvania State Univer- sity, January-March, 2004.

Experience: Instructor, Intermediate Microeconomics (ECON 302), Fall 2001, Spring 2002. Research Assistant for Vijay Krishna, Fall 2002—Summer 2004; for Motty Perry, Summer 2001 Teaching Assistant for Undergraduate courses in Microeconomics, Fall 1999— Spring 2001