Pure strategy equilibria of single and double auctions with interdependent values

We prove the existence of monotonic pure strategy equilibrium for many kinds of asymmetric auctions with n bidders and unitary demands, interdependent values and independent types. The assumptions require monotonicity only in the own bidder's type. The payments can be a function of all bids. Thus, we provide a new equilibrium existence result for asymmetrical double auctions and a small number of bidders. The generality of our setting requires the use of special tie-breaking rules. We present an example of a double auction with interdependent values where all equilibria are trivial, that is, they have zero probability of trade. This is related to Akerlof's “market for lemmons” example and to the “winner's curse,” establishing a connection between them. However, we are able to provide sufficient conditions for non-trivial equilibrium existence.     

Equilibrium transformations and the Revenue Equivalence Theorem

I develop a method that transforms an equilibrium strategy profile from one auction to another. The method is constructive and does not require complicated computation. This provides a new approach to revenue equivalence and extends the theorem to domains where it had not previously been known, in particular to simultaneous multiple object auctions with complete information and to auction environments having correlated private values and common values.     

A comparison of first price multi-object auctions

We compare simultaneous multi-object English-type ascending price auctions with first price sealed bid auctions in private values environments with multi-object demands. Special attention is paid to the effect of closing rules on ascending auctions’ outcomes. We find that simultaneous ascending auctions with the soft closing rule are the most efficient, while the sealed bid auctions generate the highest revenue. Ascending auctions with the hard closing rule display a significant amount of late bidding, resulting in the lowest among the three institutions revenue and efficiency.

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The communication requirements of social choice rules and supporting budget sets

The paper examines the communication requirements of social choice rules when the (sincere) agents privately know their preferences. It shows that for a large class of choice rules, any minimally informative way to verify that a given alternative is in the choice rule is by verifying a “budget equilibrium”, i.e., that the alternative is optimal to each agent within a “budget set” given to him. Therefore, any communication mechanism realizing the choice rule must find a supporting budget equilibrium. We characterize the class of choice rules that have this property. Furthermore, for any rule from the class, we characterize the minimally informative messages (budget equilibria) verifying it. This characterization is used to identify the amount of communication needed to realize a choice rule, measured with the number of transmitted bits or real variables. Applications include efficiency in convex economies, exact or approximate surplus maximization in combinatorial auctions, the core in indivisible-good economies, and stable many-to-one matchings.     

Late and multiple bidding in second price Internet auctions: Theory and evidence concerning different rules for ending an auction

In second price Internet auctions with a fixed end time, such as those on eBay, many bidders submit their bids in the closing minutes or seconds of an auction. We propose an internet auction model, in which very late bids have a positive probability of not being successfully submitted, and show that late bidding in a fixed deadline auction can occur at equilibrium in auctions both with private values and with uncertain, dependent values. Late bidding may also arise out of equilibrium, as a best reply to incremental bidding. However, the strategic advantages of late bidding are severely attenuated in auctions that apply an automatic extension rule such as auctions conducted on Amazon. Field data show that there is more late bidding on eBay than on Amazon, and this difference grows with experience. We also study the incidence of multiple bidding, and its relation to late bidding.     

Catalog competition and Nash equilibrium in nonlinear pricing games

We model strategic competition in a market with asymmetric information as a noncooperative game in which each seller competes for a buyer of unknown type by offering the buyer a catalog of products and prices. We call this game a catalog game. Our main objective is to show that catalog games have Nash equilibria. The Nash existence problem for catalog games is particularly contentious due to payoff discontinuities caused by tie-breaking. We make three contributions. First, we establish under very mild conditions on primitives that no matter what the tie-breaking rule, catalog games are uniformly payoff secure, and therefore have mixed extensions which are payoff secure. Second, we show that if the tie-breaking rule awards the sale to firms which value it most (i.e., breaks ties in favor of firms which stand to make the highest profit), then firm profits are reciprocally upper semicontinuous (i.e., the mixed catalog game is reciprocally upper semincontinuous). This in turn implies that the mixed catalog game satisfies Reny’s condition of better-reply security—a condition sufficient for existence (Reny in Econometrica 67:1029–1056, 1999). Third, we show by example that if the tie-breaking rule does not award the sale to firms which value it most (for example, if ties are broken randomly with equal probability), then the catalog game has no Nash equilibrium. This paper was written while the second author was Visiting Professor, Centre d’Economie de la Sorbonne, Universite Paris 1, Pantheon-Sorbonne. The second author thanks CES and Paris 1, and in particular, Bernard Cornet and Cuong Le Van for their support and hospitality. The second author also thanks the C&BA and EFLS at the University of Alabama for financial support. Both authors are grateful to Monique Florenzano and to participants in the April 2006 Paris 1 NSF/NBER Decentralization Conference for many helpful comments on an earlier version of the paper. Finally, both authors are especially grateful to an anonymous referee whose thoughtful comments led to substantial improvements in the paper. Monteiro acknowleges the financial support of Capes-Cofecub 468/04.     

Variational convergence: Approximation and existence of equilibria in discontinuous games

We introduce a notion of variational convergence for sequences of games and we show that the Nash equilibrium map is upper semi-continuous with respect to variationally converging sequences. We then show that for a game G with discontinuous payoff, some of the most important existence results of Dasgupta and Maskin, Simon, and Reny are based on constructing approximating sequences of games that variationally converge to G. In fact, this notion of convergence will help simplify these results and make their proofs more transparent. Finally, we use our notion of convergence to establish the existence of a Nash equilibrium for Bertrand-Edgeworth games with very general forms of tie-breaking and residual demand rules.     

Bidding for the future: signaling in auctions with an aftermarket

This paper considers auctions where bidders compete for an advantage in future strategic interactions. When bidders wish to exaggerate their private information, equilibrium bidding functions are biased upwards as bidders attempt to signal via the winning bid. Signaling is most prominent in second-price auctions where equilibrium bids are “above value.” In English and first-price auctions, signaling is less extreme since the winner incurs the cost of her signaling choice. The opportunity to signal lowers bidders’ payoffs and raises revenue. When bidders understate their private information, separating equilibria need not exist and the auction may not be efficient.     

Learning in games with strategic complementarities revisited

Fictitious play is a classical learning process for games, and games with strategic complementarities are an important class including many economic applications. Knowledge about convergence properties of fictitious play in this class of games is scarce, however. Beyond games with a unique equilibrium, global convergence has only been claimed for games with diminishing returns [V. Krishna, Learning in games with strategic complementarities, HBS Working Paper 92-073, Harvard University, 1992]. This result remained unpublished, and it relies on a specific tie-breaking rule. Here we prove an extension of it by showing that the ordinal version of strategic complementarities suffices. The proof does not rely on tie-breaking rules and provides some intuition for the result.     

Tie-breaks and bid-caps in all-pay auctions

We revisit the two bidder complete information all-pay auction with bid-caps introduced by Che and Gale (1998), dropping their assumption that tie-breaking must be symmetric. Any choice of tie-breaking rule leads to a different set of Nash equilibria. Compared to the optimal bid-cap of Che and Gale we obtain that in order to maximize the sum of bids, the designer prefers to set a less restrictive bid-cap combined with a tie-breaking rule which slightly favors the weaker bidder. Moreover, the designer is better off breaking ties deterministically in favor of the weak bidder than symmetrically.     

Equilibrium theory with asymmetric information and with infinitely many commodities

The traditional deterministic general equilibrium theory with infinitely many commodities cannot cover economies with private information constraints on the consumption sets. We bring the level of asymmetric information equilibrium theory at par with that of the deterministic one. In particular, we establish results on equilibrium existence for exchange economies with asymmetric (differential) information and with an infinite dimensional commodity space. Our new equilibrium existence theorems include, as a special case, classical results, e.g. Bewley [Existence of equilibria in economies with infinitely many commodities, J. Econ. Theory 4 (1972) 514-540] or Mas-Colell [The price equilibrium existence problem in topological vector lattices, Econometrica 54 (1986) 1039-1053].     

Equilibria in first price auctions with participation costs

This paper characterizes the equilibria of first price auctions with participation costs in the independent private values environment. Bidders use cutoff strategies to decide whether they will participate in the auction. It is shown that, when bidders are homogeneous, there always exists a unique symmetric equilibrium, and further, there is no other equilibrium when valuation distribution functions are inelastic. When distribution functions are elastic at the symmetric equilibrium, there exists an asymmetric equilibrium. Inelasticity/elasticity includes concavity/convexity of distribution functions as a special case. We find similar results when bidders are heterogeneous.     

Strategy-proofness of the plurality rule over restricted domains

We give a complete characterization of preference domains over which the plurality rule is strategy-proof. In case strategy-proofness is required to hold under all tie-breaking rules, strategy-proof domains coincide with top-trivial ones where the range of the plurality rule admits at most two alternatives. This impossibility virtually prevails when strategy-proofness is weakened so as to hold under at least one tie-breaking rule: unless there are less than five voters, the top-triviality of a domain is equivalent to the (weak) non-manipulability of the plurality rule. We also characterize the cases with two, three or four voters. I am grateful to Attila Tasnadi for his thorough reading of the paper and pointing to an error in an earlier version. I thank Levent Kutlu, Clemens Puppe, Arunava Sen, an anonymous referee and an anonymous AE for their comments. This research is part of the project “Social Perception—A Social Choice Perspective” supported by Istanbul Bilgi University Research Fund. I also acknowledge the support of the Turkish Academy of Sciences Distinguished Young Scientist Award Program (TUBA-GEBIP).     

Learning to bid: The design of auctions under uncertainty and adaptation

We examine auction design in a context where symmetrically informed adaptive agents with common valuations learn to bid for a good. Despite the absence of private valuations, asymmetric information, or risk aversion, bidder strategies do not converge to the Bertrand–Nash equilibrium strategies even in the long run. Deviations from equilibrium strategies depend on uncertainty regarding the value of the good, auction structure, the agents? learning model, and the number of bidders. Although individual agents learn Nash bidding strategies in isolation, the learning of each agent, by flattening the best-reply correspondence of other agents, blocks common learning. These negative externalities are more severe in second-price auctions, auctions with many bidders, and auctions where the good has an uncertain value ex post.     

Endogenous fluctuations under evolutionary pressure in Cournot competition

An evolutionary game theoretic model of Cournot competition is investigated. Individuals choose from a finite set of different behavioral rules. Each rule specifies the quantity to be produced in the current period as a function of past quantities. Using more sophisticated rules may require extra information costs. Based upon realized payoffs, the fractions of the population choosing a certain behavioral rule are updated according to the replicator equation with noise. The long-run behavior of the evolutionary system consisting of the population dynamics coupled with the quantity dynamics of the Cournot game may be complicated and endogenous fluctuations may arise. We consider a typical example where firms can choose between two rules: the Nash rule and the best-reply rule. We show that a homoclinic tangency between the stable and unstable manifold of the equilibrium occurs as evolutionary pressure increases, implying bifurcation routes to complicated dynamics and strange attractors.     

Mixed bundling auctions

We study multi-object auctions where agents have private and additive valuations for heterogeneous objects. We focus on the revenue properties of a class of dominant strategy mechanisms where a weight is assigned to each partition of objects. The weights influence the probability with which partitions are chosen in the mechanism. This class contains efficient auctions, pure bundling auctions, mixed bundling auctions, auctions with reserve prices and auctions with pre-packaged bundles. For any number of objects and bidders, both the pure bundling auction and separate, efficient auctions for the single objects are revenue-inferior to an auction that involves mixed bundling.     

Dynamic price formation in a futures market via double auctions

Summary We report an exploratory study of the process of price formation in a speculative market in the absence of liquidity traders. Traders exchange a futures contract because they interpret information differently. We formulate trading as a sequence of anonymous double auctions and introduce a notion of bounded rationality in which traders use approximate models of market response in forming their bids. We prove existence of a perfect equilibrium in the sequential anonymous auctions game, and show that the equilibrium has a no-regret property. After learning the market price, a trader regrets neither the bid that he made nor the position that he holds. We show that trading volume is related to changes in the distribution of information in the economy. We also show that volume and expected change in price are related to two different attributes of the pattern of private information flow. Fundamentally, no particular relationship between the time series of these variables is always valid for all futures contracts. This point is emphasized by an example.I am thankful for useful comments made by Avraham Beja, James Gammil, Chi-fu Huang, David Scharfstein and three anonymous referees. Financial support from Stanford Graduate School Faculty Fellowship is gratefully acknowledged.     

Existence of voting-market equilibria

We show the existence of an equilibrium in a model where private goods are allocated by markets and the public good (bad) is allocated by majority rule voting.     

Bid rotation and collusion in repeated auctions

This paper studies bidder collusion with communication in repeated auctions when no side transfer is possible. It presents a simple dynamic bid rotation scheme which coordinates bids based on communication history and enables intertemporal transfer of bidders’ payoffs. The paper derives a sufficient condition for such a dynamic scheme to be an equilibrium and characterizes the equilibrium payoffs in a general environment with affiliated signals and private or interdependent values. With IPV, it is shown that this dynamic scheme yields a strictly higher payoff to the bidders than any static collusion scheme which coordinates bids based only on the current reported signals.     

Collusion Without Conspiracy: An Experimental Study of One-Sided Auctions

The study investigates collusion-nonfacilitating features of one-sided auctions. We report the results of oral nondescending bid auction experiments in which the strict bid improvement rule was absent, and compare them with the results of sealed bid auction experiments. In the sealed bid experiments the outcomes converged to the competitive equilibrium. In the oral auctions, collusive outcomes emerged and were sustained with bidders using bid matching strategies. We conclude that oral auctions provide opportunities for tacit coordination and collusion enforcement that do not exist under the sealed bid. Therefore, the strict bid improvement rule becomes critical for breaking collusion.