Equilibrium uniqueness with perfect complements

The present note analyzes the simultaneous ascending-bid auction with arbitrarily many asymmetric bidders with decreasing marginal valuations under complete information. We show that the game is solvable by iterated elimination of weakly dominated strategies if the efficient allocation assigns at least one unit to every player and if bid increments are sufficiently small. In that unique equilibrium, bidders immediately reduce their demand to the efficient allocation, and the auction ends in the first round of bidding.We would like to thank seminar participants at UC Davis, in particular Klaus Nehring and Louis Makowski for comments. Financial support by the Alexander–von–Humboldt Foundation through a Feodor–Lynen grant and by the Deutsche Forschungsgemeinschaft, SFB Transregio 15, “Governance and Efficiency of Economic Systems” is gratefully acknowledged.     

Price flexibility and market performance in experimental markets

Summary Posted offer and double auction pricing institutions are examples of fixed and flexible pricing institutions, respectively. In laboratory settings, double auction markets dominate posted offer institutions in terms of the amount of potential surplus which traders are able to extract. These results are invariant with respect to the experience of traders or the perishability of the good traded. Performance differences across institutions may be due to information differences. The introduction of a second price-posting in the posted offer institution results in a pricing institution which is closer to the double auction's flexible pricing environment. Laboratory results suggest that under otherwise comparable conditions, this innovation is sufficient to remove the performance differences which have been demonstrated to exist between the posted offer and double auction institutions.We acknowledge the support of the Social Sciences and Humanities Research Council of Canada and the Arts Research Board of McMaster University. We thank Robert Moir and Susan Sabourin for computational work on this project and the fifteen undergraduates who worked as recorders and recruiters. We also thank Charles Plott and seminar participants at the Economic Science Association, Canadian Economics Association, and Waterloo University for the comments received on recent versions of this work.     

Collusion in one-shot second-price auctions

Summary. Most of the literature on collusive behavior in auctions ignores two important issues that make collusion difficult to sustain at least in one-shot interactions: the detection of cheating and the verification of bids. Colluding bidders may deceive each other by using shill bidders. Also, if the identities of the bidders and their bids are not published then it would be difficult to verify the bid of a colluding bidder. This paper addresses these problems in one shot second price auctions where one bidder offers another bidder a side payment in exchange for not participating in the auction, while the number of other bidders is stochastic. In spite of the barriers to collusion mentioned above, a simple side payment mechanism which depends only on the auction price is introduced. It induces a successful collusion, eliminates the verification problem, provides no incentive for the use of shill bidders and guarantees that the proponent obtains ex-post non-negative payoff. The colluding bidders are ex-ante strictly better off compared with the competitive case, irrespective of their types.Received: 27 November 2002, Revised: 28 January 2005, JEL Classification Numbers: C72, D44, D82.Yair Tauman: Correspondence toWe would like to thank an anonymous referee for very valuable comments and suggestions that significantly improved the paper. We thank Shmuel Zamir for a helpful discussion.     

A choice of auction format in seller cheating: a signaling game analysis

In this paper, we examine which auction format, first-price or second-price, a seller will choose when he can profitably cheat in a second price auction by observing all bids by possible buyers and submitting a shill bid as pretending to be a buyer. We model this choice of auction format in seller cheating as a signaling game in which the buyers may regard the selection of a second price auction by the seller as a signal that he is a shill bidder. By introducing trembling-hand perfectness as a refinement of signaling equilibrium, we find two possible strictly perfect signaling equilibria. One is a separating equilibrium in which a noncheating honest seller selects a first price auction and a cheating seller does a second price auction. In another pooling equilibrium, however, both cheating and non-cheating sellers select a second price auction. The conclusion that a seller chooses a second price auction even if he cannot cheat is in contrast to the previous literature, which focused on the case of independent values. We thank an anonymous referee for useful comments that have improved the paper. This research was partially supported by the Ministry of Education, Science, Sports and Culture, Grant-in-Aid for Scientific Research (B) 15310023 and (C) 18530139.     

Existence of an equilibrium in first price auctions

Summary We study the first price auction game with an arbitrary number of bidders when the bidders' valuations are independent from each other. In technical words, we work within the independent private value model. We show that if the supports of the valuation probability distributions have the same minimum and if this minimum is not a mass point of any of these distributions, then a Nash equilibrium of the first price auction exists. We then modify the first price auction game by adding a closed interval of messages. Every bidder has to send a message with the bid he submits. These messages are used in the resolution of the ties. The winner of the auction is chosen randomly among the highest bidders with the highest value of the message among the highest bidders. In the general case, we prove the existence of a Nash equilibrium for this augmented first price auction.I wish to thank Mamoru Kaneko and a referee for their comments on an earlier draft.     

Asset price bubbles and crashes with near-zero-intelligence traders

Summary. We examine whether a simple agent-based model can generate asset price bubbles and crashes of the type observed in a series of laboratory asset market experiments beginning with the work of Smith, Suchanek and Williams (1988). We follow the methodology of Gode and Sunder (1993, 1997) and examine the outcomes that obtain when populations of zero-intelligence (ZI) budget constrained, artificial agents are placed in the various laboratory market environments that have given rise to price bubbles. We have to put more structure on the behavior of the ZI-agents in order to address features of the laboratory asset bubble environment. We show that our model of near-zero-intelligence traders, operating in the same double auction environments used in several different laboratory studies, generates asset price bubbles and crashes comparable to those observed in laboratory experiments and can also match other, more subtle features of the experimental data.Received: 15 July 2003, Revised: 28 September 2004, JEL Classification Numbers: D83, D84, G12. Correspondence to: John DuffyWe would like to thank an Anonymous referee, Guillaume Frechette, David Laibson, Al Roth and participants in Harvard Experimental and Behavioral Economics Workshop for their comments, and Charles Noussair for providing his data set.     

Private provision of a discrete public good: efficient equilibria in the private-information contribution game

We study the voluntary provision of a discrete public good via the contribution game. Players independently and simultaneously make nonrefundable contributions to fund a discrete public good, which is provided if and only if contributions cover the cost of production. We characterize nonconstant continuous symmetric equilibria, giving sufficient conditions for their existence. We show the common normalization by which players’ values are distributed over [0, 1] is not without loss of generality: if the distribution over this interval has continuous density f with f(0) >  0, then no (nonconstant) continuous symmetric equilibrium exists. We study in detail the case in which players’ private values are uniformly distributed, showing that, generically, when one continuous equilibrium exists, a continuum of continuous equilibria exists. For any given cost of the good, multiple continuous equilibria cannot be Pareto ranked. Nevertheless, not all continuous equilibria are interim incentive efficient. The set of interim incentive efficient equilibria is exactly determined. The authors thank Manfred Dix, George Mailath, Andrew Postlewaite, and an anonymous referee for their comments.     

Overconfidence and Market Efficiency with Heterogeneous Agents

We study financial markets in which both rational and overconfident agents coexist and make endogenous information acquisition decisions. We demonstrate the following irrelevance result when a positive fraction of rational agents (endogenously) decides to become informed in equilibrium, prices are set as if all investors were rational, and as a consequence the overconfidence bias does not affect informational efficiency, price volatility, rational traders’ expected profits or their welfare. Intuitively, as overconfidence goes up, so does price informativeness, which makes rational agents cut their information acquisition activities, effectively undoing the standard effect of more aggressive trading by the overconfident. The main intuition of the paper, if not the irrelevance result, is shown to be robust to different model specifications.We would like to thank Alberto Bisin, Xavier Freixas, Ken French, Moshe Kim, Jose Marín, and Terrance Odean for comments on an early draft, as well as an anonymous referee and seminar participants at HEC Geneva, the 2004 EFA meetings, the 2004 European Econometric Society meetings and the 2005 SAET conference. Diego García and Branko Urošević gratefully acknowledge financial support by SECCF (Belgrade).     

Commitment in sequential auctioning: advance listings and threshold prices

This paper models sequential auctioning of two perfect substitutes by a strategic seller, who learns about demand from the first-auction price. The seller holds the second auction only when the remaining demand is strong enough to cover her opportunity cost. Bidding in anticipation of such a contingent future auction is characterized, including a sufficient condition for existence of an invertible (increasing symmetric pure-strategy) bidding equilibrium that facilitates the seller’s learning. A unique invertible bidding equilibrium exists for the Dutch auction format, but only when the second auction is sufficiently discounted by the bidders. In the equilibrium, high-valuation bidders shade their bids down as if the second auction were guaranteed. To counter such strategic bidding, the seller would value ex-ante commitment to hold the second auction less often. Three forms of such commitment are analyzed: commitment to list future auctions in advance, commitment to not hold the second auction unless the first price exceeds a publicly announced threshold, and commitment to a reserve-price in the second auction. I would like to thank Georgios Katsenos, Thomas Jeitschko, Miguel Villas-Boas, George Deltas, and an anonymous referee for thorough and insightful feedback.     

Auctions with Financial Externalities

We study auctions with financial externalities, i.e., auctions in which losers care about how much the winner pays. In the first-price auction, larger financial externalities result in a lower expected price; in the second-price auction, the effect is ambiguous. Although the expected price in the second-price auction may increase if financial externalities increase, the seller is not able to gain more revenue by guaranteeing the losers a fraction of the auction revenue. With a reserve price, we find that both auctions may have pooling at the reserve price. This finding suggests that identical bids need not be a signal of collusion, in contrast to what is sometimes argued in anti-trust cases. We gratefully acknowledge financial support from the Dutch National Science Foundation (NWO 510.010.501 and NWO-VICI 453.03.606). For valuable discussions and comments, we would like to thank Eric van Damme, Jacob Goeree, Thomas Kittsteiner, Marta Kolodziejczyk, seminar participants at Tilburg University, Humboldt University Berlin, and National University of Singapore, and audiences at ESEM 2001 in Lausanne, and the FEEM 2002 conference in Milan on auctions and market design. The suggestions of an anonymous referee of this Journal greatly improved the article. The usual disclaimer applies.     

Make a guess: a robust mechanism for King Solomon’s dilemma

We introduce endogenous fees for participating in second-price auction which we use for a two-stage mechanism to solve King Solomon’s dilemma. They are positive for all agents. They are nonetheless shown to maintain the agents’ incentives for truthful bidding and guarantee participation by the highest-value agent. This feature of the endogenous fees is powerful enough for the efficient outcome to uniquely result from one round elimination of weakly dominated strategies, followed by at most four rounds of iterative elimination of strictly dominated stage-strategies. We provide an extension to cases with n agents and k identical prizes. We thank Gary Charness, Harrison Cheng, Jeffrey Ely, Rod Garratt, Mamoru Kaneko, Xiao Luo, Albert Ma, and Steven Morris for helpful discussions. We also thank an anonymous referee for comments that have helped to improve the paper.     

Finding a Nash equilibrium in spatial games is an NP-complete problem

Summary. We consider the class of (finite) spatial games. We show that the problem of determining whether there exists a Nash equilibrium in which each player has a payoff of at least k is NP-complete as a function of the number of players.Received: 15 September 2002, Revised: 9 March 2003, JEL Classification Numbers: C72.Correspondence to: H. HallerWe thank a referee for helpful comments. The hospitality of the Institute of Economics, University of Copenhagen, and the Institute for Advanced Studies, Vienna, is gratefully acknowledged by the third author.     

Trust and trustworthiness in games: An experimental study of intergenerational advice

This paper investigates the development of conventions of trust in what we call intergenerational games, i.e., games played by a sequence of non-overplapping agents, who pass on advice on how to play the game across adjacent generations of players. Using the trust game of Berg et al. (1995) as our experimental decision problem, advice seems to decrease the amount of trustthat evolves when this game in played in an inter-generational manner in that it decreases the amount of money sent from Senders to Returners. Ironically, advice increases trustworthinessin that Returners tend to send more back. Further, subjects appear to follows conventions of reciprocity in that they tend to Send more if they think the Returners acted in a “kind” manner, where kind means the Sender sent more money than the receiver expected. Finally, while we find a causal relationship running from trustworthiness to trust, the opposite can not be established. We note that many of our results can only be achieved using the tools offered by inter-generational games. The inter-generational advice offered provides information not available when games are played in their static form. Combining that information with elicited beliefs of the Senders and Returners adds even more information that can be used to investigate the motives that subjects have for doing what they do. Electronic supplementary material Electronic supplementary material is available for this article at and accessible for authorised users. JEL Classification C91 · C72 Resources for this research were provided by National Science Foundation grants SBR-9709962 and SBR-9709079 and by both the Center for Experimental Social Science and the C.V. Starr Center for Applied Economics at New York University. We would like to thank Shachar Kariv for both his comments and research assistance. We also thank Mikhael Shor and Judy Goldberg for research assistance, and Yevgeniy Tovshteyn for computer programming.     

Endogenous market segmentation with heterogeneous agents

Summary. An explanation is provided for the evolution of segmented marketplaces in a pairwise exchange economy. Large traders operating in a pairwise exchange market prefer to meet other similar traders, because this enables them to trade their endowments in a smaller number of encounters. Large and small traders, however, cannot be distinguished a priori, and the existence of the small traders imposes a negative externality on the large traders. We show that, under conditions which are not very restrictive, establishing a separate market (perhaps with an entry fee) designated for the large traders induces the two types of traders to segment themselves. However, this segmentation is not necessarily welfare improving. Received: January 12, 2001; revised version: July 17, 2002 RID="*" ID="*" I wish to thank the participants in the Friday Theory Workshop at the University of Sydney, and the participants at the 17th Australian Theory Workshop at the University of Melbourne for comments and discussion. John Hillas and Stephen King pointed out an omission in an earlier version, and Catherine de Fontenay and Hodaka Morita made extensive comments on earlier drafts. This work was initiated while I was a short-term visitor at the University of Southern California.     

Optimal sale across venues and auctions with a buy-now option

We characterize the optimal selling mechanism for a seller who faces demand demarcated by a high and a low end and who can access an (online) auction site (by paying an access cost) in addition to using his own store that can be used as a posted price selling venue. We first solve for the optimal mechanism of a direct revelation game in which there is no venue-restriction constraint. We find that the direct optimal mechanism must necessarily incorporate a certain kind of pooling. We then show that even with the venue constraint, the seller can use a two stage indirect mechanism that implements the allocation rule from the optimal direct mechanism, and uses the venues in an optimal fashion. The first stage of the indirect mechanism is a posted price at the store. If the object is not sold, we move to stage two, which involves an auction at the auction site. A feature of this auction is a buy-now option which is essential for implementing the pooling feature of the optimal direct mechanism. We also show that the buy-now option in the optimal mechanism is of a “temporary” variety, and that a “permanent” buy-now option, in contrast, cannot implement the optimal mechanism. Auctions with a temporary buy-now option are in widespread use on eBay. We thank the Associate Editor, George Deltas, for his insightful comments. We also thank seminar participants at the University of Basel and the SAET conference 2007.     

Bertrand Games and Sharing Rules

We consider asymmetric Bertrand games with arbitrary payoffs at ties or sharing rules, and identify sufficient conditions for the zero-profit outcome and the existence of Nash equilibria. Subject to some technical conditions on non-tied payoffs the following hold. If the sharing rule is strictly tie-decreasing all players but one receive zero equilibrium payoffs, while everybody does so if non-tied payoffs are symmetric. Mixed (pure) strategy Nash equilibria exist if the sharing rule is (norm) tie-decreasing and coalition-monotone. I would like to thank Fernando Branco, the audience at Pompeu Fabra (Barcelona), ISEG (Lisbon), University of Mannheim, ESEM 2003 (Venice), EARIE 2005 (Porto), two anonymous referees, and the editor Dan Kovenock for very useful comments. This research received financial support under project POCTI/ECO/37925/2001 of FCT and FEDER.     

Petty corruption

This paper analyzes a petty corruption model in which the entrepreneur’s type is drawn from an absolutely continuous probability distribution function F over [0, 1], and perfect Bayesian equilibrium is adopted as the solution concept for a one-stage game. In the one-stage game, if there is more than one bureaucrat, no project is approved with a strictly positive probability. For an infinitely repeated game, I show that the single window policy strictly increases the social benefits in a socially optimal equilibrium. I would like to thank Mukul Majumdar for valuable guidance and encouragement. I am also grateful to Kaushik Basu, Fernando Vega-Redondo, an anonymous referee, seminar participants at the international meeting for public economic theory (PET07) and especially Ani Guerdjikova and Roy Radner for helpful comments. Thanks are due to Hideaki Goto and Eunkyeong Lee for useful conversation.     

Externalities, monopoly and the objective function of the firm

This paper provides a theory of general equilibrium with externalities and/or monopoly. We assume that the firm’s decisions are based on the preferences of shareholders and/or other stakeholders. Under these assumptions a firm will produce fewer negative externalities than the comparable profit maximising firm. In the absence of externalities, equilibrium with a monopoly will be Pareto efficient if the firm can price discriminate. The equilibrium can be implemented by a two-part tariffWe would like to thank John Fender, Herakles Polemarchakis, Les Reinhorn, John Roberts, John Roemer, Colin Rowat, Erkan Yalcin two anonymous referees and participants in seminars at the Universities of Birmingham, Durham, Heidelberg, Mannheim and Queens, the Public Economic Theory conference at the University of Warwick, July 2000, and the EEA congress, Lausanne 2001 for comments and suggestions     

Speculative trading with rational beliefs and endogenous uncertainty

Summary. This paper introduces the framework of rational beliefs of Kurz (1994), which makes the assumptions of heterogeneous beliefs of Harrison and Kreps (1978) and Morris (1996) more plausible. Agents hold diverse beliefs that are “rational” in the sense of being compatible with ample observed data. In a non-stationary environment the agents only learn about the stationary measure of observed data, but their beliefs can remain non-stationary and diverse. Speculative trading then stems from disagreements among traders. In a Markovian framework of dividends and beliefs, we obtain analytical results to show how the speculative premium depends on the extent of heterogeneity of beliefs. In addition, we demonstrate that there exists a unique Rational Belief Equilibrium (RBE) generically with endogenous uncertainty (as defined by Kurz and Wu, 1996) and that the RBE price is higher than the rational expectation equilibrium price (REE) under some general conditions Received: March 15, 2001; revised version: April 26, 2002 RID="*" ID="*" We are deeply grateful to Mordecai Kurz for his constant encouragement and inspiring guidance over the years. We wish to express our gratitude to an anonymous referee for the very valuable comments provided. We also thank Kenneth Arrow, Peter Hammond, Roko Aliprantis and Nicholas Yannelis for their helpful suggestions and Academia Sinica and the National Science Council of the R.O.C. for their indispensable support. Correspondence to: H.-M. Wu     

Existence of Sparsely Supported Correlated Equilibria

We show that every N-player K ₁ × ... × K _N game possesses a correlated equilibrium with at least zero entries. In particular, the largest N-player K × ... × K games with unique fully supported correlated equilibrium are two-player games. We thank an anonymous referee for most useful comments. The first author acknowledges financial support from Spanish Ministry of Science and Technology, grant SEJ2004-03619, and in form of a Ramón y Cajal fellowship. The second author acknowledges support by the PASCAL Network of Excellence under EC grant no.506778, as well as from Spanish Ministry of Science and Technology and FEDER, grant BMF2003-03324. Both authors also acknowledge financial support from BBVA grant “Aprender a jugar.”