Sequential location equilibria under incomplete information

Summary We consider a simple model of incomplete information in location theory. Two firms compete in a two stage framework: a sequential location stage and a price competition stage. Firm 1 knows both its own constant marginal cost technology and that of Firm 2, whereas the latter has incomplete information about firm 1's technology. The location stage turns out to be a monotonic signaling game and theunique D1 equilibrium is a pure strategy separating equilibrium if firm 1's cost advantage is below some bound, and otherwise a pooling equilibrium if the prior probability that Firm 1 is of the low cost type is high, or a semi-pooling equilibrium if it is low. This surprising result is due to the fact that the location gap between the two types of Firm 1 is bounded because of natural economic reasons, which may prevent the separation of the two types. Hence, incomplete information matters: the equilibrium locations differ quite significantly from the full information equilibrium locations.We would like to thank an anonymous referee for very helpful comments and also the participants in seminars at GREQE (Marseille), Université de Montréal, UBC, HEC (Paris), in the Location Theory session of the World Congress of the Econometric Society (Barcelona) and in the Game Theory Conference at the University of Western Ontario for their comments. We remain, of course, solely responsible for the content of the paper. Financial support from FCAR (Québec), SSHRCC (Canada) and CNRS (France) is gratefully acknowledged.     

Rationalizable foresight dynamics

This paper proposes and studies the rationalizable foresight dynamics. A normal form game is repeatedly played in a random matching fashion by a continuum of agents who make decisions at stochastic points in time. A rationalizable foresight path is a feasible path of action distribution along which each agent takes an action that maximizes his expected discounted payoff against another path which is in turn a rationalizable foresight path. We consider a set-valued stability concept under this dynamics and compare it with the corresponding concept under the perfect foresight dynamics.     

Testing, hold up and the dynamics of preferences

This paper presents an overlapping generations model with cultural transmission of preferences in an economy in which players face a hold up problem. One of the players, the firm, can use a testing technology which allows him to imperfectly monitor his partner's behaviour. This technology is completely useless with homogeneous preferences. We obtain that in the stable steady state of the economy there is a mixed distribution of preferences where both selfish and other-regarding preferences are present in the population. Moreover, with a good testing technology, the steady state is characterized by the first-best result in the investment decisions. JEL Classification: C78, D23, D63 The authors acknowledge financial support from the Spanish Ministry of Science and Technology project SEC2001-2763. This paper has also benefited from comments of participants in the XXVIII Spanish Symposium of Economic Analysis in Seville (Spain), in the International Workshop on Social and Behavioral Economics in Valencia (Spain), in the 2nd World Congress of the Game Theory Society in Marseille and in the VI Spanish Meeting on Game Theory and Practice in Elche (Spain).     

Non-implementation of rational expectations as a perfect Bayesian equilibrium

Summary. We point out several conceptual difficulties of the rational expectations equilibrium concept. In particular we show that such an equilibrium need not be incentive compatible and need not be implementable as a perfect Bayesian equilibrium . A comparison of rational expectations equilibria with the private core is also provided. We conclude that the private core is a more appropriate concept to capture the idea of contracts under asymmetric information.Received: 15 December 2003, Revised: 18 November 2004, JEL Classification Numbers: C71, C72, D5, D82. Correspondence to: Nicholas C. YannelisWe wish to thank Dr A. Hadjiprocopis for his invaluable help with the implementation of Latex in a Unix environment.     

Bayesian–Walrasian equilibria: beyond the rational expectations equilibrium

In general rational expectations equilibrium (REE), as introduced in Radner (Econometrica 47:655–678, 1978) in an Arrow–Debreu–McKenzie setting with uncertainty, does not exist. Moreover, it fails to be fully Pareto optimal and incentive compatible and is also not implementable as a perfect Bayesian equilibrium of an extensive form game (Glycopantis et al. in Econ Theory 26:765–791, 2005). The lack of all the above properties is mainly due to the fact that the agents are supposed to predict the equilibrium market clearing price (as agent’s expected maximized utility is conditioned on the information that equilibrium prices reveal), which leads inevitably to the presumption that agents know all the primitives in the economy, i.e., random initial endowments, random utility functions and private information sets. To get around this problematic equilibrium notion, we introduce a new concept called Bayesian–Walrasian equilibrium (BWE) which has Bayesian features. In particular, agents try to predict the market-clearing prices using Bayesian updating and evaluate their consumption in terms of Bayesian price estimates, which are different for each individual. In this framework agents maximize expected utility conditioned on their own private information about the state of nature, subject to a Bayesian estimated budget constraint. Market clearing is not an intrinsic part of the definition of BWE. However, both in the case of perfect foresight and in the case of symmetric information BWE leads to a statewise market clearing; it then becomes an ex post Walrasian equilibrium allocation. This new BWE exists under standard assumptions, in contrast to the REE. In particular, we show that our new BWE exists in the well-known example in Kreps (J Econ Theory 14:32–43, 1977), where REE fails to exist. This work was done in the Spring of 2005, when EJB was a visiting professor at the University of Illinois.     

Reviews

Book reviewed in this article:
Risk, Organizations, and Society. Studies in Risk and Uncertainty, edited by M. Shubik
Human Development Reports 1990-92, United Nations Development Program, published for the UNDP by Oxford University Press
Mastering Risk: Environment, Markets & Politics in Australian Economic History, by Colin White
Applied Econometric Techniques, by Keith Cuthbertson, Stephen G. Hall and Mark P. Taylor
Issues in Contemporary Economics, Proceedings of the Ninth World Congress of the International Economic Association, edited by Congress Editor, Amartia Sen. Volume 4: Women s Work in the World Economy, edited by Nancy Folbre, Barbara Bergman, Bina Agrawal, and Maria Floro
A Short History of Socialist Money, G. Peebles
Money in the People's Republic of China: A Comparative Perspective, G. Peebles
Growth through Competition, Competition through Growth: Strategic Management and the Economy in Japan, by Hiroyuki Odagiri
Commodities in Crisis, by Alfred Maizcls
Taxation and the Global Economy, edited by A. Razin and J. Slemrod
Conflicts and Cooperation in Managing Environmental Resources, edited by Rudiger Pethig
The Economics of Business Culture Game Theory, Transaction Costs, and Economic Performance, by Mark Casson     

Why is there money? Endogenous derivation of `money' as the most liquid asset: a class of examples

Summary. The monetary character of trade, use of a common medium of exchange, is shown to be an outcome of an economic general equilibrium. Monetary structure can be derived from price theory in a modified Arrow-Debreu model. Two constructs are added: transaction costs and market segmentation in trading posts (with a separate budget constraint at each transaction). Commodity money arises endogenously as the most liquid (lowest transaction cost) asset. Government-issued fiat money has a positive equilibrium value from its acceptability for tax payments. Scale economies in transaction cost account for uniqueness of the (fiat or commodity) money in equilibrium. Received: February 15, 2002; revised version: August 12, 2002 RID="*" ID="*" This paper has benefited from seminars and colleagues' helpful remarks at the University of California - Santa Barbara, University of California - San Diego, NSF-NBER Conference on General Equilibrium Theory at Purdue University, Society for the Advancement of Behavioral Economics at San Diego State University, Econometric Society at the University of Wisconsin - Madison, SITE at Stanford University-2001, Federal Reserve Bank of Kansas City, Federal Reserve Bank of Minneapolis, Midwest Economic Theory Conference at the University of Illinois - Urbana Champaign, University of Iowa, Southern California Economic Theory Conference at UC - Santa Barbara, Midwest Macroeconomics Conference at University of Iowa, University of California - Berkeley, European Workshop on General Equilibrium Theory at University of Paris I, Society for Economic Dynamics at San Jose Costa Rica, World Congress of the Econometric Society at University of Washington, Cowles Foundation at Yale University. It is a pleasure to acknowledge comments of Henning Bohn, Harold Cole, James Hamilton, Mukul Majumdar, Harry Markowitz, Chris Phelan, Meenakshi Rajeev, Wendy Shaffer, Bruce Smith, and Max Stinchcombe.     

Monotonicity and Rationalizability in a Large First Price Auction

This paper proves that the monotonicity of bidding strategies together with the rationality of bidders implies that the winning bid in a first price auction converges to the competitive equilibrium price as the number of bidders increases ( Wilson, 1977 ). Instead of analysing the symmetric Nash equilibrium, we examine rationalizable strategies ( Bernheim (1984) , Pearce (1984) ) among the set of monotonic bidding strategies to prove that any monotonic rationalizable bidding strategy must be within a small neighbourhood of the „truthful” valuation of the object, conditioned on the signal received by the bidder. We obtain an information aggregation result similar to that of Wilson (1977) , while dispensing with almost all symmetric assumptions and using a milder solution concept than the Nash equilibrium. In particular, if every bidder is ex ante identical, then any rationalizable bidding strategy must be within a small neighbourhood of the symmetric Nash equilibrium. In a symmetric first price auction, the symmetry of outcomes is implied rather than assumed.     

An ascending double auction

We show why the failure of the affiliation assumption prevents the double auction from achieving efficient outcomes when values are interdependent. This motivates the study of an ascending price version of the double auction. It is shown that when there is a sufficiently large, but still finite, number of sellers, this mechanism has an approximate perfect Bayesian equilibrium in which traders continue bidding if and only if their true estimates of the ‘value’ of the object being traded exceed the current price. This equilibrium is ex post efficient and has a rational expectations property in the sense that along the equilibrium path traders appear to have made the best possible trades conditional on information revealed by the trading process. We thank two anonymous referees and Dan Kovenock, the Editor, whose detailed comments and suggestions have allowed us to substantially improve the paper. We also thank seminar participants at University of Toronto, University of Wisconsin-Madison, Summer 2003 North American Meetings of the Econometric Society, 2004 NSF Decentralization Conference for their comments.     

Comparing finite mechanisms

Summary. This paper obtains finite analogues to propositions that a previous literature obtained about the informational efficiency of mechanisms whose possible messages form a continuum. Upon reaching an equilibrium message, to which all persons “agree”, a mechanism obtains an action appropriate to the organization's environment. Each person's privately observed characteristic (a part of the organization's environment) enters her agreement rule. An example is the Walrasian mechanism in an exchange economy. There a message specifies a proposed trade vector for each trader as well as a price for each non-numeraire commodity. A trader agrees if the price of each non-numeraire commodity equals her marginal utility for that commodity (at the proposed trades) divided by her marginal utility for the numeraire. At an equilibrium message, the mechanism's action consists of the trades specified in that message, and (for classic economies) those trades are Pareto-optimal and individually rational. Even though the space of environments (characteristics) is a continuum, mechanisms with a continuum of possible messages are unrealistic, since transmitting every point of a continuum is impossible. In reality, messages have to be rounded off and the number of possible messages has to be finite. Moreover, reaching a continuum mechanism's equilibrium message typically requires infinite time and that difficulty is absent if the number of possible messages is finite. The question therefore arises whether results about continuum mechanisms have finite counterparts. If we measure a continuum mechanism's communication cost by its message-space dimension, then our corresponding cost measure for a finite mechanism is the (finite) number of possible equilibrium messages. We find that if two continuum mechanisms yield the same action but the first has higher message-space dimension, then a sufficiently fine finite approximation of the first has larger error than an approximation of the second if the cost of the first approximation is no higher than the cost of the second approximation. An approximation's “error” is the largest distance between the continuum mechanism's action and the approximation's action. We obtain bounds on error. We also study the performance of Direct Revelation (DR) mechanisms relative to “indirect” mechanisms, both yielding the same action, when the environment set grows. We find that as the environment-set dimension goes to infinity, so does the extra cost of the DR approximation, if the error of the DR approximation is at least as small as the error of the indirect approximation. While the paper deals with information-processing costs and not incentives, it is related to the incentive literature, since the Revelation Principle is central to much of that literature and one of our main results is the informational inefficiency of finite Direct Revelation mechanisms. Received: May 21, 2001; revised version: December 14, 2001 RID="*" ID="*" Earlier versions of this paper were presented at the Decentralization Conference, Washington University, St Louis, April 2000 and at the Eighth World Congress of the Econometric Society, August 2000. We are grateful for comments received on those occasions. The second author gratefully acknowledges support from National Science Foundation grant #IIS9712131. Correspondence to: T. Marschak     

Asymmetric information and rationalizability

We study how asymmetric information affects the set of rationalizable solutions in a linear setup where the outcome is determined by forecasts about this same outcome. The unique rational expectations equilibrium is also the unique rationalizable solution when the sensitivity of the outcome to agents’ forecasts is less than one, provided that this sensitivity is common knowledge. Relaxing this common knowledge assumption, multiple rationalizable solutions arise when the proportion of agents who know the sensitivity is large, and the uninformed agents believe it is possible that the sensitivity is greater than one. Instability is equivalent to existence of some kind of sunspot equilibria.     

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).     

Implementing Arrow-Debreu equilibria by trading infinitely-lived securities

Summary. We show that Arrow-Debreu equilibria with countably additive prices in infinite-time economy under uncertainty can be implemented by trading infinitely-lived securities in complete sequential markets under two different portfolio feasibility constraints: wealth constraint, and essentially bounded portfolios. Sequential equilibria with no price bubbles implement Arrow-Debreu equilibria, while those with price bubbles implement Arrow-Debreu equilibria with transfers. Transfers are equal to price bubbles on initial portfolio holdings. Price bubbles arise in sequential equilibrium under the wealth constraint if some securities are in zero supply or negative prices are permitted, but cannot arise with essentially bounded portfolios.Received: 19 November 2003, Revised: 24 February 2004, JEL Classification Numbers: D50, G12, E44.Correspondence to: Jan WernerWe acknowledge helpful discussions with Roko Aliprantis, Subir Chattopaydhyay, Steve LeRoy, Manuel Santos, and seminar participants at Brown University, University of Pennsylvania, NBER Workshop in General Equilibrium Theory, SITE 2000, the 2000 World Congress of the Econometric Society, and Federal Reserve Bank of Kansas City. The views expressed herein are those of the authors and do not necessarily reflect the views of Federal Reserve Bank of Kansas City or the Federal Reserve System.     

Purification in the infinitely-repeated prisoners' dilemma

This paper investigates the Harsanyi [Harsanyi, J.C., 1973. Games with randomly disturbed payoffs: A new rationale for mixed-strategy equilibrium points. International Journal of Game Theory 2 (1), 1–23]-purifiability of mixed strategies in the repeated prisoners' dilemma with perfect monitoring. We perturb the game so that in each period, a player receives a private payoff shock which is independently and identically distributed across players and periods. We focus on the purifiability of one-period memory mixed strategy equilibria used by Ely and Välimäki [Ely, J.C., Välimäki, J., 2002. A robust folk theorem for the prisoner's dilemma. Journal of Economic Theory 102 (1), 84–105] in their study of the repeated prisoners' dilemma with private monitoring. We find that any such strategy profile is not the limit of one-period memory equilibrium strategy profiles of the perturbed game, for almost all noise distributions. However, if we allow infinite memory strategies in the perturbed game, then any completely-mixed equilibrium is purifiable.     

ROBUST RATIONALIZABILITY UNDER ALMOST COMMON CERTAINTY OF PAYOFFS

An action is robustly rationalizable if it is rationalizable for every type who has almost common certainty of payoffs. We illustrate by means of an example that an action may not be robustly rationalizable even if it is weakly dominant, and argue that robust rationalizability is a very stringent refinement of rationalizability. Nonetheless, we show that every strictly rationalizable action is robustly rationalizable. We also investigate how permissive robust rationalizability becomes if we require that players be fully certain of their own payoffs.     

Connecting the cooperative and competitive structures of the multiple-partners assignment game

Multiple-partners assignment game is the name used by Sotomayor [The multiple partners game, in: M. Majumdar (Ed.), Equilibrium and Dynamics: Essays in Honor of David Gale, The Macmillan Press Ltd., New York, 1992; The lattice structure of the set of stable outcomes of the multiple partners assignment game, Int. J. Game Theory 28 (1999) 567-583] to describe the cooperative structure of the many-to-many matching market with additively separable utilities. Stability concept is proved to be different from the core concept. An economic structure is proposed where the concept of competitive equilibrium payoff is introduced in connection to the equilibrium concept from standard microeconomic theory. The paper examines how this equilibrium concept compares with the cooperative equilibrium concept. Properties of interest to the cooperative and competitive markets are derived.     

Trembles in extensive games with ambiguity averse players

We introduce and analyze three definitions of equilibrium for finite extensive games with imperfect information and ambiguity averse players. In a setting where players’ preferences are represented by maxmin expected utility, as characterized in Gilboa and Schmeidler (J Math Econ 18(2):141–153, 1989), our definitions capture the intuition that players may consider the possibility of slight arbitrary mistakes. This generalizes the idea leading to trembling-hand perfect equilibrium as introduced in Selten (Int J Game Theory 4(1):25–55, 1975), by allowing for ambiguous trembles characterized by sets of distributions. We prove existence for two of our equilibrium notions and relate our definitions to standard equilibrium concepts with expected utility maximizing players. Our analysis shows that ambiguity aversion can lead to behavioral implications that are distinct from those attained under expected utility maximization, even if ambiguous beliefs only arise from the possibility of slight mistakes in the implementation of unambiguous strategies.     

On non-revealing rational expectations equilibrium

It is shown that a non-revealing rational expectations equilibrium may not be coalitionally Bayesian incentive compatible, may not be implementable as a perfect Bayesian equilibrium and may not belong to the weak fine core and thus may not be fully Pareto optimal. These negative results lead us to conclude the non-revealing rational expectations equilibrium is not a sensible solution concept. We wish to thank Dr A. Hadjiprocopis for his invaluable help with the implementation of Latex in a Unix environment. We also thank a referee for several, constructive suggestions.     

The structure of information networks

We develop a strategic model of information acquisition in networks where agents pay for all the pieces of information they acquire, including those through indirect links. The cost of information depends on the distance it traverses in the network. We consider two possibilities in this context: (1) costs increasing with distance, and (2) costs decreasing with distance. The paper also examines situations where it is more expensive to acquire information of higher value. We show that there is almost no divergence between the efficient and Nash equilibrium information architectures. We then study the effect of decay in networks where information through longer paths is cheaper. Finally, we also examine a model with costly link formation that combines both types of cost related assumptionsWe are grateful to Hans Haller, Rob Gilles, Susanne Maria Schmidt, Sumit Joshi, Georg Erber, Beth Allen, Mark Machina, Bob Martin, Johanna Francis, Cheryl Long, Raja Kali, Kaz Miyagiwa, Micheal Kosfeld, Bibhudutta Panda and two anonymous referees for helpful suggestions. The paper has further benefited from the comments of participants at Game Theory and Applications Mumbai 2003, Royal Economic Society Meetings 2003, SED 2004 and GAMES 2004. Sudipta Sarangi acknowledges the hospitality of DIW Berlin where a part of this research was carried out. Rajgopal Kannan acknowledges the support of NSF grants IIS-0329738 and IIS-0312632