The evolution of conventions under incomplete information

Summary. We formulate an evolutionary learning process with trembles for static games of incomplete information. For many games, if the amount of trembling is small, play will be in accordance with the games (strict) Bayesian equilibria most of the time. This supports the notion of Bayesian equilibrium. Often the process will select a specific equilibrium. We study an extension to incomplete information of the prototype conflict known as Chicken and find that the equilibrium selection by evolutionary learning may well be in favor of inefficient Bayesian equilibria where some types of players fail to coordinate.Received: 17 March 2003, Revised: 3 December 2003, JEL Classification Numbers: C72.     

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.     

Resolving Indeterminacy in Coordination Games: A New Approach Applied to a Pay-as-you-go Pension Scheme

This paper shows that when players ignore what outcome will emerge because of the presence of multiple equilibria, they can coordinate their expectations by forming an initial belief based on the principle of indifference followed by a process of reasoning that updates this belief. Since this procedure describes a natural way to form beliefs under indeterminacy, it is reasonable for every agent to conjecture that all the others form their beliefs according to the same logic. Exactly the fact that agents are aware that they form their beliefs following the same procedure allows them to successfully coordinate their expectations.      

Transient and asymptotic dynamics of reinforcement learning in games

Reinforcement learners tend to repeat actions that led to satisfactory outcomes in the past, and avoid choices that resulted in unsatisfactory experiences. This behavior is one of the most widespread adaptation mechanisms in nature. In this paper we fully characterize the dynamics of one of the best known stochastic models of reinforcement learning [Bush, R., Mosteller, F., 1955. Stochastic Models of Learning. Wiley & Sons, New York] for 2-player 2-strategy games. We also provide some extensions for more general games and for a wider class of learning algorithms. Specifically, it is shown that the transient dynamics of Bush and Mosteller's model can be substantially different from its asymptotic behavior. It is also demonstrated that in general—and in sharp contrast to other reinforcement learning models in the literature—the asymptotic dynamics of Bush and Mosteller's model cannot be approximated using the continuous time limit version of its expected motion.     

Learning to Accept in Ultimatum Games: Evidence from an Experimental Design that Generates Low Offers

Focusing on responder behavior, we report panel data findings from both low and high stakes ultimatum bargaining games. Whereas Slonim and Roth (1998) find that offers are rejected fairly equally across rounds in both low and high stakes games, we find that learning does take place, but only when there is sufficient money on the table. The disparate results can be reconciled when one considers the added power that our experimental design provides-detecting subtle temporal differences in responder behavior requires a data generation process that induces a significant number of proportionally low offers.     

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.     

A Note on Best Response Dynamics

We investigate the relationship between the continuous time best response dynamic, its perturbed version, and evolutionary dynamics in relation to mixed strategy equilibria. We find that as the level of noise approaches zero, the perturbed best response dynamic has the same qualitative properties as a broad class of evolutionary dynamics. That is, stability properties of equilibria are robust across learning dynamics of quite different origins and motivations. Journal of Economic Literature Classification Numbers: C72, D83.     

Bank runs as coordination failures: An experimental study

We use experimental methods to investigate what factors contribute to breakdowns in coordination among a bank’s depositors. Subjects in our experiment decide whether to leave their money deposited in a bank or withdraw it early; a bank run occurs when there are too many early withdrawals. We explore the effects of adding uncertainty about fundamental withdrawal demand and of changing the number of opportunities subjects have to withdraw. Our results show that (i) bank runs are rare when fundamental withdrawal demand is known but occur frequently when it is stochastic, and (ii) subjects are more likely to withdraw when given multiple opportunities to do so than when presented with a single decision. For the multiple-opportunity case, we evaluate individual withdrawal decisions according to a set of simple cutoff rules. We find that the cutoff rule corresponding to the payoff-dominant equilibrium of the game, which involves Bayesian updating of probabilities, explains subject behavior better than other rules.     

Continuous-time games of timing

We address the question of existence of equilibrium in general timing games with complete information. Under weak assumptions, any two-player timing game has a Markov subgame perfect -equilibrium, for each >0. This result is tight. For some classes of games (symmetric games, games with cumulative payoffs), stronger existence results are established.     

Learning,diversification and the nature of risk

In many economic settings, faster learning is achievable only through greater exposure to risk. We study this conflict in the context of project choice, where a risk-averse agent must choose whether to invest in two projects of the same type (focus) or of different types (diversification). Focus enables faster learning across periods, but is riskier due to common type-specific shocks. Optimal choice involves balancing these two considerations. We show that focus is preferred for intermediate learning speeds, and that higher prior uncertainty may encourage focus. Thus, what matters for the focus-diversification choice is not only the level of risk, but also whether the risk is permanent or can be “learned away.” We are grateful to two anonymous referees, Serkan Bahceci, Steve Cecchetti, Adam Jaffe, Bojan Jovanovic, Kala Krishna, Blake LeBaron, Rachel McCulloch, Yana Rodgers, participants at the 2004 AEA-CSWEP Meetings and the 2003 North East Universities Development Conference, and seminar participants at Brandeis University for comments and suggestions.     

A geometric approach to the price of anarchy in nonatomic congestion games

We present a short, geometric proof for the price-of-anarchy results that have recently been established in a series of papers on selfish routing in multicommodity flow networks and on nonatomic congestion games. This novel proof also facilitates two new types of theoretical results: On the one hand, we give pseudo-approximation results that depend on the class of allowable cost functions. On the other hand, we derive stronger bounds on the inefficiency of equilibria for situations in which the equilibrium costs are within reasonable limits of the fixed costs. These tighter bounds help to explain empirical observations in vehicular traffic networks. Our analysis holds in the more general context of nonatomic congestion games, which provide the framework in which we describe this work.     

Brown–von Neumann–Nash dynamics: The continuous strategy case

Brown and von Neumann introduced a dynamical system that converges to saddle points of zero sum games with finitely many strategies. Nash used the mapping underlying these dynamics to prove existence of equilibria in general games. The resulting Brown–von Neumann–Nash dynamics are a benchmark example for myopic adjustment dynamics that, in contrast to replicator dynamics, allow for innovation, but require less rationality than the best response dynamics. This paper studies the BNN dynamics for games with infinitely many strategies. We establish Nash stationarity for continuous payoff functions. For negative semidefinite games (that include zero sum games), we generalize the results of Brown and von Neumann. In addition, we show that evolutionarily robust Nash equilibria are asymptotically stable. A complete stability analysis for doubly symmetric games is also obtained.     

Strategic knowledge sharing in Bayesian games

This paper provides a model for the study of direct, public and strategic knowledge sharing in Bayesian games. We propose an equilibrium concept which takes into account communication possibilities of exogenously certifiable statements and in which beliefs off the equilibrium path are explicitly deduced from consistent possibility correspondences, without making reference to perturbed games. Properties of such an equilibrium and of revised knowledge are examined. In particular, it is shown that our equilibrium is always a sequential equilibrium of the associated extensive form game with communication. Finally, sufficient conditions for the existence of perfectly revealing or non-revealing equilibria are characterized in some classes of games. Several examples and economic applications are investigated.     

Reinforcement-Based Adaptive Learning in Asymmetric Two-Person Bargaining with Incomplete Information

The sealed bidk-double auction is a mechanism used to structure bilateral bargaining under two-sided incomplete information. This mechanism is tested in two experiments in which subjects are asked to bargain repeatedly for 50 rounds with the same partner under conditions of information disparity favoring either the buyer (Condition BA) or seller (Condition SA). Qualitatively, the observed bid and offer functions are in agreement with the Bayesian linear equilibrium solution (LES) constructed by Chatterjee and Samuelson (1983). A trader favored by the information disparity, whether buyer or seller, receives a larger share of the realized gain from trade than the other trader. Comparison with previous results reported by Daniel, Seale, and Rapoport (1998), who used randomly matched rather than fixed pairs, shows that when reputation effects are present this advantage is significantly enhanced. A reinforcement-based learning model captures the major features of the offer and bid functions, accounting for most of the variability in the round-to-round individual decisions.     

Imperfect information, Bayesian learning, and capital accumulation

This paper examines the consequences of informational imperfections for economic growth in an overlapping generations model in which agents learn the technological parameters in a Bayesian fashion. Under mild sufficient conditions, beliefs converge to the true value of the technological parameters. Nevertheless, even short-lived informational imperfections could have lasting effects, as they alter the long-run equilibrium levels of the capital stock. Therefore, learning dynamics may explain some of the observed differences in the performance of countries with otherwise similar economic characteristics.