Inferring repeated-game strategies from actions: evidence from trust game experiments

Summary. Combining a strategy model, an inference procedure and a new experimental design, we map sequences of observed actions in repeated games to unobserved strategies that reflect decision-makers’ plans. We demonstrate the method by studying two institutional settings with distinct theoretical predictions. We find that almost all strategies inferred are best responses to one of the inferred strategies of other players, and in one of the settings almost all of the inferred strategies, which include triggers to punish non-cooperators, are consistent with equilibrium strategies. By developing a method to infer unobserved repeated-game strategies from actions, we take a step toward making game theory a more applied tool, bridging a gap between theory and observed behavior.Received: 23 December 2002, Revised: 19 April 2005, JEL Classification Numbers: C72, C80, C90.The authors are indebted for discussions with Ray Battalio, David Cooper, Robin Dubin, John Duffy, Ellen Garbarino, Susan Helper, Margaret Meyer, John Miller, Jim Rebitzer, Mari Rege, Al Roth, and John Van Huyck. The authors also benefited from discussants at economic department seminars at Case Western Reserve, McMaster and McGill University, University of Pittsburgh, SUNY-Stony Brook, and Texas A&M, and participants at the 2002 European Winter Meeting of the Econometric Society. We are grateful for the financial support provided by the Department of Economics at the University of Pittsburgh and Case Western Reserve University.     

Non-computable strategies and discounted repeated games

Summary A number of authors have used formal models of computation to capture the idea of bounded rationality in repeated games. Most of this literature has used computability by a finite automaton as the standard. A conceptual difficulty with this standard is that the decision problem is not closed. That is, for every strategy implementable by an automaton, there is some best response implementable by an automaton, but there may not exist any algorithm forfinding such a best response that can be implemented by an automaton. However, such algorithms can always be implemented by a Turing machine, the most powerful formal model of computation. In this paper, we investigate whether the decision problem can be closed by adopting Turing machines as the standard of computability. The answer we offer is negative. Indeed, for a large class of discounted repeated games (including the repeated Prisoner's Dilemma) there exist strategies implementable by a Turing machine for whichno best response is implementable by a Turing machine.The work was begun while Nachbar was a visitor at The Center for Mathematical studies in Economics and Management Science at Northwestern University; he is grateful for their hospitality. We are also grateful to Robert Anderson and Neil Gretsky and to seminar audiences at UCLA for useful comments, and to the National Science Foundation and the UCLA Academic Senate Committee on Research for financial support. This paper is an outgrowth of work reported in Learning and Computability in Discounted Supergames.     

Nash equilibrium when players account for the complexity of their forecasts

Nash equilibrium is often interpreted as a steady state in which each player holds the correct expectations about the other players' behavior and acts rationally. This paper investigates the robustness of this interpretation when there are small costs associated with complicated forecasts. The model consists of a two-person strategic game in which each player chooses a finite machine to implement a strategy in an infinitely repeated 2×2 game with discounting. I analyze the model using a solution concept called Nash Equilibrium with Stable Forecasts (ESF). My main results concern the structure of equilibrium machine pairs. They provide necessary and sufficient conditions on the form of equilibrium strategies and plays. In contrast to the “folk theorem,” these structural properties place severe restrictions on the set of equilibrium paths and payoffs. For example, only sequences of the one-shot Nash equilibrium can be generated by any ESF of the repeated game of chicken.     

A folk theorem for minority games

We study a particular case of repeated games with public signals. In the stage game an odd number of players have to choose simultaneously one of two rooms. The players who choose the less crowded room receive a reward of one euro (whence the name “minority game”). The players in the same room do not recognize each other, and between the stages only the current majority room is publicly announced. We show that in the infinitely repeated game any feasible payoff can be achieved as a uniform equilibrium payoff, and as an almost sure equilibrium payoff. In particular we construct an inefficient equilibrium where, with probability one, all players choose the same room at almost all stages. This equilibrium is sustained by punishment phases which use, in an unusual way, the pure actions that were played before the start of the punishment.     

Repeated signaling games

I analyze a class of repeated signaling games in which the informed player's type is persistent and the history of actions is perfectly observable. In this context, a large class of possibly complex sequences of signals can be supported as the separating equilibrium actions of the “strong type” of the informed player. I characterize the set of such sequences. I also characterize the sequences of signals in least cost separating equilibria (LCSE) of these games. In doing this, I introduce a state variable that can be interpreted as a measure of reputation. This gives the optimization problem characterizing the LCSE a recursive structure. I show that, in general, the equilibrium path sequences of signals have a simple structure. The shapes of the optimal sequences depend critically on the relative concavities of the payoff functions of different types, which measure the relative preferences towards payoff smoothing.     

Staff rotation as an anti-corruption policy: an experimental study

Within the German federal government regular staff rotation is a precautionary measure against corruption in public administration. To study the effect of this policy, an experiment was conducted where pairs of potential bribers and public officials were randomly re-matched in every round. The outcome is compared to the case where the identity of the pairs interacting remained fixed. The conclusion is that rotation of interacting pairs significantly reduces the levels of bribes as well as the frequency of inefficient decisions due to bribery.     

Stated beliefs versus inferred beliefs: A methodological inquiry and experimental test

Belief elicitation in game experiments may be problematic if it changes game play. We experimentally verify that belief elicitation can alter paths of play in a two-player repeated asymmetric matching pennies game. Importantly, this effect occurs only during early periods and only for players with strongly asymmetric payoffs, consistent with a cognitive/affective effect on priors that may serve as a substitute for experience. These effects occur with a common scoring rule elicitation procedure, but not with simpler (unmotivated) statements of expected choices of opponents. Scoring rule belief elicitation improves the goodness of fit of structural models of belief learning, and prior beliefs implied by such models are both stronger and more realistic when beliefs are elicited than when they are not. We also find that “inferred beliefs” (beliefs estimated from past observed actions of opponents) can predict observed actions better than the “stated beliefs” from scoring rule belief elicitation.     

Communication equilibrium payoffs in repeated games with imperfect monitoring

We characterize the set of communication equilibrium payoffs of any undiscounted repeated matrix-game with imperfect monitoring and complete information. For two-player games, a characterization is provided by Mertens, Sorin, and Zamir (Repeated games, Part A (1994) CORE DP 9420), mainly using Lehrer's (Math. Operations Res. (1992) 175) result for correlated equilibria. The main result of this paper is to extend this characterization to the n-player case. The proof of the characterization relies on an analogy with an auxiliary 2-player repeated game with incomplete information and imperfect monitoring. We use Kohlberg's (Int. J. Game Theory (1975) 7) result to construct explicitly a canonical communication device for each communication equilibrium payoff.     

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 learning-based model of repeated games with incomplete information

This paper tests a learning-based model of strategic teaching in repeated games with incomplete information. The repeated game has a long-run player whose type is unknown to a group of short-run players. The proposed model assumes a fraction of ‘short-run’ players follow a one-parameter learning model (self-tuning EWA). In addition, some ‘long-run’ players are myopic while others are sophisticated and rationally anticipate how short-run players adjust their actions over time and “teach” the short-run players to maximize their long-run payoffs. All players optimize noisily. The proposed model nests an agent-based quantal-response equilibrium (AQRE) and the standard equilibrium models as special cases. Using data from 28 experimental sessions of trust and entry repeated games, including 8 previously unpublished sessions, the model fits substantially better than chance and much better than standard equilibrium models. Estimates show that most of the long-run players are sophisticated, and short-run players become more sophisticated with experience.     

外贸企业开展绿色营销的必要性及策略分析

随着全球保护意识的增强，世界各国都在实施可持续发展战略，强调经济发展应与环境保护相协调，作为绿色保护运动的一个重要组成部分绿色营销业正成为社会和企业认真研究的热门话题。外贸企业开展绿色营销是走向世界的重要途径。     

Minimax after money-max: why major league baseball players do not follow optimal strategies

Using real-life sports data of Major League Baseball, this paper investigates whether professional players follow the minimax theorem in their strategies. Our empirical results using the 2010 regular-season data show that baseball players do not optimize their strategies: there is a significant difference in their payoffs across strategies, and the sequence of their strategy choices is predictable from their previous actions. Further analysis using individual salaries and key contract variables indicates that a higher salary has a positive impact on following minimax strategies in the regular season. By contrast, a longer contract decreases a player’s incentive to pursue optimal strategies in the postseason. These results have important implications for compensation practices in various fields.     

The dynamics of optimal abatement strategies for multiple pollutants—An illustration in the Greenhouse

We investigate optimal abatement strategies for cumulative and interacting pollutants. We show that different decay rates can cause non-monotonic behavior in the optimal paths of emissions, the aggregate level of pollution, and even the relative optimal price for emissions. This contrasts strikingly with the case of a single pollutant. The results are illustrated by numerical simulations, first for instructive fictitious and second for more realistic parameters mimicking the greenhouse problem. The results add to the skepticism existing about whether the concept of global warming potential is a useful indicator for the optimal relative abatement of different GHGs over time. In fact, we show that a constant index suitable for comparing dynamically different pollutants with respect to their economic harmfulness does not exist.     

Exploiting moral wiggle room: experiments demonstrating an illusory preference for fairness

This paper explores whether generosity in experiments is truly evidence of concern for desirable social outcomes. We conduct an experiment using a binary version of the dictator game. We introduce several treatments in which subjects are able to leave the relationship between their actions and resulting outcomes uncertain, either to themselves or to another subject influenced by those actions, thus giving subjects the moral “wiggle room” to behave self-interestedly. We find significantly less generous behavior in these manipulations, relative to a baseline in which the relationship between actions and outcomes is transparent. We conclude that many subjects behave fairly in the baseline case mainly because they intrinsically dislike appearing unfair, either to themselves or others. We thank Cristina Bicchieri; Iris Bohnet; Colin Camerer; Robyn Dawes; Ernst Fehr; George Loewenstein; John Patty; Charlie Plott; Matthew Rabin; seminar participants at Carnegie Mellon, Berkeley, Cornell, Emory, and Princeton; participants at the 2003 Public Choice, 2003 ESA, 2003 SJDM, and 2004 BDRM meetings; and anonymous referees for helpful comments and suggestions. We greatly appreciate the access to resources at the Pittsburgh Experimental Economics Laboratory (PEEL) at the University of Pittsburgh. This research was funded by a Carnegie Mellon Berkman Faculty Development Grant to Weber and a Russell Sage Behavioral Economics Small Grant to Dana.     

Approachability with bounded memory

We study Blackwell's approachability in repeated games with vector payoffs when the approaching player is restricted to use strategies with bounded memory: either strategies with bounded recall, or strategies that can be implemented by finite automata. Our main finding is that the following three statements are equivalent for closed sets. (i) The set is approachable with bounded recall strategies. (ii) The set is approachable with strategies that can be implemented with finite automata. (iii) The set contains a convex approachable set. Using our results we show that (i) there are almost-regret-free strategies with bounded memory, (ii) there is a strategy with bounded memory to choose the best among several experts, and (iii) Hart and Mas-Colell's adaptive learning procedure can be achieved using strategies with bounded memory.     

Reputation and Reciprocity: Consequences for the Labour Relation

Recent evidence highlights the importance of social norms in many economic relations. However, many of these relationships are long term and provide repeated game incentives for performance. We experimentally investigate interaction effects of reciprocity and repeated game incentives in two treatments (one-shot and repeated) of a gift-exchange game. In both treatments we observe reciprocity, which is strengthened in the repeated game. A detailed analysis shows that, in the repeated game, some subjects imitate reciprocity. Thus, reciprocity and repeated game incentives reinforce each other. Observed behaviour is robust against experience. We conclude that long-term interaction is a "reciprocity-compatible" contract enforcement device.
JEL classification : J 30; C 91     

知识产权质押融资风险补偿基金:运作模式与发展策略

知识产权质押融资风险补偿基金是近年来我国引导金融机构拓展知识产权质押融资业务,缓解中小微科技企业融资难的积极探索。目前,四川、山东、广东、辽宁4省份已开展了风险补偿基金试点工作。鉴于当前风险补偿基金在基金来源、主体结构、管理机制和责任配置等方面的运作模式,从扩展补偿范围、优化合作模式、健全评价体系、完善风险控制4个方面入手,构建风险补偿基金的政策支撑与发展策略,最终形成政府引导、市场运作、风险共担、合理容错的基金运营格局。     

Learning by Trial and Error: A Case for Moot Courts

This article describes the development of a computer tutorial for use in a history of economic ideas class. An early version of the tutorial contained ten topics, ranging from early Mercantilist thought to Jevons's marginal utility analysis. These concepts were presented in three ways: verbally, graphically, and in summary form. Student critiques were used to extend the content and revise the mode of presentation.     

Airing Your Dirty Laundry: A Quick Marketable Pollution Permits Game for the Classroom

In this article, the authors describe a simple classroom game that demonstrates the advantage of tradable emissions permits in regulating environmental pollution. Students take on the role of polluters who must consider the costs of complying with a uniform reduction and a tradable permits program. The class is divided into high-cost polluters and low-cost polluters so students can observe the gains from trade as high-cost students purchase pollution rights from the low-cost students in the tradable permits scenario. A major advantage of the game is that it can be conducted within as little as 20 minutes and does not require that students have prior knowledge of economics or regulatory policies. This makes the game appropriate for economics and noneconomics courses alike.