Learning and sophistication in coordination games

This paper studies the role of strategic teaching in coordination games and whether changing the incentives of players to teach leads to more efficient coordination. We ran experiments where subjects played one of four coordination games in constant pairings, where the incentives to teach were varied along two dimensions—the short run cost of teaching and the long run benefit to teaching. We show which aspects of the game lead subjects to adopt long run teaching strategies, and show that subjects try to manipulate their opponent’s actions to pull them out of a situation of coordination failure. We also show that extending a model of decision making by introducing a forward-looking component helps to track teachers’ behaviour more accurately, and describes the way players behave in a more unified way across both teachers and learners.     

Coordination of strategic responses to security threats: Laboratory evidence

Laboratory experiments are used to evaluate the extent to which players in games can coordinate investments that diminish the probability of losses due to security breaches or terrorist attacks. In this environment, economically sensible investments may be foregone if their potential benefits are negated by failures to invest in security at other sites. The result is a coordination game with a desirable high-payoff, high-security equilibrium and an undesirable low-security equilibrium that may result if players do not expect others to invest in security. One unique feature of this coordination situation is that investment in security by one player generates a positive externality such that all other players’ expected payoffs are increased, regardless of those other players’ investment decisions. Coordination failures are pervasive in a baseline experiment with simultaneous decisions, but coordination is improved if players are allowed to move in an endogenously determined sequence. In addition, coordinated security investments are observed more often when the largest single security threat to individuals is preventable by their own decisions to invest in security. The security coordination game is a “potential game,” and the success of coordination on the more secure equilibrium is related to the notion of potential function maximization and basin of attraction.      

Coordination and transfer

We study the ability of subjects to transfer principles between related coordination games. Subjects play a class of order statistic coordination games closely related to the well-known minimum (or weak-link) and median games (Van Huyck et al. in Am Econ Rev 80:234–248, 1990, Q J Econ 106(3):885–910, 1991). When subjects play a random sequence of games with differing order statistics, play is less sensitive to the order statistic than when a fixed order statistic is used throughout. This is consistent with the prediction of a simple learning model with transfer. If subjects play a series of similar stag hunt games, play converges to the payoff dominant equilibrium when a convention emerges, replicating the main result of Rankin et al. (Games Econ Behav 32:315–337, 2000). When these subjects subsequently play a random sequence of order statistic games, play is shifted towards the payoff dominant equilibrium relative to subjects without previous experience. The data is consistent with subjects absorbing a general principle, play of the payoff dominant equilibrium, and applying it in a new related setting.     

Are we nice(r) to nice(r) people?—An experimental analysis

We experimentally investigate whether individuals can reliably detect cooperators (the nice(r) people) in an anonymous decision environment involving “connected games.” Participants can condition their choices in an asymmetric prisoners’ dilemma and a trust game on past individual (their partner’s donation share to a self-selected charity) and social (whether their partner belongs to a group with high or low average donations) information. Thus, the two measures of niceness are the individual donation share in the donation task, and the cooperativeness of one’s choice in the two games. We find that high donors achieve a higher-than-average expected payoff by cooperating predominantly with other high donors. Group affiliation proved to be irrelevant. Electronic Supplementary Material Supplementary material is available in the online version of this article at . JEL Classification C91, C72, D3     

Voice matters in a dictator game

We conducted a laboratory experiment to study the effects of communication in a dictator game, while maintaining subjects’ anonymity. In the experiment, the recipient has an opportunity to state a payoff-irrelevant request for his/her share before the dictator dictates his/her offer. We found that the independence hypothesis that voice does not matter is rejected. In particular, if the request is for less than half of the pie, the dictator’s offer increases as the recipient’s request increases. Additionally, there is no dictator who is other-regarding and, at the same time, does not react to the recipient’s request.

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Electronic Supplementary Material  The online version of this article () contains supplementary material, which is available to authorized users.      

Behavior in one-shot traveler’s dilemma games: model and experiments with advice

We report results of one-shot traveler’s dilemma game experiments to test the predictions of a model of introspection. The model describes a noisy out-of-equilibrium process by which players reach a decision of what to do in one-shot games. To test the robustness of the model and to compare it to other models of introspection without noise, we introduce non-binding advice. Advice has the effect of coordinating all players’ beliefs onto a common strategy. Experimentally, advice is implemented by asking subjects who participated in a repeated traveler’s dilemma game to recommend an action to subjects playing one-shot games with identical parameters. In contrast to observations, models based on best-response dynamics would predict lower claims than the advised. We show that our model’s predictions with and without advice are consistent with the data.      

Dissertation abstract: Experiments on fairness and reputation

The three dissertation essays investigate different aspects of reputation in games where fairness is an important consideration. The first essay studies the effects of reputation on indirect reciprocity in different dictator games. The first experiment places dictators in two environments where they can either give money to the paired player or take money away from them: in one treatment the paired player is a stranger and in the other treatment the dictator has information on the paired player’s reputation. Contrary to anecdotal evidence, the statistical tests show that the dictators’ behavior towards a stranger is not statistically significantly different from their behavior towards an individual with an established reputation. The findings arise because a high proportion of dictators acted purely in their own self interest in both treatments. The data also provides evidence that dictators are more generous when they know that their choices (but not their identities) will be revealed in the future. In the second experiment the dictators’ choices were restricted to only generous actions. In such environment the dictators sent more money on average to recipients with a reputation for being generous than to recipients without a reputation. The second essay explores the ways in which information about others’ actions affects one’s own behavior in a dictator game. The experimental design discriminates behaviorally between three possible effects of recipient’s within-game reputation on the dictator’s decision: reputation causing indirect reciprocity, social influence, and identification. The separation of motives helps to identify the mechanisms of social transmission of impulses towards selfish or generous behavior. The data analysis reveals that the reputation effects have a stronger impact on dictators’ actions than social influence and identification. In the third essay¹ we examine the reputation effects in a labor market setting by analyzing the influence of negative technological shocks on long run relationships between firms and workers. The positive correlation between wage and effort in static conditions has been demonstrated in many experimental studies and has been one of the prominent explanations for the existence of wage rigidity. We subject these findings to further tests in a non-stationary environment that better corresponds to outside-the-lab market conditions. We observe the positive correlation of wages and effort but do not find support for downward wage rigidity in our data. Once the shocks occur, firms lower the wages and relationships often break down. The workers who accept a lower wage respond with exerting a lower effort. JEL Classification C70, C91, D63, D64 ¹Co-authored with Ninghua Du. Dissertation Committee: Dissertation Advisor: James C. Cox Martin Dufwenberg, Price V. Fishback, Ronald L. Oaxaca     

On equilibrium existence in payoff secure games

We propose a single framework for studying the existence of approximate and exact pure strategy equilibria in payoff secure games. Central to the framework is the notion of a multivalued mapping with the local intersection property. By means of the Fan-Browder collective fixed point theorem, we first show an approximate equilibrium existence theorem that covers a number of known games. Then a short proof of Reny’s (Econometrica 67:1029–1056, 1999) equilibrium existence theorem is provided for payoff secure games with metrizable strategy spaces. We also give a simple proof of Reny’s theorem in its general form for metric games in an appendix for the sake of completeness.     

Dictator game giving: altruism or artefact?

Experimental dictator games have been used to explore unselfish behaviour. Evidence is presented here, however, that subjects’ generosity can be reversed by allowing them to take a partner’s money. Dictator game giving therefore does not reveal concern for consequences to others existing independently of the environment, as posited in rational choice theory. It may instead be an artefact of experimentation. Alternatively, evaluations of options depend on the composition of the choice set. Implications of these possibilities are explored for experimental methodology and charitable donations respectively. The data favour the artefact interpretation, suggesting that demand characteristics of experimental protocols merit investigation, and that economic analysis should not exclude context-specific social norms.

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Electronic Supplementary Material  The online version of this article () contains supplementary material, which is available to authorized users.      

Observability and overcoming coordination failure in organizations: An experimental study

Motivated by problems of coordination failure in organizations, we examine how overcoming coordination failure and maintaining coordination depend on the ability of individuals to observe others’ choices. Subjects’ payoffs depend on coordinating at high effort levels in a weak-link game. Treatments vary along two dimensions. First, subjects either start with low financial incentives for coordination, which typically leads to coordination failure, and then are switched to higher incentives or start with high incentives, which usually yield effective coordination, and are switched to low incentives. Second, as the key treatment variable, subjects either observe the effort levels chosen by all individuals in their experimental group (full feedback) or observe only the minimum effort (limited feedback). We find three primary results: (1) When starting from coordination failure the use of full feedback improves subjects’ ability to overcome coordination failure, (2) When starting with good coordination the use of full feedback has no effect on subjects’ ability to avoid slipping into coordination failure, and (3) History-dependence, defined as dependence of current effort levels on past incentives, is strengthened by the use of full feedback. Electronic Supplementary Material Supplementary material is available in the online version of this article at . JEL Classification C92, D23, J31, L23, M52     

Coordination between a sophisticated and fictitious player

Successful coordination is a common and important social problem. Achieving it relies on the players’ ability to accurately anticipate future choices from known information. Individuals may not only lack this cognitive ability, but differ in it. Fictitious Play is an adaptive behavior where a myopic best response to the historical play of an opponent is selected. I consider the interaction between a fictitious player and a sophisticated player in 2 × 2 coordination games. The existence of coordination, the selection of equilibria, and the payoffs generated are analyzed.     

A generalization of Condorcet’s Jury Theorem to weighted voting games with many small voters

We extend Condorcet’s Jury Theorem (Essai sur l’application de l’analyse à la probabilité des décisions rendues à la pluralité des voix. De l’imprimerie royale, 1785) to weighted voting games with voters of two kinds: a fixed (possibly empty) set of ‘major’ voters with fixed weights, and an ever-increasing number of ‘minor’ voters, whose total weight is also fixed, but where each individual’s weight becomes negligible. As our main result, we obtain the limiting probability that the jury will arrive at the correct decision as a function of the competence of the few major players. As in Condorcet’s result the quota q = 1/2 is found to play a prominent role. I wish to thank Maurice Koster, Moshé Machover, Guillermo Owen and two anonymous referees for helpful comments.     

Equilibrium existence and approximation of regular discontinuous games

Many conditions have been introduced to ensure equilibrium existence in games with discontinuous payoff functions. This paper introduces a new condition, called regularity, that is simple and easy to verify. Regularity requires that if there is a sequence of strategies converging to s* such that the players’ payoffs along the sequence converge to the best-reply payoffs at s*, then s* is an equilibrium. We show that regularity is implied both by Reny’s better-reply security and Simon and Zame’s endogenous sharing rule approach. This allows us to explore a link between these two distinct methods. Although regularity implies that the limits of e{\epsilon}-equilibria are equilibria, it is in general too weak for implying equilibrium existence. However, we are able to identify extra conditions that, together with regularity, are sufficient for equilibrium existence. In particular, we show how regularity allows the technique of approximating games both by payoff functions and space of strategies.     

Aggregative games and best-reply potentials

This paper introduces quasi-aggregative games and establishes conditions under which such games admit a best-reply potential. This implies existence of a pure strategy Nash equilibrium without any convexity or quasi-concavity assumptions. It also implies convergence of best-reply dynamics under some additional assumptions. Most of the existing literature’s aggregation concepts are special cases of quasi-aggregative games, and many new situations are allowed for. An example is payoff functions that depend on own strategies as well as a linear combination of the mean and the variance of players’ strategies.     

An evolutionary edge of knowing less (or: On the ‘curse’ of global information)

Consider a population of farmers who live around a lake. Each farmer engages in trade with his two adjacent neighbors. The trade is governed by a prisoner’s dilemma ‘rule of engagement.’ A farmer’s payoff is the sum of the payoffs from the two prisoner’s dilemma games played with his two neighbors. When a farmer dies, his son takes over. The son decides whether to cooperate or defect by considering the actions taken and the payoffs received by the most prosperous members of the group comprising his own father and a set of his father’s neighbors. The size of this set, which can vary, is termed the ‘span of information.’ It is shown that a larger span of information can be detrimental to the stable coexistence of cooperation and defection, and that in well-defined circumstances, a large span of information leads to an end of cooperation, whereas a small span does not. Conditions are outlined under which, when individuals’ optimization is based on the assessment of less information, the social outcome is better than when optimization is based on an assessment of, and a corresponding response to, more information.     

Equilibrium selection through incomplete information in coordination games: an experimental study

We perform an experiment on a pure coordination game with uncertainty about the payoffs. Our game is closely related to models that have been used in many macroeconomic and financial applications to solve problems of equilibrium indeterminacy. In our experiment, each subject receives a noisy signal about the true payoffs. This game (inspired by the “global” games of Carlsson and van Damme, Econometrica, 61, 989–1018, 1993) has a unique strategy profile that survives the iterative deletion of strictly dominated strategies (thus a unique Nash equilibrium). The equilibrium outcome coincides, on average, with the risk-dominant equilibrium outcome of the underlying coordination game. In the baseline game, the behavior of the subjects converges to the theoretical prediction after enough experience has been gained. The data (and the comments) suggest that this behavior can be explained by learning. To test this hypothesis, we use a different game with incomplete information, related to a complete information game where learning and prior experiments suggest a different behavior. Indeed, in the second treatment, the behavior did not converge to equilibrium within 50 periods in some of the sessions. We also run both games under complete information. The results are sufficiently similar between complete and incomplete information to suggest that risk-dominance is also an important part of the explanation.      

Stochastic strategy adjustment in coordination games

Summary. The paper explores a model of equilibrium selection in coordination games, where agents from an infinite population stochastically adjust their strategies to changes in their local environment. Instead of playing perturbed best-response, it is assumed that agents follow a rule of ‘switching to better strategies with higher probability’. This behavioral rule is related to bounded-rationality models of Rosenthal (1989) and Schlag (1998). Moreover, agents stay with their strategy in case they successfully coordinate with their local neighbors. Our main results show that both strict Nash equilibria of the coordination game correspond to invariant distributions of the process, hence evolution of play is not ergodic but instead depends on initial conditions. However, coordination on the risk-dominant equilibrium occurs with probability one whenever the initial fraction contains infinitely many agents, independent of the spatial distribution of these agents. Received: March 14, 2000; revised version: June 21, 2001     

Communication and coordination in the laboratory collective resistance game

This paper presents a laboratory collective resistance (CR) game to study how different forms of non-binding communication among responders can help coordinate their collective resistance against a leader who transgresses against them. Contrary to the predictions of analysis based on purely self-regarding preferences, we find that non-binding communication about intended resistance increases the incidence of no transgression even in the one-shot laboratory CR game. In particular, we find that the incidence of no transgression increases from 7 percent with no communication up to 25–37 percent depending on whether communication occurs before or after the leader’s transgression decision. Responders’ messages are different when the leaders can observe them, and the leaders use the observed messages to target specific responders for transgression.

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Electronic Supplementary Material The online version of this article () contains supplementary material, which is available to authorized users.      

In search of an evolutionary edge: trading with a few, more, or many

Consider a population of farmers who live around a lake. Each farmer engages in trade with his m adjacent neighbors, where m is termed the “span of interaction.” Trade is governed by a prisoner’s dilemma “rule of engagement.” A farmer’s payoff is the sum of the payoffs from the m prisoner’s dilemma games played with his ^m/₂^m/_2 neighbors to the left, and with his ^m/₂^m/_2 neighbors to the right. When a farmer dies, his son takes over. The son who adheres to his father’s span of interaction decides whether to cooperate or defect by considering the actions taken and the payoffs received by the most prosperous member of the group comprising his father and his father’s m trading partners. Under a conventional structure of payoffs, it is shown that a large span of interaction is detrimental to the long-run coexistence of cooperation and defection, and conditions are provided under which the social outcome associated with the expansion of trade when individuals trade with a few is better than that when they trade with many. Under the stipulated conditions it is shown, by means of a static comparative analysis of the steady state configurations of the farmer population, that an expansion of the market can be beneficial in one context, detrimental in another.     

A network experiment in continuous time: The influence of link costs

In recent work on non-cooperative network formation star-shaped networks play an important role. In a particular theoretical model of Bala and Goyal (2000) center-sponsored stars are the only strict Nash networks. In testing this theoretical model, Falk and Kosfeld (2003) do not find experimental evidence that players select the center-sponsored star. Based on a slight modification of Bala and Goyal’s model, we design a network formation experiment in which, depending on link costs, periphery-sponsored stars and the empty network are the only strict Nash networks. We observe that almost all groups not only reach a strict Nash network once but also switch the center player in periphery-sponsored stars several times. The main innovation in our experiment is to use a continuous time framework which we believe to be a more realistic setting to study behavior in network formation situations and which makes coordination on stars much easier than simultaneous strategy adaptation in discrete time. Electronic Supplementary Material Supplementary material is available in the online version of this article at http://dx.doi.org/10.1007/s10683-006-9125-1. JEL Classification C72 · C92 · D81