Rent, risk, and replication: Preference adaptation in winner-take-all markets

We study the evolution of an economy where agents who are heterogeneous with respect to risk attitudes can either earn a certain income or enter a risky rent-seeking contest. We assume that agents behave rationally given their preferences, but that the population distribution of preferences evolves over time in response to material payoffs. We show that, in particular, initial distributions with full support converge to stationary states where all types are still present. Although rents are perfectly dissipated in material terms at a steady state, efficiency is greater than if everybody had been risk neutral, since risk lovers specialize in rent seeking.     

How to talk to multiple audiences

Stochastic fictitious play (SFP) assumes that agents do not try to influence the future play of their current opponents, an assumption that is justified by appeal to a setting with a large population of players who are randomly matched to play the game. However, the dynamics of SFP have only been analyzed in models where all agents in a player role have the same beliefs. We analyze the dynamics of SFP in settings where there is a population of agents who observe only outcomes in their own matches and thus have heterogeneous beliefs. We provide conditions that ensure that the system converges to a state with homogeneous beliefs, and that its asymptotic behavior is the same as with a single representative agent in each player role.     

How sensitive are bargaining outcomes to changes in disagreement payoffs?

We use a human-subjects experiment to investigate how bargaining outcomes are affected by changes in bargainers’ disagreement payoffs. Subjects bargain against changing opponents, with randomly drawn asymmetric disagreement outcomes that vary over plays of the game, and with complete information about disagreement payoffs and the cake size. We find that subjects only respond about half as much as theoretically predicted to changes in their own disagreement payoff and to changes in their opponent’s disagreement payoff. This effect is observed in a standard Nash demand game and a related unstructured bargaining game, in both early and late rounds, and is robust to moderate changes in stake sizes. We show theoretically that standard models of expected utility maximisation are unable to account for this under-responsiveness, even when generalised to allow for risk aversion. We also show that quantal-response equilibrium has, at best, mixed success in characterising our results. However, a simple model of other-regarding preferences can explain our main results.     

Long-term relationships as safeguards

We analyze a repeated prisoners’ dilemma game played in a community setting with heterogeneous types. The setting is such that individuals choose whether to continue interacting with their present partner, or separate and seek a new partner. Players’ types are not directly observed, but may be imperfectly inferred from observed behavior. We focus on a class of equilibria that satisfy zero tolerance and fresh start. We find that the punishment for defecting and the reward for cooperating are driven by the formation and the dissolution of long-term, high-paying relationships: an individual who defects, aborts a long-term relationship that he is in, or that he might have entered into, is thrown into short-term interactions with individuals who are likely to defect and, consequently, receives low payoffs. On the flip side, an individual who cooperates, enters into or prolongs a long-term interaction with a partner who cooperates and, consequently, receives high payoffs.     

Social dilemmas,time preferences and technology adoption in a commons problem

Agents interacting on a body of water choose between technologies to catch fish. One is harmless to the resource, as it allows full recovery; the other yields high immediate catches, but low(er) future catches. Strategic interaction in one ‘objective’ resource game may induce several ‘subjective’ games in the class of social dilemmas. Which unique ‘subjective’ game is actually played depends crucially on how the agents discount their future payoffs. We examine equilibrium behavior and its consequences on sustainability of the common-pool resource system under exponential and hyperbolic discounting. A sufficient degree of patience on behalf of the agents may lead to equilibrium behavior averting exhaustion of the resource, though full restraint (both agents choosing the ecologically or environmentally sound technology) is not necessarily achieved. Furthermore, if the degree of patience between agents is sufficiently dissimilar, the more patient is exploited by the less patient one in equilibrium. We demonstrate the generalizability of our approach developed throughout the paper. We provide recommendations to reduce the enormous complexity surrounding the general cases.     

On the convergence of reinforcement learning

This paper examines the convergence of payoffs and strategies in Erev and Roth's model of reinforcement learning. When all players use this rule it eliminates iteratively dominated strategies and in two-person constant-sum games average payoffs converge to the value of the game. Strategies converge in constant-sum games with unique equilibria if they are pure or if they are mixed and the game is 2×2. The long-run behaviour of the learning rule is governed by equations related to Maynard Smith's version of the replicator dynamic. Properties of the learning rule against general opponents are also studied.     

Games superheroes play: Teaching game theory with comic book favorites

The valuable insights of game theory sometimes remain out of reach for students who are overwhelmed by the subject's complexity. Comic book applications of game theory, with superheroes as players, can facilitate enthusiasm and classroom interaction to enhance the learning of game theory. Drawing from content in superhero movies and books, the authors construct games to illustrate pure-strategy Nash equilibrium, Bayes-Nash equilibrium, mixed strategies, sub-game perfection, and perfect Bayesian equilibrium. To help instructors build students' skills in finding and interpreting game solutions, they translate comic book scenarios into specific game forms; however, not all scenarios are obvious so they suggest instructors help students develop their own game-theoretic judgments to determine what game forms, payoffs, and solution concepts might be appropriate for understanding a situation.     

Spatial Evolution of Social Norms in a Common-Pool Resource Game

We study the conditions for the emergence of cooperation in a spatial common-pool resource (CPR) game. We consider three types of agents: cooperators, defectors and enforcers. The role of enforcers is to punish defectors for overharvesting the resource. Agents are located on a circle and they only observe the actions of their two nearest neighbors. Their payoffs are determined by both local and global interactions and they modify their actions by imitating the strategy in their neighborhood with the highest average payoffs on average. Using theoretical and numerical analysis, we find a large diversity of equilibria to be the outcome of the game. In particular, we find conditions for the occurrence of equilibria in which the three strategies coexist. We also derive the stability of these equilibria. Finally, we show that introducing resource dynamics in the system favors the occurrence of cooperative equilibria.      

Reciprocity, Trust, and Payoff Privacy in Extensive Form Bargaining

We report decision making in two-person extensive form game trees, using six treatments that vary matching protocol, payoffs, and payoff information. Our objective is to examine game theoretic hypotheses of decision making based on dominance and backward induction in comparison with the culturally or biologically derived hypothesis that reciprocity supports more cooperation than predicted by game theory. We find strong support for cooperation under complete information, even in single-play treatments and in games of trust, unreinforced by the prospect of punishment for defection from reciprocity. Only under private information do we observe strong support for noncooperative game theory.Journal of Economic LiteratureClassification Numbers: C78, C92.     

Experimental study of repeated team-games

We report an experiment in which the Intergroup Prisoner's Dilemma (IPD) game was contrasted with a structurally identical (single-group) Prisoner's Dilemma (PD). The games were played repeatedly for 40 rounds. We found that subjects were initially more likely to cooperate in the IPD game than in the PD game. However, cooperation rates decreased as the game progressed and, as a result, the differences between the two games disappeared. This pattern is consistent with the hypothesis that subjects learn the structure of the game and adapt their behavior accordingly. Computer simulations based on a simple learning model by Roth and Erev (Learning in extensive-form games: Experimental data and simple dynamic models in the intermediate term, Games and Economic Behavior 8, 164–212, 1995) support this interpretation.     

Coexistence of Strategies and Culturally-Specific Common Knowledge: An Evolutionary Analysis

We analyze social dynamics in a continuous population where randomly matched individuals have to choose between two pure strategies only ('cooperate' (C) and 'not cooperate' (NC)). Individual payoffs associated with the possible outcomes of each interaction may differ across groups, depending on the specific social and cultural context to which each agent belongs. In particular, it is assumed that three sub-populations are initially present, 'framing' the game according to the prisoner's dilemma (PD), assurance game (AG) and other regarding (OR) payoff configurations, respectively. In other words, we assume that common knowledge about the payoffs of the game is 'culturally-specific'. In this context, we examine both the adoption process of strategies C and NC within each sub-population and the diffusion process of 'types' (PD, AG and OR) within the overall community. On the basis of an evolutionary game-theoretic approach, the paper focuses on the problem of coexistence of PD, AG and OR groups as well as of 'nice' (C) and 'mean' (NC) strategies. We show that coexistence between C and NC is possible in the heterogeneous community under examination, even if it is ruled out in homogeneous communities where only one of the three types is present. This revised version was published online in July 2006 with corrections to the Cover Date.     

The optimal degree of cooperation in the repeated Prisoners' Dilemma with side payments

In the infinitely repeated Prisoners' Dilemma with side payments, we characterize the Pareto frontier of the set of subgame perfect equilibrium payoffs for all possible combinations of discount factors. Play paths implementing Pareto dominant equilibrium payoffs are uniquely determined in all but the first period. Full cooperation does not necessarily implement these payoffs even when it maximizes total stage game payoffs. Rather, when the difference in players' discount factors is sufficiently large, Pareto dominant equilibrium payoffs are implemented by partial cooperation supported by repeated payments from the impatient to the patient player. When both players are sufficiently patient, such payoffs, while implemented via full cooperation, are supported by repeated payments from the impatient to the patient player. We characterize conditions under which public randomization has no impact on the Pareto dominant equilibrium payoffs and conditions under which such payoffs are robust to renegotiation.     

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.     

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.     

Stable games and their dynamics

We study a class of population games called stable games. These games are characterized by self-defeating externalities: when agents revise their strategies, the improvements in the payoffs of strategies to which revising agents are switching are always exceeded by the improvements in the payoffs of strategies which revising agents are abandoning. We prove that the set of Nash equilibria of a stable game is globally asymptotically stable under a wide range of evolutionary dynamics. Convergence results for stable games are not as general as those for potential games: in addition to monotonicity of the dynamics, integrability of the agents' revision protocols plays a key role.     

Noisy Contagion Without Mutation

In a local interaction game agents play an identical stage game against their neighbours over time. For nearest neighbour interaction, it is established that, starting from a random initial configuration in which each agent has a positive probability of playing the risk dominant strategy, a sufficiently large population coordinates in the long-run on the risk dominant equilibrium almost surely. Our result improves on Blume (1995), Ellison (2000), and Morris (2000) by showing that the risk dominant equilibrium spreads to the entire population in a two dimensional lattice and without the help of mutation, as long as there is some randomness in the initial configuration.     

Economic Games Among the Amazonian Tsimane: Exploring the Roles of Market Access, Costs of Giving, and Cooperation on Pro-Social Game Behavior

This paper reports the results of the Ultimatum Game (UG), Dictator Game (DG) and Public Goods Game (PGG) played among the Tsimane, a group of forager-horticulturalists living in the Bolivian Amazon. Game results differ significantly from those commonly reported among modern, westernized populations. Without a long history of anonymous interactions, it is highly suspect whether the Tsimane or other traditional populations play economic games under assumptions of anonymity and one-shot exposure. Employing a behavioral ecology framework, I test predictions that differential market exposure, costs of giving, and experience with cooperation can help explain much of the variance in game outcomes. While these factors sometimes act as important predictors of game behavior, the most significant predictor is village membership. Implications for understanding the role of markets, frequent interaction with strangers, and payoffs to cooperation in daily life can help us better understand cross-cultural variation in pro-social behavior.     

Does the reliability of institutions affect public good contributions? Evidence from a laboratory experiment

Reliable institutions, i.e., institutions that live up to the norms that agents expect them to keep foster cooperative behavior. We experimentally confirm this hypothesis in a public goods game with a salient norm that cooperation was socially demanded and corruption ought not to occur. When nevertheless corruption attempts came up, groups that were told that ‘the system’ had fended off the attempts made considerably higher contributions to the public good than groups that learned that attempts only did not affect their payoffs or that were not exposed to corruption at all.     

Learning aspiration in repeated games

We study infinitely repeated symmetric 2×2 games played by bounded rational agents who follow a simple rule of thumb: each agent continues to play the same action if the current payoff exceeds the average of the past payoffs, and switches to the other action with a positive probability otherwise. By applying the stochastic approximation technique, we characterize the asymptotic outcomes for all 2×2 games. In the prisoners’ dilemma game, for example, the players cooperate in the limit if and only if the gain from defecting against cooperation is “modest.”     

Endogenous transfers in the Prisoner's Dilemma game: An experimental test of cooperation and coordination

We test a two-stage compensation mechanism for promoting cooperation in Prisoner's Dilemma games. Players first simultaneously choose binding non-negative amounts to pay their counterparts for cooperating, and then play the induced game knowing these amounts. In our games, all payment pairs consistent with mutual cooperation in subgame-perfect equilibrium transform these games into coordination games, with both mutual cooperation and mutual defection as Nash equilibria in the second stage. When endogenous transfer payments are not permitted, cooperation is much less likely. Mutual cooperation is most likely when the (sufficient) payments are identical, and it is also substantially more likely with payment pairs that bring the mutual-cooperation payoffs closer together. Both the Fehr–Schmidt and Charness–Rabin models predict that transfers that make final payoffs closer are preferred; however, they do not explain why equal transfers are particularly effective. Transfers are also effective in sustaining cooperation even when they are imposed and not chosen.