The folk theorem for irreducible stochastic games with imperfect public monitoring

This paper introduces stochastic games with imperfect public signals. It provides a sufficient condition for the folk theorem when the game is irreducible, thus generalizing the results of Dutta (1995) [5] and Fudenberg, Levine, and Maskin (1994) [9]. To do this, the paper extends the concept of self-generation (Abreu, Pearce, and Stacchetti, 1990 [1]) to “return generation,” which explicitly tracks actions and incentives until the next time the state returns to its current value, and asks that players not wish to deviate given the way their continuation payoffs from the time of this return depend on the public signals that have been observed.     

Restricted feedback in long term relationships

This paper studies long term relationships, modeled as repeated games, with restricted feedback. Players condition current play on summary statistics of past play rather than the entire history, as may be the case in online markets. Our state strategy equilibrium framework allows for arbitrary restrictions on strategies. We derive a recursive characterization for the set of equilibrium payoffs similar to that of Abreu, Pearce, and Stacchetti (1986, 1990) [2], [3] for perfect public equilibria and show that the set of equilibrium payoffs is the largest fixed point of a monotone operator. We use our characterization to derive necessary and sufficient conditions for efficient trade in a repeated product choice game where costumers condition their purchase decisions only on the last performance signal.     

Breaking the norm: An empirical investigation into the unraveling of good behavior

We present results from an artefactual field experiment conducted in rural Peru that considers whether observing non-reciprocal behavior influences an individual's decision to reciprocate. Specifically, we consider the behavior of second movers in a trust game, assessing whether their decision to reciprocate is influenced by the observed reciprocity of others. In documenting the impact of an external shock to observed reciprocity, this paper shows that small increases in non-reciprocal behavior result in an unraveling of the norm of reciprocity. Survey data is used to explore mechanisms by which this occurred. Results are not consistent with learning effects, suggesting that preferences may be changed by observing others deviating from a norm of reciprocity. These results suggest that investing in encouraging trustworthy behavior can have large benefits in situations where individuals are observing each other's behavior, such as may be the case in a new market institution.     

A theory of sequential reciprocity

Many experimental studies indicate that people are motivated by reciprocity. Rabin [Amer. Econ. Rev. 83 (1993) 1281] develops techniques for incorporating such concerns into game theory and economics. His theory is developed for normal form games, and he abstracts from information about the sequential structure of a strategic situation. We develop a theory of reciprocity for extensive games in which the sequential structure of a strategic situation is made explicit, and propose a new solution concept—sequential reciprocity equilibrium—for which we prove an equilibrium existence result. The model is applied in several examples, and it is shown that it captures very well the intuitive meaning of reciprocity as well as certain qualitative features of experimental evidence.     

Psychology at work

We compare three wage–effort psychological games. In the first game, the agent’s motivation hinges on a degree of altruism influenced by the surprise generated by the principal’s wage offer. The agent works harder when the wage is greater than expected and vice versa when the wage is smaller than expected. Consistent equilibrium beliefs oblige the principal to randomize in order to surprise the agent, which results in the principal being worse off than if she were dealing with an unemotional agent. We then consider an intention-based reciprocity model and a model of guilt aversion. We find that guilt aversion may potentially yield the best outcome for the principal.     

Mindreading and endogenous beliefs in games

We argue that a Bayesian explanation of strategic choices in games requires introducing a psychological theory of belief formation. We highlight that beliefs in epistemic game theory are derived from the actual choice of the players, and cannot therefore explain why Bayesian rational players should play the strategy they actually chose. We introduce the players’ capacity of mindreading in a game theoretical framework with the simulation theory, and characterise the beliefs that Bayes rational players could endogenously form in games. We show in particular that those beliefs need not be ratifiable, and therefore that rational players can form action-dependent beliefs.     

Bayesian learning and convergence to Nash equilibria without common priors

Summary. Consider an infinitely repeated game where each player is characterized by a “type” which may be unknown to the other players in the game. Suppose further that each player's belief about others is independent of that player's type. Impose an absolute continuity condition on the ex ante beliefs of players (weaker than mutual absolute continuity). Then any limit point of beliefs of players about the future of the game conditional on the past lies in the set of Nash or Subjective equilibria. Our assumption does not require common priors so is weaker than Jordan (1991); however our conclusion is weaker, we obtain convergence to subjective and not necessarily Nash equilibria. Our model is a generalization of the Kalai and Lehrer (1993) model. Our assumption is weaker than theirs. However, our conclusion is also weaker, and shows that limit points of beliefs, and not actual play, are subjective equilibria. Received: March 3, 1995; revised version: February 17, 1997     

Incorporating rigidity and commitment in the timing structure of macroeconomic games

This paper proposes a novel framework that generalizes the timing structure of games. Building on alternating move games and models of rational inattention, the players' actions may be rigid, i.e., infrequent. This rigidity in the timing of moves makes the game more dynamic and asynchronous, acting as a commitment mechanism. Therefore, it can enhance cooperation and often eliminate inefficient equilibrium outcomes present in the static (normal form) game. Interestingly, (i) this can happen even in a finite game (possibly as short as two periods) and (ii) without reputation building. Furthermore, (iii) the required degree of commitment may be under some circumstances arbitrarily low and under others infinitely high. Our main example comes from macroeconomics in which various rigidities have played a central role. Investigating the Kydland–Prescott–Barro–Gordon monetary policy game, we derive the necessary and sufficient degree of long-term policy commitment to low inflation under which the influential time-inconsistency problem is eliminated.     

Peer pressure,social spillovers,and reciprocity: an experimental analysis

This study examines whether reciprocity is affected by what others know and do. Two types of social effects are investigated within the framework of a modified investment game. On the one hand, we assess the role played by the awareness that own choices are observed by another trustee—i.e., peer pressure. On the other hand, we measure the interaction between trustees’ choices—i.e., social spillovers. We find that peer pressure fosters reciprocity and, to a lesser extent, so do social spillovers.     

Between-game rule learning in dissimilar symmetric normal-form games

Rule learning posits that decision makers, rather than choosing over actions, choose over behavioral rules with different levels of sophistication. Rules are reinforced over time based on their historically observed payoffs in a given game. Past works on rule learning have shown that when playing a single game over a number of rounds, players can learn to form sophisticated beliefs about others. Here we are interested in learning that occurs between games where the set of actions is not directly comparable from one game to the next. We study a sequence of ten thrice-played dissimilar games. Using experimental data, we find that our rule learning model captures the ability of players to learn to reason across games. However, this learning appears different from within-game rule learning as previously documented. The main adjustment in sophistication occurs by switching from non-belief-based strategies to belief-based strategies. The sophistication of the beliefs themselves increases only slightly over time.     

Proper belief revision and equilibrium in dynamic games

We present a theory of rationality in dynamic games in which players, during the course of the game, may revise their beliefs about the opponents’ utility functions. The theory is based upon the following three principles: (1) the players’ initial beliefs about the opponents’ utilities should agree on some profile u of utility functions, (2) every player should believe, at each of his information sets, that his opponents are carrying out optimal strategies and (3) a player at information set h should not change his belief about an opponent's ranking of strategies a and b if both a and b could have led to h. Scenarios with these properties are called preference conjecture equilibria for the profile u of utility functions. We show that every normal form proper equilibrium for u induces a preference conjecture equilibrium for u, thus implying existence of preference conjecture equilibrium.     

Efficient budget balancing cartel equilibria with imperfect monitoring

Summary We modify the infinitely repeated Cournot game with imperfect monitoring of Green and Porter (1984) and Abreu, Pearce and Stacchetti (1986) to include heterogenous products and the possibility of balanced budget side payments (Holmström 1982). It is shown that a transfer mechanism which induces the efficient outcome exists under a reasonable technical assumption in contrast to the preceding authors. Intuitively, the existence of an observable random price vector rather than a single price makes it possible to identify likely defectors, eliminating the need for collective punishments.This paper is based on the last chapter of Demougin's dissertation at the University of Western Ontario.     

Imitation and the Incentive to Contribute Early in a Sequential Public Good Game

Whether motivated by reciprocity or conformity, imitation is common in public good contexts. We consider the incentive for an agent to contribute to a public good if he expects imitation from others. Using a sequential public good game with exogenous ordering, we show that agents early enough in the sequence who believe imitation to be sufficiently likely would want to contribute. By contributing, they expect total contributions to increase significantly. We also show that preferences determine how early an agent need be, that the observed share of imitators in experiments is sufficiently high to warrant contribution and that an increase in group size reduces the incentive to contribute.     

Kernel-based type spaces

Type space is of fundamental importance in epistemic game theory. This paper shows how to build type space if players approach the game in a way advocated by Bernheim's justification procedure. If an agent fixes a strategy profile of her opponents and ponders which of their beliefs about her set of strategies make this profile optimal, such an analysis is represented by kernels and yields disintegrable beliefs. Our construction requires that underlying space is Polish.     

A theory of reciprocity

People are reciprocal if they reward kind actions and punish unkind ones. In this paper we present a formal theory of reciprocity. It takes into account that people evaluate the kindness of an action not only by its consequences but also by its underlying intention. The theory is in line with the relevant stylized facts of a wide range of experimental games, such as the ultimatum game, the gift-exchange game, a reduced best-shot game, the dictator game, the prisoner's dilemma, and public goods games. Furthermore, it predicts that identical consequences trigger different reciprocal responses in different environments. Finally, the theory explains why outcomes tend to be fair in bilateral interactions whereas extremely unfair distributions may arise in competitive markets.     

Does truth win when teams reason strategically?

We study behavior in the race game with the aim of assessing whether teams can create synergies. The race game has the advantage that the optimal strategy depends neither on beliefs about other players nor on distributional or efficiency concerns. Our results reveal that teams not only outperform individuals but that they can also beat the “truth-wins” benchmark. In particular, varying the length of the race game we find that the team advantage increases with the complexity of the game.     

Motives for sharing in social networks

What motivates people in rural villages to share? We first elicit a baseline level of sharing using a standard, anonymous dictator game. Then using variants of the dictator game that allow for either revealing the dictator's identity or allowing the dictator to choose the recipient, we attribute variation in sharing to three different motives. The first of these, directed altruism, is related to preferences, while the remaining two are incentive-related (sanctions and reciprocity). We observe high average levels of sharing in our baseline treatment, while variation across individuals depends importantly on the incentive-related motives. Finally, variation in measured reciprocity within the experiment predicts observed ‘real-world’ gift-giving, while other motives measured in the experiment do not predict behavior outside the experiment.     

How to identify trust and reciprocity

This paper uses a three-games (or triadic) design to identify trusting and reciprocating behavior. A large literature on single-game trust and reciprocity experiments is based on the implicit assumption that subjects do not have altruistic or inequality-averse other-regarding preferences. Such experimental designs test compound hypotheses that include the hypothesis that other-regarding preferences do not affect behavior. In contrast, experiments with the triadic design do discriminate between transfers resulting from trust or reciprocity and transfers resulting from other-regarding preferences that are not conditional on the behavior of others. Decomposing trust from altruism and reciprocity from altruism or inequality aversion is critical to obtaining empirical information that can guide the process of constructing models that can increase the empirical validity of game theory.     

Network Interaction with Material and Relational Goods: An Exploratory Simulation

We model interaction within the members of a work group, who carry on different projects and can choose to cooperate with each other or to exploit the work of the others. We focus our attention on the dynamic interdependence between their material performance and the structure of the relations among them. In particular, we assume that their satisfaction depends both upon the material outcome of their projects and upon the quality of their reciprocal relations (so that these two aspects are seen, to a certain extent, as psychological substitutes). As dissatisfaction may generate the disruption of social links, the consequent failure of the projects is a possible outcome. In turn, satisfactory relations and stable cooperation may loop positively together. Our model merges game theory and network representation of the relations among actors, thus opening a route of research that, to our knowledge, has not yet been explored.     

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.