We use simple learning models to track the behavior observed in experiments concerning three extensive form games with similar perfect equilibria. In only two of the games does observed behavior approach the perfect equilibrium as players gain experience. We examine a family of learning models which possess some of the robust properties of learning noted in the psychology literature. The intermediate term predictions of these models track well the observed behavior in all three games, even though the models considered differ in their very long term predictions. We argue that for predicting observed behavior the intermediate term predictions of dynamic learning models may be even more important than their asymptotic properties.

Demands in the Ultimatum Game in its traditional form with one proposer and one responder are compared with demands in an Ultimatum Game with responder competition. In this modified form one proposer faces three responders who can accept or reject the split of the pie. Initial demands in both ultimatum games are quite similar, however in the course of the experiment, demands in the ultimatum game with responder competition are significantly higher than in the traditional case with repeated random matching. Individual round-to-round changes of choices that are consistent with directional learning are the driving forces behind the differences between the two learning curves and cannot be tracked by an adjustment process in response to accumulated reinforcements. The importance of combining reinforcement and directional learning is addressed. Moreover, learning transfer between the two ultimatum games is analyzed.

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Models have been put forward recently that seem to be successful in explaining apparently anomalous experimental results in the Ultimatum Game, where responders reject positive offers. While imparting fixed preference orders to fully rational agents, these models depart from traditional models by assuming preferences that take account not only of the material payoff to oneself, but also of that which is given to others. However, they leave open the question of how is an agent’s economic survival helped by a preference order that advises him to leave money on the table. Our answer is that, indeed, doing so does not help. But that the same envious preference order that ill advises in some circumstances to reject an “insultingly ” small offer, advises well in other circumstances, when it helps the same agent to overcome his risk-aversion and to offer a risky, tough offer that yields him a higher expected dollar gain. We show the existence of population distributions where the two effects exactly balance out across different preference types. These distributions are asymptotically stable, stationary, and inefficient, in which different preferences are represented, and where, as commonly observed in an Ultimatum Game, positive offers are made, of which some are rejected with positive probability. Our theory yields new testable hypotheses.

Abstract. This paper develops a theory of revealed preferences over one’s own and others’monetary payo¤s. We introduce “more altruistic than”(MAT), a partial ordering over preferences, and interpret it with known parametric models. We also introduce and illustrate “more generous than ” (MGT), a partial ordering over opportunity sets. Several recent discussions of altruism focus on two player extensive form games of complete information in which the …rst mover (FM) chooses a more or less generous opportunity set for the second mover (SM). Here reciprocity can be formalized as the assertion that an MGT choice by the FM will elicit MAT preferences in the SM and, furthermore, that the e¤ect on preferences is stronger for acts of commision than acts of ommision by FM. We state and prove propositions on the observable consequences of these assertions. Then we test those propositions using existing data from investment games with dictator controls and Stackelberg games and new data from Stackelberg mini-games. The test results provide support for the theory of revealed altruism.

This paper investigates the implications of normal- and extensive-form quantal response equilibrium (QRE) models (McKelvey and Palfrey, 1995, Games and Economic Behavior; 1998, Experimental Economics) in the ultimatum bargaining game, assuming that players maximize expected monetary payo ¯ s. It is shown that normal-form QRE can select a non-sequential equilibrium, and that the selection depends crucially on the noise structure. The normal-form QRE describes the main qualitative features of experimental subjects ' behavior better than extensiveform QRE even in experiments with extensive-form games. Journal of Economic Literature Classi ± cation Number: C79, C92

Human players in our laboratory experiment received flow payoffs over 120 seconds each period from a standard Hawk-Dove bimatrix game played in continuous time. Play converged closely to the symmetric mixed Nash equilibrium under a one-population matching protocol. When the same players were matched in a two-population protocol with the same bimatrix, they showed clear movement towards an asymmetric (and very inequitable) pure Nash equilibrium of the game. These findings support distinctive predictions of evolutionary game theory. Keywords: Evolutionary dynamics, Hawk-Dove game, game theory, laboratory experiment, continuous time game.

In this paper we propose a model for human learning and decision making in environments of repeated Cliff-Edge (CE) interactions. In CE environments, which include common daily interactions, such as sealed-bid auctions and the Ultimatum Game (UG), the probability of success decreases monotonically as the expected reward increases. Thus, CE environments are characterized by an underlying conflict between the strive to maximize profits and the fear of causing the entire deal to fall through. We focus on the behavior of people who repeatedly compete in one-shot CE interactions, with a different opponent in each interaction. Our model, which is based upon the Deviated Virtual Reinforcement Learning (DVRL) algorithm, integrates the Learning Direction Theory with the Reinforcement Learning algorithm. We also examined several other models, using an innovative methodology in which the decision dynamics of the models were compared with the empirical decision patterns of individuals during their interactions. An analysis of human behavior in auctions and in the UG reveals that our model fits the decision patterns of far more subjects than any other model.

Strong reciprocity refers to the willingness to sacrifice one's own material selfinterest to punish others for opportunistic actions. This propensity provides a decentralized mechanism for the enforcement of social norms, but its extent and persistence poses a theoretical puzzle. Since opportunistic individuals choose optimally to comply with or violate norms based on the likelihood and severity of sanctioning they anticipate, such individuals will always outperform reciprocators within any group. The presence of reciprocators in a group can, however, alter the behavior of opportunists in such a manner as to benefit all members of the group (including reciprocators). We show that under these circumstances, reciprocators can invade a population of opportunists when groups dissolve and are formed anew according to a process of purely random (non-assortative) matching. Furthermore, even when these conditions are not satisfied (so that an opportunistic population is stable) there may exist additional stable population states in which reciprocators are present.

Abstract: This paper reports an experiment comparing three stag hunt games that have the same best-response correspondence and the same expected payoff from the mixed equilibrium, but differ in the incentive to play a best response rather than an inferior response. In each game, risk dominance conflicts with payoff dominance and selects an inefficient pure strategy equilibrium. We find statistically and economically significant evidence that the differences in the incentive to optimize help explain observed behavior.