The power of ESS: An experimental study

Abstract. Our experimental design mimics a traditional evolutionary game framework where players are matched pairwise to play a symmetric 33 bimatrix game that has two Nash equilibria. One equilibrium is an evolutionary stable state, or ESS; the other is an equilibrium in dominated strategies. Our primary experimental result is the observation that the ESS becomes extremely attractive when subjects have minimal information about the payoff functions, although the dominated equilibrium assures the highest equilibrium payoff. The attractiveness of the ESS is only moderate when players are completely informed about the 33 payoff matrix. Correspondence to: S.K. Berninghaus     

Optimal privatization in a mixed duopoly with consistent conjectures

We show that partially privatizing a public firm alters underlying conjectures, in turn, changing the optimal degree of privatization. The consistent conjectures equilibrium (CCE) generates substantially greater optimal privatization than does any conjecture shared between the firms including the standard Cournot–Nash equilibrium (CNE). Yet, when the private rival is foreign, the CCE generates substantially less privatization than the CNE. The optimal extent of privatization with a domestic rival exceeds that with a foreign rival in the CCE as well as in the CNE.     

Learning with perfect information

For extensive form games with perfect information, consider a learning process in which, at any iteration, each player unilaterally deviates to a best response to his current conjectures of others' strategies; and then updates his conjectures in accordance with the induced play of the game. We show that, for generic payoffs, the outcome of the game becomes stationary, and is consistent with Nash equilibrium. In general, if payoffs have ties or if players observe more of each others' strategies than is revealed by plays of the game, the same result holds provided a rationality constraint is imposed on unilateral deviations: no player changes his moves in subgames that he deems unreachable, unless he stands to improve his payoff there. Moreover, with this constraint, the sequence of strategies and conjectures also becomes stationary, and yields a self-confirming equilibrium.     

Evolutionarily stable conjectures and other regarding preferences in duopoly games

We study the evolutionary selection of conjectures in duopoly games when players have other regarding preferences, i.e. preferences over payoff distributions. In both the Cournot and Bertrand duopoly games, the consistent conjectures are independent of other regarding preferences. Both duopoly games have evolutionarily stable conjectures that depend on other regarding preferences but that do not coincide with the consistent conjectures. For increasingly spiteful preferences, the evolutionarily stable conjectures implicate low quantities in the Cournot game and high prices in the Bertrand game, whereas the inverse relationships hold for the consistent conjectures. We discuss our findings in the context of ultimate and proximate causation.     

Finitely repeated games with monitoring options

We study finitely repeated games where players can decide whether to monitor the other players? actions or not every period. Monitoring is assumed to be costless and private. We compare our model with the standard one where the players automatically monitor each other. Since monitoring other players never hurts, any equilibrium payoff vector of a standard finitely repeated game is an equilibrium payoff vector of the same game with monitoring options. We show that some finitely repeated games with monitoring options have sequential equilibrium outcomes which cannot be sustained under the standard model, even if the stage game has a unique Nash equilibrium. We also present sufficient conditions for a folk theorem, when the players have a long horizon.     

An experimental study of information and mixed-strategy play in the three-person matching-pennies game

Summary. Recent experiments on mixed-strategy play in experimental games reject the hypothesis that subjects play a mixed strategy even when that strategy is the unique Nash equilibrium prediction. However, in a three-person matching-pennies game played with perfect monitoring and complete payoff information, we cannot reject the hypothesis that subjects play the mixed-strategy Nash equilibrium. Given this support for mixed-strategy play, we then consider two qualitatively different learning theories (sophisticated Bayesian and naive Bayesian) which predict that the amount of information given to subjects will determine whether they can learn to play the predicted mixed strategies. We reject the hypothesis that subjects play the symmetric mixed-strategy Nash equilibrium when they do not have complete payoff information. This finding suggests that players did not use sophisticated Bayesian learning to reach the mixed-strategy Nash equilibrium. Received: August 9, 1996; revised version: October 21, 1998     

Consistent conjectures and voluntary contributions to public goods: why the conventional theory does not work

The conventional theory of voluntary contributions to public goods assumes that each person maximises his utility taking other people's behaviour as given (Nash conjectures). It is now coming to be recognised that this theory is inconsistent with observed behaviour. This paper argues that the problems of the theory are compounded — and not, as some economists have argued, resolved — by relaxing the assumption of Nash conjectures. If individuals' conjectures are consistent, each person will hold the (correct) belief that other people's contributions are inversely related to his own. Under most reasonable assumptions, equilibrium is a state in which no-one contributes anything.     

Voluntary Contributions to Multiple Public Projects

The problem of financing a set of discrete public goods (facilities, projects) by private contributions is studied. The corresponding cooperative game, the realization game , is shown to be convex. For the noncooperative setting we study a realization scheme that induces a strategic game. This contribution game is shown to be a generalized ordinal potential game ; a best–response in the contribution game implies a best response in a coordination game in which the payoff to all players is the utilitarian collective welfare function, i.e., the sum of the utility functions of the players. Strategy profiles maximizing utilitarian welfare are strong Nash equilibria of the contribution game. Each strong Nash equilibrium corresponds in a natural way with a core element of the realization game, and vice versa. Moreover, each strong Nash equilibrium is coalitional proof.     

Evolutionary Stability in the Finitely Repeated Prisoner 's Dilemma Game

The dynamic evolutionary stability of mutual defection is proven for the repeated prisoner 's dilemma game where payoffs are cumulative and the number of repetitions is known. This agrees with the classical result that the only Nash equilibrium outcome is to defect at all stages of this repeated game. Moreover, it is shown that, for any initial polymorphic population, the evolutionary dynamic converges to a unique Nash equilibrium strategy that depends on the original polymorphism. Both these results confirm earlier conjectures concerning the application of evolutionary game theory to the repeated prisoner 's dilemma.Journal of Economic LiteratureClassification Numbers: C70, C72.     

Heterogeneous quantal response equilibrium and cognitive hierarchies

We explore an equilibrium model of games where behavior is given by logit response functions, but payoff responsiveness and beliefs about others' responsiveness are heterogeneous. We study two substantively different ways of extending quantal response equilibrium (QRE) to this setting: (1) Heterogeneus QRE, where players share identical correct beliefs about the distribution of payoff responsiveness; and (2) Truncated QRE, where players have downward looking beliefs, systematically underestimating others' responsiveness. We show that the cognitive hierarchy model is a special case of Truncated QRE. We conduct experiments designed to differentiate these approaches. We find significant evidence of payoff responsive stochastic choice, and of heterogeneity and downward looking beliefs in some games.     

Testing the TASP: An experimental investigation of learning in games with unstable equilibria

We report experiments studying mixed strategy Nash equilibria that are theoretically stable or unstable under learning. The Time Average Shapley Polygon (TASP) predicts behavior in the unstable case. We study two versions of Rock-Paper-Scissors that include a fourth strategy, Dumb. The unique Nash equilibrium is identical in the two games, but the predicted frequency of Dumb is much higher in the game where the NE is stable. Consistent with TASP, the observed frequency of Dumb is lower and play is further from Nash in the high payoff unstable treatment. However, Dumb is played too frequently in all treatments.     

Beliefs and Public Good Provision with Anonymous Contributors

We analyze a static game of public good contributions where finitely many anonymous players have heterogeneous preferences about the public good and heterogeneous beliefs about the distribution of preferences. In the unique symmetric equilibrium, the only individuals who make positive contributions are those who most value the public good and who are also the most pessimistic; that is, according to their beliefs, the proportion of players who most like the public good is smaller than it would be according to any other possible belief. We predict whether the aggregate contribution is larger or smaller than it would be in an analogous game with complete information and heterogeneous preferences, by comparing the beliefs of contributors with the true distribution of preferences. A trade‐off between preferences and beliefs arises if there is no individual who simultaneously has the highest preference type and the most pessimistic belief. In this case, there is a symmetric equilibrium, and multiple symmetric equilibria occur only if there are more than two preference types.     

Institutions,assortative matching and cultural evolution

This paper studies the role of institution in a Darwinian evolutionary process of cultural selection. The primary function of an institution is to determine how citizens in a society are matched pairwisely to interact. We examine three different types of institutions: utilitarian, egalitarian, and Nash. Two cultural types stand out in the long run through the evolutionary process. The opportunistic cultural type maximizes individual payoff against another opportunistic cultural type, while the civic-minded cultural type maximizes the total payoff of a pair. We show that the structure of the underlying interactions among citizens plays a critical role. On the one hand, the evolutionary stability of the civic-minded cultural type requires supermodularity of the citizens’ payoff function under the utilitarian institution and log-supermodularity in addition under the Nash institution. On the other hand, the evolutionary stability of the opportunistic cultural type requires submodularity of the citizens’ payoff function under the utilitarian institution and log-submodularity under the Nash institution. Neither type’s evolutionary stability is guaranteed under the egalitarian institution.     

N-PERSON NASH BNARGAINING WITH VARIABEL THREATS*

We consider two models of n-person bargaining problems with the endogenous determination of disagreement points. In the first model, which is a direct extension of Nash's variable threat bargaining model, the disagreement point is determined as an equilibrium threat point. In the second model, the disagreement point is given as a Nash equilibrium of the underlying noncooperative game. These models are formulated as extensive games, and axiomatizations of solutions are given for both models. It is argued that for games with more than two players, the first bargaining model does not preserve some important properties valid for two-person games, e.g., the uniqueness of equilibrium payoff vector. We also show that when the number of players is large, any equilibrium threat point becomes approximately a Nash equilibrium in the underlying noncooperative game, and vice versa. This result suggests that the difference between the two models becomes less significant when the number of players is large.     

Participation and the provision of discrete public goods: a strategic analysis

This paper considers the Nash equilibria to a game where a discrete public good is to be provided. Each individual may participate by making a fixed contribution. If a sufficient number of contributions are made, the good is provided. Otherwise, the good is not provided. One variant of the rules allows for contributions to be refunded when the good is not provided. For pure strategies, we find that the Nash equilibria with a refund are a superset of those without a refund. For both rules, the efficient number of players contributing is an equilibrium. For mixed strategies, to every equilibrium without a refund there is a corresponding equilibrium with a refund with a higher number of expected contributors. Mixed strategy equilibria ‘disappear’ as the number of players grows large. Some results reported in the experimental literature are discussed in light of these theoretical results.     

Repeated games with present-biased preferences

We study infinitely repeated games with perfect monitoring, where players have β-δ preferences. We compute the continuation payoff set using recursive techniques and then characterize equilibrium payoffs. We then explore the cost of the present-time bias, producing comparative statics. Unless the minimax outcome is a Nash equilibrium of the stage game, the equilibrium payoff set is not monotonic in β or δ. Finally, we show how the equilibrium payoff set is contained in that of a repeated game with smaller discount factor.     

SEQUENTIAL TWO‐PLAYER GAMES WITH AMBIGUITY*

If players' beliefs are strictly nonadditive, the Dempster–Shafer updating rule can be used to define beliefs off the equilibrium path. We define an equilibrium concept in sequential two‐person games where players update their beliefs with the Dempster–Shafer updating rule. We show that in the limit as uncertainty tends to zero, our equilibrium approximates Bayesian Nash equilibrium. We argue that our equilibrium can be used to define a refinement of Bayesian Nash equilibrium by imposing context‐dependent constraints on beliefs under uncertainty.     

Tariff Revenue Competition in a Free Trade Area: The Case of Asymmetric Large Countries

The striking result has been shown by Richardson that tariff revenue competition between two symmetric member countries of a free trade area (FTA) results in complete elimination of external tariffs if there exists a pure‐strategy Nash equilibrium at all. Richardson also conjectures without building a model that if member countries are asymmetric in market size, there exists a pure‐strategy Nash equilibrium in which both countries set positive external tariffs. We explicitly extend his tariff competition model into the case of asymmetric FTA member countries, and confirm his conjecture. We also show that there exists a mixed‐strategy Nash equilibrium in the case of symmetric countries.     

Optimal Individual Decisions with a Psycho-materialistic Utility Function

This paper proposes a theoretical model aimed at interpreting the outcome of some games. It is often observed that players do not play the Nash equilibrium, when their utility is modelled on their payoff only. However, other models which include psychological variables fail to describe the behaviours observed. Here I propose a utility function which encompasses both monetary and psychological payoffs in such a way that the predictions fit the actual decisions of the players observed in experiments.     

Robustness to strategic uncertainty

We introduce a criterion for robustness to strategic uncertainty in games with continuum strategy sets. We model a player's uncertainty about another player's strategy as an atomless probability distribution over that player's strategy set. We call a strategy profile robust to strategic uncertainty if it is the limit, as uncertainty vanishes, of some sequence of strategy profiles in which every player's strategy is optimal under his or her uncertainty about the others. When payoff functions are continuous we show that our criterion is a refinement of Nash equilibrium and we also give sufficient conditions for existence of a robust strategy profile. In addition, we apply the criterion to Bertrand games with convex costs, a class of games with discontinuous payoff functions and a continuum of Nash equilibria. We show that it then selects a unique Nash equilibrium, in agreement with some recent experimental findings.