A common observation in the informal literature of economics (and elsewhere) is that in multistage “games, ” players may seek early in the game to acquire a reputation for being “tough ” or “benevolent ” or something else. But this phenomenon is not observed in some formal game-theoretic analyses of finite games, such as Selten’s finitely repeated chain-store game or in the finitely repeated prisoners ’ dilemma. We reexamine Selten’s model, adding to it a “small ” amount of imperfect (or incomplete) information about players ’ payoffs, and we find that this addition is sufficient to give rise to the “reputation effect ” that one intuitively expects. Journal of Economic Literature, Classification Numbers: 026. 2 13, 6 11. 1.

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According to conventional wisdom, Nash equilibrium in a game “involves” common knowl-edge of the payoff functions, of the rationality of the players, and of the strategies played. The basis for this wisdom is explored, and it turns out that considerably weaker conditions suffice. First, note that if each player is rational and knows his own payoff function, and the strategy choices of the players are mutually known, then these choices form a Nash equilibrium. The other two results treat the mixed strategies of a player not as conscious randomization of that player, but as conjectures of the other players about what he will do. When n = 2, mutual knowledge of the payoff functions, of rationality, and of the conjectures yields Nash equilibrium. When n ≥ 3, mutual knowledge of the payoff functions and of rationality, and common knowl-edge of the conjectures yield Nash equilibrium when there is a common prior. Examples are provided showing these results to be sharp.

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The problem of incentives for correct revelation in a collective decision model is presented as a game with incomplete information. Two ap-proaches to incomplete information are used, a first where the individual beliefs are not introduced and a second where they are. In the first ap-proach it is recalled that the mechanisms for which the solution to the incentive problem is in dominant strategies lead in general to a budgetary problem for the central agency. For these mechanisms a uniqueness prop-erty is demonstrated. In the second approach it is shown that if a com-patibility condition is imposed on the individual beliefs and if a Bayesian solution is given to the incentive problem, then it is possible to avoid the budgetary problem. 1

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We propose an explanation of cooperation among unrelated members of a social group, in which providing group benefits evolves because it constitutes an honest signal of the member’s quality as a mate, coalition partner or competitor, and therefore results in advantageous alliances for those signaling in this manner. Our model is framed as an n-player game that involves no repeated or assortative interactions, and assumes a payoff structure that would conform to an n-player public goods game in which non-cooperation would be a dominant strategy if there were no signaling benefits. We show that honest signaling of underlying quality by providing a public good to group members can be evolutionarily stable. We also show that this behavior is capable of proliferating in a population in which it is initially rare. Our model applies to a range of cooperative interactions, including providing individually consumable resources, participating in group raiding or defense, and punishing free-riding or other violations of social norms. Our signaling model is distinctive in applying to group rather than dyadic interactions and in determining endogenously the fraction of the group that signals high quality in equilibrium.