Abstract: In Fujiwara-Greve and Okuno-Fujiwara (2009), an evolutionary stability concept was defined by allowing mutations of any strategy. However, in human societies, not all strategies are likely to be tried out when a player considers what happens in the future. In this paper we introduce the "shared belief" of potential continuation strategies, generated and passed on in a society, and mutations are restricted only among best responses against the shared belief. We show that a myopic strategy becomes a part of a bimorphic equilibrium under a shared belief and contributes to a higher payoff than ordinary neutrally stable distributions'.

Abstract: In many repeated interactions, repetition is not guaranteed but instead must be agreed upon. We formulate a model of voluntary repetition by introducing outside options to a repeated Prisoner's Dilemma and investigate how the structure of outside options affects the sustainability of mutual cooperation. When the outside option is deterministic and greater than the value of mutual defection, the lower bound of the discount factors that sustain repeated cooperation is greater than the one for ordinary repeated Prisoner's Dilemma, making cooperation more difficult. However, stochastic outside options with the same mean may reduce the lower bound of discount factors as compared to the deterministic case. This is possible when the stochasticity of the options increases the value of the cooperation phase more than the value of the punishment phase. Necessary and sufficient conditions for this positive effect are given under various option structures.

This paper studies a class of infinitely repeated games with two players in which the action space of each player is an interval, and the one-shot payoff of each player is additively separable in their actions. We define an immediately reactive equilibrium (IRE) as a pure-strategy subgame perfect equilibrium such that each player’s action in each period is a stationary function of the other player’s last action. We completely characterize IREs and their dynamics in terms of certain indifference curves. In a special case we establish a folk-type theo-rem using only IREs that are continuous and punish deviations in a minimal way. Our results are used to show that in a prisoners’ dilemma game with observable mixed strategies, gradual cooperation occurs when the players are sufficiently patient, and that in a certain duopoly game, kinked demand curves emerge naturally.

Ordinary repeated games do not apply to real societies where one can cheat and escape from partners. We formulate a model of endogenous relationships that a player can unilaterally end and start with a randomly assigned new partner with no information flow. Focusing on two-person, two-action Prisoner’s Dilemma, we show that the endogenous duration of partnerships generates a significantly dif-ferent evolutionary stability structure from ordinary random matching games. Monomorphic equilibria require initial trust building, while a polymorphic equilibrium includes earlier cooperators than any strat-egy in monomorphic equilibria and is thus more efficient. This is due to the non-linearity of average payoffs. 1.

This paper studies the possibility of whole population cooperation based on playerspreferences. Consider the following in…nitely repeated game, similar to Ghosh and Ray (1996). At each stage, uncountable numbers of players are randomly matched without information about their partnerspast actions and play a prisoners dilemma game. The players have the option to continue their relationship, and they all have the same discount factor. Also, they have two possible types: high ability player (H) or low ability player (L). H can produce better outcomes for its partner as well as for itself than L can. I look for an equilibrium that is robust against both pair-wise deviation and individual deviation and call such equilibrium a social equilibrium. In this setting, long-term cooperative behavior among the whole population can take place in a social equilibrium because of the playerspreferences for their partnerstypes. In addition, a folk theorem of this model is proposed.

This paper studies a class of infinitely repeated games with two players in which the action space of each player is an interval, and the one-shot payoff of each player is additively separable in their actions. We define an immediately reactive equilibrium (IRE) as a pure-strategy subgame perfect equilibrium such that the action of each player is a stationary function of the last action of the other player. We show that the set of IREs in the simultaneous move game is identical to that in the alternating move game. In both games, IREs are com-pletely characterized in terms of indifference curves associated with what we call effective payoffs. A folk-type theorem using only IREs is established in a special case. Our results are applied to a prisoner’s dilemma game with observable mixed strategies and a duopoly game. In the latter game, kinked demand curves with a globally stable steady