Takako Fujiwara-Greve

Repeated Cooperation with Outside Options

If agents can choose when to end a repeatable interaction as well as what to do within the partnership, cooperation incentives are interrelated with the ending decision. Using a Prisoners Dilemma with outside options, we investigate how the structure of outside options affects the minimum discount factor that sustains mutual cooperation. One-sided outside options do not make cooperation easier than ordinary repeated games, but uncertainty of options reduces the difficulty by enabling cooperation until a good option realizes. Two-sided and very low outside options make cooperation easier than ordinary repeated games. Economic implications are also discussed.

Bayesian Nash Equilibrium

In this chapter, we explain Harsanyi’s Bayesian framework for games with incomplete information. For normal-form games with incomplete information, Bayesian games and Bayesian Nash equilibrium are defined.

Voluntarily separable repeated Prisonerʼs Dilemma with reference letters

We consider the voluntarily separable repeated Prisonerʼs Dilemma model in which players randomly meet and form pairs to repeatedly play Prisonerʼs Dilemma only by mutual agreement. While the literature has dealt with the case of no information flow across partnerships, we consider the case in which players can issue a “reference letter” to verify at least that the partnership entered the cooperation phase. We show that such reference letters can be voluntarily provided by the partners even at some cost, and that the sheer existence of a letter shortens the trust-building periods and thus improves social efficiency.

Diverse Behavior Patterns in a Symmetric Society with Voluntary Partnerships

In the literature of voluntarily repeated Prisoners Dilemma, the focus is on how long-term cooperation is established, when newly matched partners cannot know the past actions of each other. In this paper we investigate how non-cooperative and cooperative players co-exist. In many incomplete information versions of a similar model, inherently non-cooperative players are assumed to exist in the society, but their long-run fitness has not been analyzed. In reality and in experiments, we also observe that some people are cooperative, while others never cooperate. We show that a bimorphic equilibrium of the most cooperative strategy and the most myopic strategy exists for sufficiently high survival rate of players, and that it is evolutionarily stable under uncoordinated mutations. For lower survival rates, adding initial periods of defection makes similar bimorphic equilibria. Both types of equilibria confirm persistence of defectors.

Subgame Perfect Equilibrium

For general extensive-form games with or without perfect information, subgame perfect equilibrium is defined. Various repeated games are analyzed, and Perfect Folk Theorem is proved.

Asymmetry of Reputation Loss and Recovery under Endogenous Partnerships: Theory and Evidence

This paper is inspired by real-world phenomena that firms lose customers based on imprecise information and take a long time to recover. If consumers are playing an ordinary repeated game with fixed partners, there is no clear reason why recovery slowly happens. However, if consumers are playing an endogenously repeated game, a class of simple efficient equilibria exhibits the asymmetry of fast loss of customers after a bad signal and slow recovery. Exit is systematic but formation of a new partnership is random. We also give empirical evidence of our equilibria at an individual-firm level.

Behavioral Diversity in Voluntarily Separable Repeated Prisoner's Dilemma

In the literature of voluntarily repeated Prisoners Dilemma type games with no information flow, the focus is on how long-term cooperation is established. In this paper we investigate how non-cooperative and cooperative players co-exist. In many incomplete information versions of a similar model, inherently non-cooperative players are assumed to exist in the society, but their long-run fitness has not been analyzed. In reality and in experiments, we also observe that some people are cooperative, while others never cooperate. We show that, although defect-and-run type strategies are vulnerable to invasion of trust-building strategies (which defect initially but cooperate later among themselves), distributions of both types of strategies can be evolutionarily stable under equilibrium entrants. Moreover, the bimorphic equilibrium of the most cooperative strategy and the most myopic strategy exists under any payoff parameter combination, while bimorphic equilibria consisting only of cooperative (trust-building) strategies may not exist. In terms of payoffs, the bimorphic equilibrium of contrary strategies is equivalent to the equilibrium of infinitely many trust-building strategies. Both equilibria confirm the persistent presence of defectors.

Exit Option Can Make Cooperation Easier

In the modern society, mobility of players has increased dramatically. It is important to extend game theory to endogenize the set of players. In this paper, we extend the repeated n-person Prisoners Dilemma by allowing some or all players to exit after each period and analyze how mobility affects endogenous cooperation in a fundamental sense: players voluntarily participate in the game and act non-myopically. We characterize the parameters (the allocation of exit options among players and the payoffs) under which eternal universal cooperation is sustained at a discount factor lower than the one for the ordinary repeated Prisoners Dilemma. Factors that enhance endogenous long-term cooperation include payoff asymmetry in the Prisoners Dilemma, multiple mobile players who have the exit option, and immobile players having low after-game payoffs. These are new insights which give policy implications on employment relationships and other voluntary partnership situations as well.

Asymmetry of Customer Loss and Recovery Under Endogenous Partnerships: Theory and Evidence

This article is inspired by real‐world phenomena that firms lose customers based on imprecise information and take a long time to recover. If consumers are playing an ordinary repeated game with fixed partners, there is no clear reason why recovery happens slowly. However, if consumers are playing an endogenously repeated game, a class of simple efficient equilibria exhibits the asymmetry of fast loss and slow recovery of customers after a bad signal. Exit is systematic, but formation of a new partnership is random. We also give empirical evidence of our equilibria at an individual‐firm level.

Games in Game Theory

This chapter explains what a “game” means in game theory and gives classifications of games. Rationality in game theory is also discussed.