8. Introduction to Repeated Games

This chapter deals with repeated games. It first considers the simplest model (with observed moves) and introduces: finitely repeated games where the payoff is the average of the stage payoffs over finitely many stages, discounted games where the payoff is the (infinite) discounted sum of the stage payoffs, and uniform games where players consider the payoff in any long enough game (or in any discounted game with low enough discount factor). We present the Folk theorem, which states that the equilibrium payoffs of a uniform game are precisely the vector payoffs which are feasible (achievable) and individually rational (above the punishment level for each player). The message is simple: if players are patient enough, any reasonable payoff can be achieved at equilibrium. Folk theorems for discounted games and finitely repeated games are also exposed and proved. The last section presents extensions of the model in three directions: repeated games with signals (imperfect observation of moves), stochastic games where the “Big Match” is studied, and repeated games with incomplete information where we prove the cav(u) theorem of Aumann and Maschler.