Degree-K subgame perfect Nash equilibria and the folk theorem

In infinitely repeated n-player games, we introduce a notion of degree-K subgame perfect Nash equilibria, in which any set of players whose size is up to K can coalitionally deviate and can transfer their payoffs within the coalition. If we only assume that players’ actions are observable, a coalitional deviation with hidden deviators who play as in the equilibrium cannot be detected by the other players. Hence we consider two models in which the hidden deviators can and cannot be detected, respectively. In the first model, there is an observer who can detect any coalitional deviation and report it to all players. We show an extension of the standard folk theorem; all feasible payoff vectors in which the sum of payoffs within any feasible coalition is strictly larger than the counterpart of the minmax value defined for the coalition arise as a degree-K subgame perfect Nash equilibrium if players are sufficiently patient. In the second model where the hidden deviators cannot be distinguished, we characterize degree-K subgame perfect equilibrium payoff vectors under patience by strategies which punish all players after any deviation. Finally, we adopt a new approach to characterize degree-n subgame perfect Nash equilibrium payoff vectors in the first model, since the punishment in the above folk theorem does not work when the grand coalition is feasible.