Dynamical Systems Associated with the $$\beta $$ β -Core in the Repeated Prisoner’s Dilemma

We consider the repeated prisoner’s dilemma (PD). We assume that players make their choices knowing only average payoffs from the previous stages. A player’s strategy is a function from the convex hull $${\mathfrak {S}}$$ S of the set of payoffs into the set $$\{C,\,D\}$$ { C , D } (C means cooperation, D—defection). Smale (Econometrica 48:1617–1634, 1980) presented an idea of good strategies in the repeated PD. If both players play good strategies then the average payoffs tend to the payoff corresponding to the profile (C, C) in PD. We adopt the Smale idea to define semi-cooperative strategies—players do not take as a referencing point the payoff corresponding to the profile (C, C), but they can take an arbitrary payoff belonging to the $$\beta $$ β -core of PD. We show that if both players choose the same point in the $$\beta $$ β -core then the strategy profile is an equilibrium in the repeated game. If the players choose different points in the $$\beta $$ β -core then the sequence of the average payoffs tends to a point in $${\mathfrak {S}}$$ S . The obtained limit can be treated as a payoff in a new game. In this game, the set of players’ actions is the set of points in $$\mathfrak {S}$$ S that corresponds to the $$\beta $$ β -core payoffs.