Dynamical systems associated to the 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 $\mathfrak{S}$ of the set of payoffs into the set $\{C,\,D\}$ ($C$ means cooperation, $D$ -- defection). S. Smale in \cite{smale} presented an idea of good strategies in the repeated PD. If both players play good strategies then the average payoffs tends 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 $\beta$-core of PD. We show that if both players choose the same point in the $\beta$-core then the strategy profile is an equilibrium in the repeated game. If the players choose different points in the $\beta$-core then the sequence of the average payoffs tends to a point in $\mathfrak{S}$. 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 $S$ that corresponds to the $\beta$-core payoffs.