We construct a statistical ensemble of games, wherein each independent subensemble we have two players playing the same game. We derive the mean payoffs per move of the representative players of the game, and we evaluate all the deterministic policies with finite memory. In particular, we show that if one of the players has a generalized tit-for-tat policy, the mean payoff per move of both players is the same, forcing the equalization of the mean payoffs per move. In the case of symmetric, noncooperative and dilemmatic games, we show that generalized tit-for-tat or imitation policies together with the condition of not being the first to defect, leads to the highest mean payoffs per move for the players. Within this approach, it can be decided which policies perform better than others. In particular, it shows that reciprocity in noncooperative iterated games forces equality of mean payoffs. We prove a simple ergodic theorem for symmetric and noncooperative games. The Prisoner's Dilemma and the Hawk–Dove games have been analyzed, and the equilibrium states of the infinitely iterated games have been determined. In infinitely iterated games with the player choosing their moves with equal probabilities, strict Nash solutions are not necessarily reachable equilibrium solutions of games.
The glassy solid as a statistical ensemble of crystalline microstates