  We construct a statistical ensemble of games, where in 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 of both players. In the case of
symmetric, non-cooperative and dilemmatic games, we show that generalized
tit-for-tat policies together with the condition of not being the first to
defect, leads to the highest mean payoffs per move for the players.
