A Note On Repeated Games With Vanishing Actions

A two-person general-sum repeated game with vanishing actions is an infinitely repeated game where the players face the following restrictions. Each action must be used by playerk ∈ {1,2}at least once in everyrk∈ ℕconsecutive stages, otherwise the action vanishes for the remaining play. We assume that the players wish to maximize their limiting average rewards over the entire time-horizon.A strategy-pair isjointly convergentif for each action pair a number exists to which the relative frequency with which this action pair is chosen, converges with probability one. A pair of feasible rewards is calledindividually rationalif each player receives at least the threat-point reward, i.e., the amount which he can guarantee himself regardless of what the opponent does givenr1,r2and the actions available in the long run. In a repeated game with vanishing actions, there may exist multiple threat points which areendogenousto the play.We prove that all individually-rational jointly-convergent pure-strategy rewards can be supported by an equilibrium. Furthermore, each convex combination of individually-rational jointly-convergent pure-strategy rewards, can be supported by an equilibrium form × n-games providedr1> m ≥ 2,r2> n ≥ 2.