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 player k ¿ {1,2} at least once in every rk ¿ ¿ consecutive 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 is jointly convergent if 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 called individually rational if each player receives at least the threat-point reward, i.e., the amount which he can guarantee himself regardless of what the opponent does given r1, r2 and the actions available in the long run. In a repeated game with vanishing actions, there may exist multiple threat points which are endogenous to 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 for m × n-games provided r1 > m ¿ 2, r2 > n ¿ 2. Keywords: Stochastic games; vanishing actions; limiting Average rewards; endogenous threats