The Dynamics of the Nash Correspondence and n-Player Stochastic Games

A quitting game is a sequential game where each player has two actions: to continue or to quit, and the game continues as long as no player quits. For every continuation payoff x we assign a one-shot game, where the payoff if everyone continues is x. We study the dynamics of the correspondence that assigns to every continuation payoff the set of equilibrium payoffs in the corresponding one shot game. The study presented here has an implication on the approach one should take in trying to prove, or disprove, the existence of an equilibrium payoff in n-player stochastic games. It also shows that the minimal length of the period of a periodic d-equilibrium in three-player quitting games needs not be uniformly bounded for d > 0.