The Dynamics of the Nash Correspondence and $n$-Player Stochastic Games
International Game Theory Review 3 (2001), 291-300.
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.
We study an example of a three player quitting game that admits a uniform equilibrium payoff, and show that the correspondence that assigns to every continuation payoff the set of Nash equilibria in the corresponding one shot game does not admit non trivial periodic points.
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 multi-player stochastic games. It also shows that the minimal length of the period of a periodic e-equilibrium in three-player quitting games needs not be uniformly bounded for e > 0.