Quitting games - An example

Quitting games are multi-player sequential games in which, at every stage, each player has the choice between continuing and quitting. The game ends as soon as at least one player chooses to quit; each player i then receives a payoff ri S , which depends on the set S of players that did choose to quit. If the game never ends, the payoff to each player is zero. We exhibit a four-player quitting game, where the “simplest” equilibrium is periodic with period two. We argue that this implies that all known methods to prove existence of an equilibrium payoff in multiplayer stochastic games are therefore bound to fail in general, and provide some geometric intuition for this phenomenon. ∗MEDS Department, Kellogg School of Management, Northwestern University, 2001 Sheridan Road, Evanston, IL 60208 and the School of Mathematical Sciences, Tel Aviv University. e-mail: [email protected] †Département Finance et Economie, HEC, 78 351 Jouy-en-Josas, France. e-mail: [email protected]