Quitting Games

Quitting games are $n$-player sequential games in which, at any tage, each player has the choice between continuing and quitting. The game ends as soon as at least one player chooses to quit; player i then receives a payoff, which depends on the set S of players that did choose to quit. If the game never ends, the payoff to each player is 0. The paper has four goals. (i) we prove the existence of a subgame perfect uniform ${{\varepsilon}}$-equilibrium, under some assumptions on the payoff structure, (ii) we study the structure of the ${{\varepsilon}}$-equilibrium strategies, (iii) we present a new method for dealing with $n$-player games, and (iv) we study an example of a four-player quitting game where the ``simplest'' equilibrium is cyclic with period 2. We also discuss the relation to Dynkin's stopping games, and provide a generalization of our result to these games.