Subgame perfect equilibria in stopping games

Stopping games (without simultaneous stopping) are multi-player sequential games in which at every stage one of the players is chosen according to a stochastic process, and that player decides whether to continue the interaction or to stop it, whereby the terminal payoff vector is obtained by another stochastic process. We prove that if the payoff process is integrable, a $delta $ -approximate subgame perfect ${epsilon }$ -equilibrium exists for every $delta ,epsilon >0$ ; that is, there exists a strategy profile that is an ${epsilon }$ -equilibrium in all subgames, except possibly in a set of subgames that occurs with probability at most $delta $ (even after deviation by some of the players).