Nash Equilibrium for Upward-Closed Objectives

We study in nite stochastic games played by n-players on a nite graph with goals speci ed by sets of in nite traces. The games are concurrent (each player simultaneously and independently chooses an action at each round), stochastic (the next state is determined by a probability distribution depending on the current state and the chosen actions), in nite (the game continues for an in nite number of rounds), nonzero-sum (the players' goals are not necessarily con icting), and undiscounted. We show that if each player has an upwardclosed objective, then there exists an "-Nash equilibrium in memoryless strategies, for every " > 0; and exact Nash equilibria need not exist. Upward-closure of an objective means that if a set Z of in nitely repeating states is winning, then all supersets of Z of in nitely repeating states are also winning. Memoryless strategies are strategies that are independent of history of plays and depend only on the current state. We also study the complexity of nding values (payo pro le) of an "-Nash equilibrium. We show that the values of an "-Nash equilibrium in nonzero-sum concurrent games with upward-closed objectives for all players can be computed by computing "-Nash equilibrium values of nonzero-sum concurrent games with reachability objectives for all players and a polynomial procedure. As a consequence we establish that values of an "-Nash equilibrium can be computed in TFNP (total functional NP), and hence in EXPTIME. This research was supported in part by the ONR grant N00014-02-1-0671, the AFOSR MURI grant F49620-00-1-0327, and the NSF grant CCR-0225610.