Two-player Nonzero-sum !-regular Games

We study in nite stochastic games played by two-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 !-regular objective expressed as a parity objective, then there exists an "-Nash equilibrium, for every " > 0. However, exact Nash equilibria need not exist. We study the complexity of nding values (payo pro le) of some "-Nash equilibrium. We show that the values of some "-Nash equilibrium in nonzero-sum concurrent parity games can be computed by solving the following two simpler problems: computing the values of zero-sum (the goals of the players are strictly con icting) concurrent parity games and computing "-Nash equilibrium values of nonzero-sum concurrent games with reachability objectives. As a consequence we establish that values of some "-Nash equilibrium can be approximated in FNP (functional NP), and hence in EXPTIME. The work was supported by the AFOSR MURI grant F49620-00-1-0327, by the ONR grant N00014-02-1-0671, and by the NSF grants CCR-0132780, CCR-0234690, CCR9988172, and CCR-0225610, and by the NSF Career grant CCR-0132780, the NSF grant CCR-0234690, and by the ONR grant N00014-02-1-0671