Abstract: 
We study infinite stochastic games played by twoplayers on a finite graph with goals specified by sets of infinite 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), infinite (the game continues for an infinite number of rounds), nonzerosum (the players' goals are not necessarily conflicting), and undiscounted. We show that if each player has an Wregular objective expressed as a paxity objective, then there exists an epsilonNash equilibrium, for every epsilon > 0. However, exact Nash equilibria need not exist. We study the complexity of finding values (payoff profile) of an epsilonNash equilibrium. We show that the values of an epsilonNash equilibrium in nonzerosum concurrent parity games can be computed by solving the following two simpler problems: computing the values of zerosum (the goals of the players axe strictly conflicting) concurrent parity games and computing epsilonNash equilibrium values of nonzerosum concurrent games with reachability objectives. As a consequence we establish that values of an epsilonNash equilibrium can be computed in TFNP (total functional NP), and hence in EXPTIME.
