Two-Player Nonzero-Sum ω-Regular Games

Abstract

We study infinite stochastic games played by two-players 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), nonzero-sum (the players’ goals are not necessarily conflicting), and undiscounted. We show that if each player has an ω-regular objective expressed as a parity objective, then there exists an ε-Nash equilibrium, for every ε> 0. However, exact Nash equilibria need not exist. We study the complexity of finding values (payoff profile) of an ε-Nash equilibrium. We show that the values of an ε-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 conflicting) concurrent parity games and computing ε-Nash equilibrium values of nonzero-sum concurrent games with reachability objectives. As a consequence we establish that values of an ε-Nash equilibrium can be computed in TFNP (total functional NP), and hence in EXPTIME.