On Nash Equilibria in Stochastic Games

Abstract

We study infinite stochastic games played by n-players on a finite graph with goals given by sets of infinite traces. The games are stochastic (each player simultaneously and independently chooses an action at each round, and 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 a reachability objective, that is, if the goal for each player i is to visit some subset R _i of the states, then there exists an ε-Nash equilibrium in memoryless strategies, for every ε >0. However, exact Nash equilibria need not exist. We study the complexity of finding such Nash equilibria, and show that the payoff of some ε-Nash equilibrium in memoryless strategies can be ε-approximated in NP.

We study the important subclass of n-player turn-based probabilistic games, where at each state at most one player has a nontrivial choice of moves. For turn-based probabilistic games, we show the existence of ε-Nash equilibria in pure strategies for games where the objective of player i is a Borel set B _i of infinite traces. However, exact Nash equilibria may not exist. For the special case of ω-regular objectives, we show exact Nash equilibria exist, and can be computed in NP when the ω-regular objectives are expressed as parity objectives.

This research was supported in part by the AFOSR MURI grant F49620-00-1-0327, ONR grant N00014-02-1-0671, NSF grants CCR-9988172 and CCR-0225610.