Fast Planning in Stochastic Games

TL;DR

This paper introduces a generalized finite-horizon value iteration algorithm for computing Nash equilibria in stochastic games, extending planning methods from MDPs to multiagent settings, and presents an approach for large or infinite state spaces.

Contribution

It develops a novel finite-horizon value iteration algorithm for stochastic games and adapts sparse sampling techniques to approximate Nash equilibria in large or infinite state spaces.

Findings

The algorithm computes Nash strategies in general-sum stochastic games.

It extends sparse sampling methods to multiagent stochastic settings.

Infinite-horizon discounted value iteration does not always converge in general-sum cases.

Abstract

Stochastic games generalize Markov decision processes (MDPs) to a multiagent setting by allowing the state transitions to depend jointly on all player actions, and having rewards determined by multiplayer matrix games at each state. We consider the problem of computing Nash equilibria in stochastic games, the analogue of planning in MDPs. We begin by providing a generalization of finite-horizon value iteration that computes a Nash strategy for each player in generalsum stochastic games. The algorithm takes an arbitrary Nash selection function as input, which allows the translation of local choices between multiple Nash equilibria into the selection of a single global Nash equilibrium. Our main technical result is an algorithm for computing near-Nash equilibria in large or infinite state spaces. This algorithm builds on our finite-horizon value iteration algorithm, and adapts the sparse sampling methods of Kearns, Mansour and Ng (1999) to stochastic games. We conclude by descrbing a counterexample showing that infinite-horizon discounted value iteration, which was shown by shaplely to converge in the zero-sum case (a result we give extend slightly here), does not converge in the general-sum case.