Value Iteration for Simple Stochastic Games: Stopping Criterion and Learning Algorithm

TL;DR

This paper introduces a stopping criterion for value iteration in simple stochastic games, enabling it to serve as an anytime algorithm with guaranteed error bounds and a simulation-based variant that avoids full graph exploration.

Contribution

It provides the first stopping criterion for value iteration in simple stochastic games and develops an asynchronous, simulation-based algorithm with guaranteed accuracy.

Findings

First stopping criterion for VI in simple stochastic games

Anytime algorithm with error bounds

Simulation-based VI avoids full graph exploration

Abstract

Simple stochastic games can be solved by value iteration (VI), which yields a sequence of under-approximations of the value of the game. This sequence is guaranteed to converge to the value only in the limit. Since no stopping criterion is known, this technique does not provide any guarantees on its results. We provide the first stopping criterion for VI on simple stochastic games. It is achieved by additionally computing a convergent sequence of over-approximations of the value, relying on an analysis of the game graph. Consequently, VI becomes an anytime algorithm returning the approximation of the value and the current error bound. As another consequence, we can provide a simulation-based asynchronous VI algorithm, which yields the same guarantees, but without necessarily exploring the whole game graph.