Solving Simple Stochastic Games with Few Random Vertices

TL;DR

This paper introduces two new algorithms for solving simple stochastic games efficiently when the number of random vertices is small, leveraging permutation strategies and fixed-parameter tractability.

Contribution

The paper presents two algorithms based on permutation strategies that solve simple stochastic games in polynomial time for fixed numbers of random vertices, improving previous methods.

Findings

Algorithms run in polynomial time with fixed random vertices

No need to transform game into a stopping game

Permutation-enumeration avoids linear programming

Abstract

Simple stochastic games are two-player zero-sum stochastic games with turn-based moves, perfect information, and reachability winning conditions. We present two new algorithms computing the values of simple stochastic games. Both of them rely on the existence of optimal permutation strategies, a class of positional strategies derived from permutations of the random vertices. The "permutation-enumeration" algorithm performs an exhaustive search among these strategies, while the "permutation-improvement" algorithm is based on successive improvements, \`a la Hoffman-Karp. Our algorithms improve previously known algorithms in several aspects. First they run in polynomial time when the number of random vertices is fixed, so the problem of solving simple stochastic games is fixed-parameter tractable when the parameter is the number of random vertices. Furthermore, our algorithms do not require the input game to be transformed into a stopping game. Finally, the permutation-enumeration algorithm does not use linear programming, while the permutation-improvement algorithm may run in polynomial time.