A Convex Programming-based Algorithm for Mean Payoff Stochastic Games with Perfect Information

TL;DR

This paper presents a convex programming approach to solve two-person zero-sum stochastic mean payoff games with perfect information, achieving pseudo-polynomial time complexity when the number of random positions is fixed.

Contribution

It introduces a convex programming-based algorithm that solves BWR-games in pseudo-polynomial time for a fixed number of random positions, addressing a long-standing open problem.

Findings

Solves BWR-games using convex programming.

Achieves pseudo-polynomial time complexity for fixed random positions.

Provides a new approach for an open problem in stochastic game theory.

Abstract

We consider two-person zero-sum stochastic mean payoff games with perfect information, or BWR-games, given by a digraph $G = (V, E)$, with local rewards $r: E \to \ZZ$, and three types of positions: black $V_B$, white $V_W$, and random $V_R$ forming a partition of $V$. It is a long-standing open question whether a polynomial time algorithm for BWR-games exists, even when $|V_R|=0$. In fact, a pseudo-polynomial algorithm for BWR-games would already imply their polynomial solvability. In this short note, we show that BWR-games can be solved via convex programming in pseudo-polynomial time if the number of random positions is a constant.