Decidability Results for Multi-objective Stochastic Games

TL;DR

This paper investigates the decidability of multi-objective stochastic games, providing algorithms for Pareto curve computation and proving finiteness of Pareto points in stopping games with two objectives, leading to decidability results.

Contribution

It introduces a new algorithm for Pareto curve computation in determined games and proves finiteness of Pareto points in stopping games with two objectives, establishing decidability.

Findings

Pareto curves contain finitely many points in stopping games with two objectives.

Decidability of the two-objective discounted-reward problem in stochastic games.

New algorithm for computing Pareto curves in determined games.

Abstract

We study stochastic two-player turn-based games in which the objective of one player is to ensure several infinite-horizon total reward objectives, while the other player attempts to spoil at least one of the objectives. The games have previously been shown not to be determined, and an approximation algorithm for computing a Pareto curve has been given. The major drawback of the existing algorithm is that it needs to compute Pareto curves for finite horizon objectives (for increasing length of the horizon), and the size of these Pareto curves can grow unboundedly, even when the infinite-horizon Pareto curve is small. By adapting existing results, we first give an algorithm that computes the Pareto curve for determined games. Then, as the main result of the paper, we show that for the natural class of stopping games and when there are two reward objectives, the problem of deciding whether a player can ensure satisfaction of the objectives with given thresholds is decidable. The result relies on intricate and novel proof which shows that the Pareto curves contain only finitely many points. As a consequence, we get that the two-objective discounted-reward problem for unrestricted class of stochastic games is decidable.