Polytopal Stochastic Games

TL;DR

This paper introduces polytopal stochastic games, extending traditional stochastic games by incorporating uncertainty in transition probabilities represented as polytopes, and explores their properties, solutions, and decision complexities.

Contribution

It formalizes polytopal stochastic games, proves their fundamental properties, and demonstrates their applicability through experiments and complexity analysis.

Findings

Existence of optimal strategies in polytopal stochastic games.

Decision problems are in NP ∩ coNP.

Finite representation of solutions is possible.

Abstract

In this paper we introduce polytopal stochastic games, an extension of two-player, zero-sum, turn-based stochastic games, in which we may have uncertainty over the transition probabilities. In these games the uncertainty over the probabilities distributions is captured via linear (in)equalities whose space of solutions forms a polytope. We give a formal definition of these games and prove their basic properties: determinacy and existence of optimal memoryless and deterministic strategies. We do this for reachability and different types of reward objectives and show that the solution exists in a finite representation of the game. We also state that the corresponding decision problems are in the intersection of NP and coNP. We motivate the use of polytopal stochastic games via a simple example. Finally, we report some experiments we performed with a prototype tool.