Polytopal Stochastic Games
Pablo F. Castro, Pedro D'Argenio
TL;DR
Polytopal stochastic games (PSG) address uncertainty in transition probabilities by representing them as polytopes and selecting distributions within those polytopes. The authors prove determinacy and the existence of optimal memoryless and deterministic strategies for reachability and a range of reward objectives, showing that PSGs admit finite representations through vertex-based discretization and can be solved via their extreme interpretations. They connect PSGs to simplicial games and establish NP ∩ coNP membership for decision problems, enabling practical algorithms despite uncountably many actions. A motivating Roborta–Rigoborto example illustrates modeling of polytope-based uncertainty, and a prototype demonstrates feasibility of the approach, with proofs relegated to the Appendix for rigor. Overall, PSGs offer a principled, tractable framework for qualitative and quantitative analysis of stochastic systems under polyhedral uncertainty, with potential for broad practical impact.
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.