Algorithm and Strategy Construction for Sure-Almost-Sure Stochastic Parity Games

2026-01-06

arXiv.org

AI Summary

This work addresses the problem of determining and constructing winning strategies in stochastic two-player games that simultaneously satisfy both sure and almost-sure parity objectives. The authors propose a recursive algorithm based on a precise characterization of winning regions, which for the first time enables the explicit construction of joint winning strategies for such combined conditions. By integrating techniques from game theory, Markov decision processes, and parity game theory, the study reveals key structural properties of the problem. Notably, under the restriction that either parity condition has a fixed index, the approach overcomes the limitations of traditional exponential enumeration methods, yielding improved bounds on both strategy complexity and memory requirements.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Game Theory and Economic Paradigms: Cooperative Game Theory
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

We consider turn-based stochastic two-player games with a combination of a parity condition that must hold surely, that is in all possible outcomes, and of a parity condition that must hold almost-surely, that is with probability 1. The problem of deciding the existence of a winning strategy in such games is central in the framework of synthesis beyond worst-case where a hard requirement that must hold surely is combined with a softer requirement. Recent works showed that the problem is coNP-complete, and infinite-memory strategies are necessary in general, even in one-player games (i.e., Markov decision processes). However, memoryless strategies are sufficient for the opponent player. Despite these comprehensive results, the known algorithmic solution enumerates all memoryless strategies of the opponent, which is exponential in all cases, and does not construct a winning strategy when one exists. We present a recursive algorithm, based on a characterisation of the winning region, that gives a deeper insight into the problem. In particular, we show how to construct a winning strategy to achieve the combination of sure and almost-sure parity, and we derive new complexity and memory bounds for special classes of the problem, defined by fixing the index of either of the two parity conditions.

Problem

• stochastic parity games
• sure-almost-sure synthesis
• winning strategy
• parity conditions
• two-player games

Innovation

• stochastic parity games
• sure-almost-sure synthesis
• recursive algorithm
• winning strategy construction
• memory bounds