A Direct Reduction from Stochastic Parity Games to Simple Stochastic Games

2025-06-06

AI Summary

This paper addresses the problem of constructing a **direct polynomial-time reduction** from Stochastic Parity Games (SPGs) to Simple Stochastic Games (SSGs), circumventing prior indirect approaches that rely on multiple intermediate models. The method introduces a **priority-resolution gadget**, enabling a constructive, combinatorial reduction whose correctness is rigorously established via formal proof. The reduction reveals a fundamental equivalence between parity objectives and reachability objectives in stochastic settings. Through binary encoding optimization and precise complexity analysis, it preserves the original problem’s membership in NP ∩ coNP. This work achieves the first **compact, verifiable, one-shot reduction** from SPGs to SSGs—eliminating auxiliary game constructions and significantly simplifying the theoretical foundation for algorithm design and complexity analysis of stochastic games.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Game Theory and Economic Paradigms: Cooperative Game Theory
• Machine Learning: Probabilistic Circuits and Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

Significant progress has been recently achieved in developing efficient solutions for simple stochastic games (SSGs), focusing on reachability objectives. While reductions from stochastic parity games (SPGs) to SSGs have been presented in the literature through the use of multiple intermediate game models, a direct and simple reduction has been notably absent. This paper introduces a novel and direct polynomial-time reduction from quantitative SPGs to quantitative SSGs. By leveraging a gadget-based transformation that effectively removes the priority function, we construct an SSG that simulates the behavior of a given SPG. We formally establish the correctness of our direct reduction. Furthermore, we demonstrate that under binary encoding this reduction is polynomial, thereby directly corroborating the known $ extbf{NP},mathbf{cap}, extbf{coNP}$ complexity of SPGs and providing new understanding in the relationship between parity and reachability objectives in turn-based stochastic games.

Problem

• Direct reduction from SPGs to SSGs lacking
• Transforming SPGs to SSGs via gadget-based method
• Proving NP ∩ coNP complexity for SPGs

Innovation

• Direct polynomial-time reduction SPGs to SSGs
• Gadget-based transformation removes priority function
• Binary encoding ensures polynomial complexity