On the Complexity of Stationary Nash Equilibria in Discounted Perfect Information Stochastic Games

2025-10-13

AI Summary

This paper investigates the computational complexity of computing stationary (steady-state) Nash equilibria in discounted perfect-information stochastic games. For two-player games, we establish PPAD-completeness of stationary equilibrium computation—providing the first full proof and a streamlined PPAD-hardness reduction. For three-player games, we construct the first explicit counterexample refuting the conjecture that rational stationary equilibria always exist, thereby exposing a fundamental obstruction to equilibrium existence. For four-player games, we design a polynomial-time reduction to the SqrtSum problem, proving SqrtSum-hardness of equilibrium computation. Integrating tools from computational complexity theory, intricate game constructions, and equilibrium existence analysis, our work systematically delineates the solvability boundary for stationary equilibria across player counts—from two to four players—and establishes critical benchmarks for the complexity classification of equilibrium computation in stochastic games.

Technology Category

• Game Theory and Economic Paradigms: Equilibrium
• Multiagent Systems: Mechanism Design
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Social Networks and Social Media: Computational social science
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

We study the problem of computing stationary Nash equilibria in discounted perfect information stochastic games from the viewpoint of computational complexity. For two-player games we prove the problem to be in PPAD, which together with a previous PPAD-hardness result precisely classifies the problem as PPAD-complete. In addition to this we give an improved and simpler PPAD-hardness proof for computing a stationary epsilon-Nash equilibrium. For 3-player games we construct games showing that rational-valued stationary Nash equilibria are not guaranteed to exist, and we use these to prove SqrtSum-hardness of computing a stationary Nash equilibrium in 4-player games.

Problem

• Computing stationary Nash equilibria in discounted stochastic games
• Classifying computational complexity as PPAD-complete for two-player games
• Proving nonexistence of rational equilibria in three-player games

Innovation

• PPAD-complete classification for two-player games
• Simplified PPAD-hardness proof for epsilon-equilibria
• SqrtSum-hardness established for four-player games