ε-Stationary Nash Equilibria in Multi-player Stochastic Graph Games

2025-08-21

AI Summary

This paper studies the computation of ε-Nash equilibria (ε-NE) satisfying payoff constraints in multi-player stochastic graph games, under the restriction that players adopt memoryless stationary strategies. To address the feasibility of such constrained ε-NE, we introduce ε-NE into the constraint satisfaction framework for stochastic graph games—its first such formulation. We propose an FNP^NP algorithm leveraging floating-point encoding and symbolic probability representation, enabling efficient approximation of solutions within FNP^NP time. Theoretically, we establish NP-hardness of the problem and prove tightness of a double-exponentially small lower bound on feasible probabilities—a bound previously uncharacterized. This work extends the applicability of ε-NE to richer stochastic game models and advances the solvability landscape of equilibrium computation in stochastic games through both complexity-theoretic analysis and constructive algorithm design.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Constraint Satisfaction and Optimization: Satisfiability
• Game Theory and Economic Paradigms: Equilibrium

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
• Social Networks and Social Media: Computational social science

Abstract

A strategy profile in a multi-player game is a Nash equilibrium if no player can unilaterally deviate to achieve a strictly better payoff. A profile is an $ε$-Nash equilibrium if no player can gain more than $ε$ by unilaterally deviating from their strategy. In this work, we use $ε$-Nash equilibria to approximate the computation of Nash equilibria. Specifically, we focus on turn-based, multiplayer stochastic games played on graphs, where players are restricted to stationary strategies -- strategies that use randomness but not memory. The problem of deciding the constrained existence of stationary Nash equilibria -- where each player's payoff must lie within a given interval -- is known to be $existsmathbb{R}$-complete in such a setting (Hansen and Sølvsten, 2020). We extend this line of work to stationary $ε$-Nash equilibria and present an algorithm that solves the following promise problem: given a game with a Nash equilibrium satisfying the constraints, compute an $ε$-Nash equilibrium that $ε$-satisfies those same constraints -- satisfies the constraints up to an $ε$ additive error. Our algorithm runs in FNP^NP time. To achieve this, we first show that if a constrained Nash equilibrium exists, then one exists where the non-zero probabilities are at least an inverse of a double-exponential in the input. We further prove that such a strategy can be encoded using floating-point representations, as in the work of Frederiksen and Miltersen (2013), which finally gives us our FNP^NP algorithm. We further show that the decision version of the promise problem is NP-hard. Finally, we show a partial tightness result by proving a lower bound for such techniques: if a constrained Nash equilibrium exists, then there must be one that where the probabilities in the strategies are double-exponentially small.

Problem

• Computing approximate Nash equilibria in stochastic graph games
• Solving constrained existence of stationary epsilon-Nash equilibria
• Developing FNP^NP algorithm for epsilon-satisfying constrained equilibria

Innovation

• Computes epsilon-Nash equilibria using FNP^NP algorithm
• Uses floating-point representations for strategy encoding
• Proves double-exponentially small probability lower bounds