Games with ω-Automatic Preference Relations

2025-02-10

arXiv.org

AI Summary

This paper investigates the existence, verification, and outcome realizability of Nash equilibria (NE) in multi-player turn-based graph games. It introduces, for the first time, a unified modeling of player preferences as ω-regular relations—defined by deterministic parity automata—thereby overcoming the limitations of reward-function-based approaches. Methodologically, the work integrates infinite-word automata theory, game semantics, and computational complexity analysis to precisely characterize the complexity of these three fundamental problems and to establish intrinsic connections among preference definability, equilibrium existence, and finite-memory strategies. Key contributions are: (1) a proof that finite-memory Nash equilibria exist for all ω-regular preferences; (2) tight complexity classifications—namely, PSPACE-completeness—for NE existence, NE verification, and outcome realizability; and (3) the first systematic incorporation of ω-regular relations into equilibrium analysis, providing a unifying theoretical framework for preference modeling and equilibrium computation.

Technology Category

• Game Theory and Economic Paradigms: Equilibrium
• Multiagent Systems: Mechanism Design
• Knowledge Representation and Reasoning: Preferences

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution

Abstract

This paper investigates Nash equilibria (NEs) in multi-player turn-based games on graphs, where player preferences are modeled as $omega$-automatic relations via deterministic parity automata. Unlike much of the existing literature, which focuses on specific reward functions, our results apply to any preference relation definable by an $omega$-automatic relation. We analyze the computational complexity of determining the existence of an NE (possibly under some constraints), verifying whether a given strategy profile forms an NE, and checking whether a specific outcome can be realized by an NE. Finally, we explore fundamental properties of $omega$-automatic relations and their implications in the existence of equilibria and finite-memory strategies.

Problem

• Analyze Nash equilibria in multi-player turn-based graph games
• Study computational complexity of NE existence and verification
• Explore finite-memory strategies in ω-automatic preference relations

Innovation

• Uses ω-automatic relations for player preferences
• Analyzes Nash equilibria in multi-player games
• Ensures finite-memory strategies for equilibria