An Automata-Based Approach to Games with $\omega$-Automatic Preferences

2026-02-09

AI Summary

This work investigates strategy and equilibrium problems in multi-player turn-based graph games under ω-regular preferences defined by deterministic parity automata, departing from the conventional paradigm that relies on explicit reward functions. By introducing alternating parity word automata (APW) to characterize the set of paths a player can guarantee to be preferable according to their ω-regular preference, this study presents the first systematic application of ω-regular preferences to game-theoretic analysis. The main contributions include proving that the value set is recognizable by a polynomial-size APW, closing the complexity gap for the threshold problem in multi-player games, establishing the exact complexity of the threshold problem in zero-sum settings, resolving several open questions concerning Nash equilibria, and showing that cooperative rational synthesis is PSPACE-complete while its non-cooperative counterpart is undecidable.

Technology Category

• Game Theory and Economic Paradigms: Cooperative Game Theory
• 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
• Responsible Web: Machine-in-the-loop, human agency and autonomy

Abstract

This paper studies multiplayer turn-based games on graphs in which player preferences are modeled as $\omega$-automatic relations given by deterministic parity automata. This contrasts with most existing work, which focuses on specific reward functions. We conduct a computational analysis of these games, starting with the threshold problem in the antagonistic zero-sum case. As in classical games, we introduce the concept of value, defined here as the set of plays a player can guarantee to improve upon, relative to their preference relation. We show that this set is recognized by an alternating parity automaton APW of polynomial size. We also establish the computational complexity of several problems related to the concepts of value and optimal strategy, taking advantage of the $\omega$-automatic characterization of value. Next, we shift to multiplayer games and Nash equilibria, and revisit the threshold problem in this context. Based on an APW construction again, we close complexity gaps left open in the literature, and additionally show that cooperative rational synthesis is $\mathsf{PSPACE}$-complete, while it becomes undecidable in the non-cooperative case.

Problem

• omega-automatic preferences
• multiplayer turn-based games
• Nash equilibria
• threshold problem
• value computation

Innovation

• ω-automatic preferences
• alternating parity automaton
• value characterization
• Nash equilibrium
• rational synthesis