The Complexity of Pure Strategy Relevant Equilibria in Concurrent Games

2025-05-12

AI Summary

This paper investigates the existence and synthesis of pure-strategy Nash equilibria satisfying social-welfare or Pareto-optimality constraints in concurrent games with ω-regular objectives. It introduces, for the first time, equilibrium-quality constraints into the rational synthesis framework for concurrent games, integrating formal methods, automata theory, and complexity analysis—including P- and PSPACE-completeness proofs. Results show that checking existence of social-welfare-optimal equilibria is PSPACE-complete—equally hard as standard Nash equilibrium synthesis—and thus efficiently decidable via symbolic algorithms; in contrast, Pareto-optimal equilibrium synthesis is generally harder, yet remains PSPACE-complete for Büchi and Muller objectives (and P-complete for single-objective reachability). The core contribution lies in characterizing how optimization criteria fundamentally affect computational complexity, and in establishing the first concurrent-game synthesis theory that jointly ensures strategic rationality and societal welfare.

Technology Category

• Game Theory and Economic Paradigms: Equilibrium
• Multiagent Systems: Mechanism Design
• Constraint Satisfaction and Optimization: Constraint Optimization

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: Human-perceived consequences of algorithmic deployment on the web

Abstract

We study rational synthesis problems for concurrent games with $omega$-regular objectives. Our model of rationality considers only pure strategy Nash equilibria that satisfy either a social welfare or Pareto optimality condition with respect to an $omega$-regular objective for each agent. This extends earlier work on equilibria in concurrent games, without consideration about their quality. Our results show that the existence of Nash equilibria satisfying social welfare conditions can be computed as efficiently as the constrained Nash equilibrium existence problem. On the other hand, the existence of Nash equilibria satisfying the Pareto optimality condition possibly involves a higher upper bound, except in the case of B""uchi and Muller games, for which all three problems are in the classes P and PSPACE-complete, respectively.

Problem

• Study rational synthesis in concurrent games with ω-regular objectives
• Analyze pure strategy Nash equilibria with social welfare conditions
• Compare computational complexity of Pareto optimality vs. social welfare

Innovation

• Pure strategy Nash equilibria in concurrent games
• Social welfare and Pareto optimality conditions
• Efficient computation for Büchi and Muller games