The Complexity of Pure Strategy Relevant Equilibria in Concurrent Games

TL;DR

This paper investigates the computational complexity of finding pure strategy Nash equilibria in concurrent games with omega-regular objectives, focusing on social welfare and Pareto optimality conditions.

Contribution

It extends previous work by analyzing the complexity of equilibria with quality conditions, providing complexity bounds for various classes of omega-regular objectives.

Findings

Existence of social welfare satisfying equilibria is computationally efficient.

Pareto optimality equilibria may have higher complexity, except for Buchi and Muller games.

All three problems are in P or PSPACE-complete for Buchi and Muller objectives.

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 Buchi and Muller games, for which all three problems are in the classes P and PSPACE-complete, respectively.