Equilibria in Quantitative Concurrent Games

TL;DR

This paper introduces quantitative concurrent graph games where agents aim to reach targets minimizing their costs, analyzes the complexity of finding Pareto-optimal Nash equilibria, and identifies tractable cases.

Contribution

It formalizes a new class of quantitative multi-agent games and provides complexity results for computing Nash equilibria and related concepts.

Findings

Checking Nash equilibrium existence is NP-complete in general.

Two-player bounded-cost games can be solved in polynomial time.

The paper offers a comprehensive complexity analysis of Pareto-optimal equilibria.

Abstract

Synthesis of finite-state controllers from high-level specifications in multi-agent systems can be reduced to solving multi-player concurrent games over finite graphs. The complexity of solving such games with qualitative objectives for agents, such as reaching a target set, is well understood resulting in tools with applications in robotics. In this paper, we introduce quantitative concurrent graph games, where transitions have separate costs for different agents, and each agent attempts to reach its target set while minimizing its own cost along the path. In this model, a solution to the game corresponds to a set of strategies, one per agent, that forms a Nash equilibrium. We study the problem of computing the set of all Pareto-optimal Nash equilibria, and give a comprehensive analysis of its complexity and related problems such as the price of stability and the price of anarchy. In particular, while checking the existence of a Nash equilibrium is NP-complete in general, with multiple parameters contributing to the computational hardness separately, two-player games with bounded costs on individual transitions admit a polynomial-time solution.