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Synthesis of finite-state controllers from high-level specifications inmulti-agent systems can be reduced to solving multi-player concurrent gamesover finite graphs. The complexity of solving such games with qualitativeobjectives for agents, such as reaching a target set, is well understoodresulting in tools with applications in robotics. In this paper, we introducequantitative concurrent graph games, where transitions have separate costs fordifferent agents, and each agent attempts to reach its target set whileminimizing its own cost along the path. In this model, a solution to the gamecorresponds to a set of strategies, one per agent, that forms a Nashequilibrium. We study the problem of computing the set of all Pareto-optimalNash equilibria, and give a comprehensive analysis of its complexity andrelated problems such as the price of stability and the price of anarchy. Inparticular, while checking the existence of a Nash equilibrium is NP-completein general, with multiple parameters contributing to the computational hardnessseparately, two-player games with bounded costs on individual transitions admita polynomial-time solution.

Equilibria in Quantitative Concurrent Games