In game theory, mechanism design is concerned with the design of incentives so that a desired outcome of the game can be achieved. In this paper, we study the design of incentives so that a desirable equilibrium is obtained, for instance, an equilibrium satisfying a given temporal logic property -- a problem that we call equilibrium design. We base our study on a framework where system specifications are represented as temporal logic formulae, games as quantitative concurrent game structures, and players' goals as mean-payoff objectives. In particular, we consider system specifications given by LTL and GR(1) formulae, and show that implementing a mechanism to ensure that a given temporal logic property is satisfied on some/every Nash equilibrium of the game, whenever such a mechanism exists, can be done in PSPACE for LTL properties and in NP/$\Sigma^{P}_{2}$ for GR(1) specifications. We also study the complexity of various related decision and optimisation problems, such as optimality and uniqueness of solutions, and show that the complexities of all such problems lie within the polynomial hierarchy. As an application, equilibrium design can be used as an alternative solution to the rational synthesis and verification problems for concurrent games with mean-payoff objectives whenever no solution exists, or as a technique to repair, whenever possible, concurrent games with undesirable rational outcomes (Nash equilibria) in an optimal way.

中文翻译：

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并发博弈的均衡设计

在博弈论中，机制设计与激励的设计有关，以便可以实现期望的博弈结果。在本文中，我们研究了激励的设计，以便获得理想的均衡，例如，满足给定时间逻辑属性的均衡——我们称之为均衡设计的问题。我们的研究基于一个框架，其中系统规范表示为时间逻辑公式，游戏表示为定量的并发游戏结构，玩家的目标表示为平均收益目标。特别是，我们考虑了由 LTL 和 GR(1) 公式给出的系统规范，并表明只要存在这种机制，就可以实施一种机制来确保在游戏的某些/每个纳什均衡上满足给定的时间逻辑属性，对于 LTL 属性，可以在 PSPACE 中完成，对于 GR(1) 规范可以在 NP/$\Sigma^{P}_{2}$ 中完成。我们还研究了各种相关决策和优化问题的复杂性，例如解决方案的最优性和唯一性，并表明所有这些问题的复杂性都在多项式层次结构中。作为一种应用，均衡设计可以用作在不存在解决方案时具有平均收益目标的并发博弈的理性综合和验证问题的替代解决方案，或者作为一种技术，在可能的情况下修复具有不良理性结果的并发博弈。纳什均衡）以最佳方式。并表明所有这些问题的复杂性都在多项式层次结构中。作为一种应用，均衡设计可以用作在不存在解决方案时具有平均收益目标的并发博弈的理性综合和验证问题的替代解决方案，或者作为一种技术，在可能的情况下修复具有不良理性结果的并发博弈。纳什均衡）以最佳方式。并表明所有这些问题的复杂性都在多项式层次结构中。作为一种应用，均衡设计可以用作在不存在解决方案时具有平均收益目标的并发博弈的理性综合和验证问题的替代解决方案，或者作为一种技术，在可能的情况下修复具有不良理性结果的并发博弈。纳什均衡）以最佳方式。