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In game theory, mechanism design is concerned with the design of incentivesso that a desired outcome of the game can be achieved. In this paper, weexplore the concept of equilibrium design, where incentives are designed toobtain a desirable equilibrium that satisfies a specific temporal logicproperty. Our study is based on a framework where system specifications arerepresented as temporal logic formulae, games as quantitative concurrent gamestructures, and players' goals as mean-payoff objectives. We consider systemspecifications given by LTL and GR(1) formulae, and show that designingincentives to ensure that a given temporal logic property is satisfied onsome/every Nash equilibrium of the game can be achieved in PSPACE for LTLproperties and in NP/{\Sigma}P 2 for GR(1) specifications. We also examine thecomplexity of related decision and optimisation problems, such as optimalityand uniqueness of solutions, as well as considering social welfare, and showthat the complexities of these problems lie within the polynomial hierarchy.Equilibrium design can be used as an alternative solution to rational synthesisand verification problems for concurrent games with mean-payoff objectives whenno solution exists or as a technique to repair concurrent games withundesirable Nash equilibria in an optimal way.

Designing Equilibria in Concurrent Games with Social Welfare and Temporal Logic Constraints