Utility Design for Two-player Normal-form Games

This study formulates a utility design problem for two-player normal-form games and proposes a novel computational algorithm to solve the problem. This study proposes an automatic and systematic algorithm for calculating a utility matrix, as a utility (payoff), that enables to achieve the desired Nash equilibrium (NE). The main contribution of this study is the utility calculation algorithm, which can be constructed using a control system design framework. The nonlinear dynamics of the correlation between an NE and a corresponding utility matrix is expressed in a discrete-time, nonlinear, autonomous dynamical system, which is derived from the Karush-Kuhn-Tucker condition. The model is theoretically guaranteed to be stable and to yield updates of the NE and the corresponding utility matrix. The analysis result enables to design the tracking control system that continues to automatically adjust the amount of displacement in the NE. Besides, numerical examples confirm that the utility design problem can be solved by the proposed algorithm.