Abstract
Nash Equilibrium (NE) plays a crucial role in game theory. The relaxation method in conjunction with the Nikaido–Isoda (NI) function, namely the NI-based relaxation method, has been widely applied to the determination of NE. Genetic Algorithm (GA) with adaptive penalty is introduced and incorporated in the original NI-based relaxation method. The GA enhances the capability in the optimization step for computing the optimum response function. The optimization of the non-convex and non-concave NI function is made possible by GA. The proposed method thus combines the advantageous feature of the GA in its optimization capability and that of the relaxation method in its implementation simplicity together. The applicability of the method is shown through the illustrative examples, including the generalized Nash Equilibrium problem with nonlinear payoff functions and coupled constraints, the game with multiple strategic variables for individual players, and the non-differentiable payoff functions. All test example results suggest the appropriate crossover and mutation rate to be 0.05 and 0.002 for use in GA. These numbers are closed to the recommended values by DeJong. The proposed method shows its capability of finding correct NEs in all test examples.
Highlights
Martins FerreiraThe determination of Nash Equilibrium (NE) is the subject of interest in game theory.Among the NE finding methods, the relaxation method that is applied in conjunction with the Nikaido–Isoda (NI) function [1], referred to as the NI-based relaxation method, has been widely recognized [2]
It is proved to converge to an NE for a wide class of problems [5,6]
The co-evolutionary approach was later combined with a ranking technique called Nash non-dominated sorting (NNDS) for the determination of NE [12]
Summary
The determination of Nash Equilibrium (NE) is the subject of interest in game theory. The co-evolutionary approach was later combined with a ranking technique called Nash non-dominated sorting (NNDS) for the determination of NE [12]. The motivation of this work is to extend the NI-based relaxation method with its simplicity of implementation to a novel method that can cope with non-linear, nondifferentiable, and combinatorial forms of payoff and NI functions. Considering the potential of GA in detecting the NE from the afore-reviewed literature and its ability in solving non-linear, non-differentiable, and combinatorial forms of payoff and NI functions, GA will be employed in the present paper as the optimization tool. The novelty of this paper is it is the first time that presents an NI-based hybrid GA and relaxation method for finding Nash Equilibrium.
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