Approximate Solution for Three-Player Mixed-Zero-Sum Nonlinear Game via ADP Structure

Abstract

In this paper, a three-player mixed-zero-sum game situation with nonlinear dynamics is proposed, and an approximate dynamic programming (ADP) learning scheme is used to solve the proposed problem. First, the problem formulation is presented. A value function for player 1 and 2 nonzero-sum game is constructed, another value function for player 1 and 3 zero-sum game is presented for three-player nonlinear game system. Because of the difficulty to solve the nonlinear Hamilton-Jacobi (HJ) equation, the single-layer critic neural networks are used to approximate the optimal value functions. Then the approximated critic neural networks (NNs) are directly used to learn the optimal solutions for three-player mixed-zero-sum nonlinear game. A novel adaptive law with the estimation performance index is proposed to estimate the unknown coefficient vector. Finally, a simulation example is presented to illustrate the proposed methods.