Strange attractors and nontrivial solutions in games with three players

Abstract

Two-player games have been sufficiently researched, but three-player games have a much greater wealth of solutions, in particular, nonstationary solutions. However, unsteady solutions of three-player games have not been studied in the literature. On the basis of numerical modeling, the dynamic behavior of a 2 × 2 × 2 game with three players is investigated in the range of payoff matrices from -1 to 1. The system of dynamic equations includes three variables (probabilities of pure strategies) and twelve parameters associated with cubic payoff matrices of players. The phase space of a three-player game is a cube (probabilities vary from 0 to 1). It was found that for some payoff matrices, there are no stable solutions in the system. In addition, the structure of the phase space is complex: the volume of the cube is a fractal repulsive region from which the phase trajectories leave, a strange attractor is located near the edge corresponding to the solution of the "node-center" type, and another strange attractor is a chaotic motion between adjacent vertices of the cube. The obtained dynamic properties of the game can be used for making decisions in economics, negotiations between three persons, and modeling relationships between biological species.