Optimal Strategies Trajectory with Multi-Local-Worlds Graph

Abstract

This paper constructs a non-cooperative/cooperative stochastic differential game model to prove that the optimal strategies trajectory of agents in a system with a topological configuration of a Multi-Local-World graph would converge into a certain attractor if the system’s configuration is fixed. Due to the economics and management property, almost all systems are divided into several independent Local-Worlds, and the interaction between agents in the system is more complex. The interaction between agents in the same Local-World is defined as a stochastic differential cooperative game; conversely, the interaction between agents in different Local-Worlds is defined as a stochastic differential non-cooperative game. We construct a non-cooperative/cooperative stochastic differential game model to describe the interaction between agents. The solutions of the cooperative and non-cooperative games are obtained by invoking corresponding theories, and then a nonlinear operator is constructed to couple these two solutions together. At last, the optimal strategies trajectory of agents in the system is proven to converge into a certain attractor, which means that strategies trajectory are certainty as time tends to infinity or a large positive integer. It is concluded that the optimal strategy trajectory with a nonlinear operator of cooperative/non-cooperative stochastic differential game between agents can make agents in a certain Local-World coordinate and make the Local-World payment maximize, and can make the all Local-Worlds equilibrated; furthermore, the optimal strategy of the coupled game can converge into a particular attractor that decides the optimal property.