Optimal Strategies Trajectory with Multi-Local-Worlds Graph

This paper constructs a non-cooperative/cooperative stochasticdifferential game model to prove that the optimal strategies trajectory ofagents in a system with a topological configuration of a Multi-Local-Worldgraph would converge into a certain attractor if the system’s configuration isfixed. Due to the economics and management property, almost all systems aredivided into several independent Local-Worlds, and the interaction betweenagents in the system is more complex. The interaction between agents inthe same Local-World is defined as a stochastic differential cooperativegame;conversely, the interaction between agents in different Local-Worldsis defined as a stochastic differential non-cooperative game. We construct anon-cooperative/cooperative stochastic differential game model to describethe interaction between agents. The solutions of the cooperative and noncooperativegames are obtained by invoking corresponding theories, and thena nonlinear operator is constructed to couple these two solutions together.At last, the optimal strategies trajectory of agents in the system is proven toconverge into a certain attractor, which means that strategies trajectory arecertainty as time tends to infinity or a large positive integer. It is concluded thatthe optimal strategy trajectory with a nonlinear operator of cooperative/noncooperativestochastic differential game between agents can make agentsin a certain Local-World coordinate and make the Local-World paymentmaximize, and can make the all Local-Worlds equilibrated;furthermore, theoptimal strategy of the coupled game can converge into a particular attractorthat decides the optimal property.