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# Optimal Strategies Trajectory with Multi-Local-Worlds Graph

Xiang Yu1, Chonghua Wang2, Xiaojing Zheng3,*, Chaoyu Zeng4, Brij B. Gupta5,6,7

1 School of Electronics and Information Engineering, Taizhou University, Taizhou, Zhejiang, 318000, China
2 China Industrial Control Systems Cyber Emergency Response Team, Beijing, 100040, China
3 School of Economics and Management, Weifang University of Science and Technology, Shouguang, Shandong, 262700, China
4 National Computer System Engineering Research Institute of China, Beijing, 102699, China
5 Department of Computer Science and Information Engineering, Asia University, Taiwan, Taichung, 413, China
6 Research and Innovation Department, Skyline University College, Sharjah, P.O. Box 1797, United Arab Emirates
7 Staffordshire University, Stoke-on-Trent, ST4 2DE, UK

* Corresponding Author: Xiaojing Zheng. Email:

Computers, Materials & Continua 2023, 75(1), 2079-2099. https://doi.org/10.32604/cmc.2023.034118

## 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.

## Keywords

X. Yu, C. Wang, X. Zheng, C. Zeng and B. B. Gupta, "Optimal strategies trajectory with multi-local-worlds graph," Computers, Materials & Continua, vol. 75, no.1, pp. 2079–2099, 2023.

This work is licensed under a Creative Commons Attribution 4.0 International License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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