A novel Cooperative-Competitive Differential Graphical Game with distributed global Nash solution

This paper investigates a differential graphical game problem for fixed communication topology networks, where simultaneous mutual Cooperative-Competitive (CC) effects exist among the agents. The objective of such a system is to design distributed control policies that synchronize agents to a leader's trajectory and achieve a Nash equilibrium. We point out that existing frameworks in multi-agent and game theory fields do not explicitly define and formulate co-existed CC effects between agents. To fill this gap, we propose a novel differential graphical game named Cooperative-Competitive Differential Graphical Game (CCDGG), where such effects are modeled for each agent by introducing a new parameter called "Collective Cooperative-Competitive Impact Factor (CCCIF)". It is shown that the performance indices proposed in the literature cannot provide a distributed solution to the CCDGG. We propose novel performance indices which generate decoupled Hamilton-Jacobi (HJ) equations when agents use the best response strategy. This will consequently lead to a distributed Nash solution for CCDGG. The closed-loop stability and Nash equilibrium properties for CCDGG are also analyzed and proved. Finally, derived theoretical results are validated through numerical simulations.