Hierarchical Cooperation in LQ Multi-Population Mean Field Game With Its Application to Opinion Evolution

Cooperation and non-cooperation are two fundamental topics in games, but in multi-population mean-field games (MPMFGs), cooperation may occur at the individual level (intra-population) or at the population level (inter-population) or at the both levels, which leads to different interesting results. In this paper, model frameworks based on linear quadratic mean-field games are developed for the above three possible hierarchical cooperation cases. The optimal controls are derived via the adjoint method and solved by forward-backward ordinary differential equations. Based on the analysis of asymmetric Riccati differential equations, sufficient conditions for the existence and uniqueness of optimal solutions are given. Then, the price of anarchy in different situations are defined to measure the efficiency of the proposed MPMFG models. Finally, simulation experiments on the opinion evolutions of two populations in social networks are performed to demonstrate the feasibility and effectiveness of the above models. Specially, it is presented that the experimental results of the intra-population non-cooperative and inter-population cooperative MPMFG model provide an explanation of the behavior of coalition governments over electoral cycles studied in the existing literature.