Equilibrium reinsurance strategies for n insurers under a unified competition and cooperation framework

We propose a unified competition and cooperation framework for n insurers and investigate the resulting reinsurance game problem. Each insurer's surplus is assumed to be a diffusion process. Each insurer can purchase the proportional reinsurance to reduce his claim risk, and the reinsurance premium is determined via the variance value principle. The objective of each insurer is to find a reinsurance strategy so as to maximize the expected utility of his terminal payoff. We establish a Hamilton–Jacobi–Bellman (HJB) equation and the corresponding verification theorem. Furthermore, we derive the explicit solutions for both the equilibrium reinsurance strategy and the value function by solving the HJB equation. Finally, numerical experiments are carried out to illustrate the influences of model parameters such as the size of a group, the number of groups on the equilibrium reinsurance strategy. The numerical results reveal some similarities and differences between the competition case and the cooperation case, and the detailed effects of different competition and cooperation patterns, which provide useful insights for reinsurance in reality.