Co-opetition in Reinsurance Markets: When Pareto Meets Stackelberg and Nash

Abstract

We develop and solve a two-layer game to model co-opetition, a strategic combination of competition and cooperation, in a reinsurance market consisting of one primary insurer and two reinsurers, in which all players are equipped with mean-variance preferences and the reinsurance contracts are priced under the variance premium principle. The insurer negotiates reinsurance contracts with the two reinsurers simultaneously, modeled by two Stackelberg games, and the two reinsurers compete for business from the same insurer by setting their own pricing rules, modeled by a non-cooperative Nash game. The combined Stackelberg-Nash game constitutes the first layer of the game model and endogenously determines the risk assumed by each reinsurer. The two reinsurers, then, participate in a cooperative risk-sharing game, forming the second layer of the game model, and seek Pareto-optimal risk-sharing rules. We obtain equilibrium strategies in closed form for both layers. The equilibrium of the Stackelberg-Nash game consists of two proportional reinsurance contracts, with the more risk-averse reinsurer assuming a smaller portion of the insurer's total risk. The Pareto-optimal risk-sharing rules further dictate that the more risk-averse reinsurer transfers a portion of its assumed risk to the less risk-averse reinsurer, at the cost of a positive side payment.

Keywords: Co-opetition, Risk sharing, Pareto optimality, Stackelberg reinsurance game, Nash equilibrium