Pareto and Bowley Reinsurance Games in Peer-to-Peer Insurance

We propose a peer-to-peer (P2P) insurance scheme comprising a risk-sharing pool and a reinsurer. A plan manager determines how risks are allocated among members and ceded to the reinsurer, while the reinsurer sets the reinsurance loading. Our work focuses on the strategic interaction between the plan manager and the reinsurer, and this focus leads to two game-theoretic contract designs: a Pareto design and a Bowley design, for which we derive closed-form optimal contracts. In the Pareto design, cooperation between the reinsurer and the plan manager leads to multiple Pareto-optimal contracts, which are further refined by introducing the notion of coalitional stability. In contrast, the Bowley design yields a unique optimal contract through a leader-follower framework, and we provide a rigorous verification of the individual rationality constraints via pointwise comparisons of payoff vectors. Comparing the two designs, we prove that the Bowley-optimal contract is never Pareto optimal and typically yields lower total welfare. In our numerical examples, the presence of reinsurance improves welfare, especially with Pareto designs and a less risk-averse reinsurer. We further analyze the impact of the single-loading restriction, which disproportionately favors members with riskier losses.