[Paper Review] Pareto and Bowley Reinsurance Games in Peer-to-Peer Insurance

Tim J. Boonen, Kenneth Tsz Hin Ng+2 more|arXiv (Cornell University)|Feb 15, 2026

Game Theory and Voting Systems0 citations

TL;DR

The paper designs two contract approaches (Pareto and Bowley) for a P2P insurance pool with a reinsurer, derives closed-form optimal contracts, and compares welfare outcomes, showing Bowley is never Pareto-optimal and typically yields lower total welfare.

ABSTRACT

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.

Motivation & Objective

• Motivate decentralized P2P insurance with an external reinsurance layer and study strategic interaction between plan manager and reinsurer.
• Develop two contract designs (Pareto and Bowley) and derive their closed-form optimal contracts.
• Assess welfare implications, coalitional stability, and individual rationality under the two designs.
• Investigate effects of safety loadings, single-loading restrictions, and reinsurer risk aversion on outcomes.

Proposed method

• Model the P2P contract as a mean-variance risk-sharing problem with a plan manager and a reinsurer integrated via a reinsurance loading p and safety loadings η.
• Define risk mutualization A among members and a proportional reinsurance scheme p with actuarial fairness and zero-conserving conditions.
• Formulate member, plan manager, and reinsurer preferences using mean-variance disutility with explicit expressions for ρi(A), u_i(A,p,η), and ρR(p).
• Set up joint optimization Problem (5) to minimize the sum of member and reinsurer disutilities subject to A μ + D(μ) p = μ and 1^T A + p^T = 1^T.
• Solve for the JP-optimal A* and p* in closed form, and characterize when p* lies in (0,1)^n; derive η* via a coalition game and core concepts.
• Formulate Bowley design as a leader–follower game, solving sequential subproblems to obtain a unique Bowley-optimal contract and verify IR constraints.

Experimental results

Research questions

• RQ1 What are the JP-optimal (Pareto) risk-sharing and reinsurance contracts in a P2P scheme with a reinsurer?
• RQ2 What is the Bowley-optimal contract when the reinsurer is the leader and the plan manager responds, and under what conditions are IR constraints satisfied?
• RQ3 How do the Pareto and Bowley designs compare in terms of total welfare and Pareto efficiency?
• RQ4 What impact do single-loading restrictions and the reinsurer’s risk aversion have on allocations and welfare?
• RQ5 How does coalitional stability (core non-emptiness) influence the selection of JP-optimal contracts?

Key findings

• Pareto design yields multiple JP-optimal contracts that can be refined by coalitional stability; the core is non-empty.
• Bowley design produces a unique optimal contract, but it is never Pareto optimal and typically yields lower total welfare.
• Reinsurance improves welfare relative to no-reinsurance, especially under Pareto designs and when the reinsurer is less risk-averse.
• Single-loading restrictions tend to favor higher-risk members, potentially increasing welfare for that group while affecting others.
• With a less risk-averse reinsurer, Bowley contracts under single-loading restrictions can yield larger total welfare compared to unrestricted Bowley designs.