This paper studies how small peer-to-peer (P2P) insurance groups can agree on who takes what risk and who pays what. The authors model a P2P reinsurance platform and several individual peers as risk-averse decision makers who bargain over a contract. Each party compares any proposed deal to its outside option. For peers that outside option is a regular centralized insurer. The central idea is to use an asymmetric Nash bargaining rule — a formal way to split gains from cooperation that can reflect different bargaining strengths — while imposing a simple fairness rule for prices.

Price fairness means each peer’s expected contribution is proportional to its own expected loss, with a common extra charge (a loading) applied to everyone. This follows the standard expected-value premium idea and limits cross-subsidy between high- and low-risk members. The paper gives an axiomatic justification for using the Nash bargaining solution: it satisfies Pareto optimality (no alternative makes everyone better off), it strictly improves on each party’s disagreement point, and it has two technical invariances that are useful in voluntary small pools.

On the technical side, the authors show the bargaining problem admits an optimal contract that is unique. They derive conditions that describe the contract in three regimes: full reinsurance (the pool or platform absorbs most losses), partial reinsurance, and zero reinsurance (peers keep their own losses). They also study subgroup formation. The paper develops an ex-post stability test and tractable sufficient conditions that rule out any subgroup of members forming a better deal, at least when the P2P reinsurer must be part of any subgroup. For some special cases, such as exponential utility (a common technical assumption for risk-averse behavior), the formulas simplify and let the authors give clearer rules for how risk and premium split.