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

Tim J. Boonen, Kenneth Tsz Hin Ng, Tak Wa Ng, Thai Nguyen

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.

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

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.
Paper Structure (42 sections, 16 theorems, 96 equations, 4 figures, 6 tables)

This paper contains 42 sections, 16 theorems, 96 equations, 4 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Contributions
  3. Related Literature
  4. Model Formulations
  5. Basic setting and risk sharing
  6. Agents' preferences
  7. Individual rationality
  8. Pareto Design
  9. Characterization of Joint-Pareto optimality
  10. Cooperative game study on premium decision
  11. Coalitional stability and the core
  12. Practical premium suggestions
  13. Bowley Design
  14. Problem formulation
  15. Plan manager's follower problem
  16. ...and 27 more sections

Key Result

Lemma 3.2

The matrix $\overline{\bm{M}}$ is positive definite and thus invertible.

Figures (4)

  • Figure 1: Illustration of the risk management mechanism.
  • Figure 2: Summary of the sequential game between the reinsurer and the P2P insurance plan manager. The reinsurer selects the risk loading $\eta$, and the insurer the proportional reinsurance strategy $\bm{p}(\bm{\eta})$.
  • Figure 3: Members' reinsurance strategies of the four contracts with different $\gamma_R$.
  • Figure 4: Comparison of total welfare gains among contracts with different $\gamma_R$.

Theorems & Definitions (40)

  • Definition 2.1: Zero conserving
  • Definition 2.2: Actuarial fairness
  • Definition 2.3: P2P insurance contract
  • Definition 3.1
  • Lemma 3.2
  • proof
  • Proposition 3.3
  • proof
  • Theorem 3.4
  • proof
  • ...and 30 more