Optimal Risk-Sharing Rules in Network-based Decentralized Insurance

TL;DR

This work develops a network-based theory of actuarially fair, linear risk-sharing for connected graphs, extending prior complete-graph results to general networks. It derives a unique optimal risk-sharing matrix $\boldsymbol{A}_*$ under the constraint that only friends share risk, with a closed-form that incorporates a sparse correction matrix $\boldsymbol{\Gamma}$ accounting for non-edges; when friends share equal shares, the solution takes the form $\boldsymbol{\hat{A}}=\mathbf{I}-\hat{c}\mathbf{L}\mathbf{M}^{-1}$, linking to the graph Laplacian $\mathbf{L}$ and offering a clean interpretation in regular, iid settings. The paper also provides necessary and sufficient nonnegativity conditions for the entries of $\boldsymbol{A}_*$ and $\boldsymbol{\hat{A}}$, illustrates effects through complete, edge-removed, and barbell networks, and discusses how network design can mitigate negative allocations. Collectively, these results establish a tractable framework for deploying P2P risk-sharing on realistic networks, with practical implications for risk pooling and contract design in decentralized insurance contexts.

Abstract

This paper studies decentralized risk-sharing on networks. In particular, we consider a model where agents are nodes in a given network structure. Agents directly connected by edges in the network are referred to as friends. We study actuarially fair risk-sharing under the assumption that only friends can share risk, and we characterize the optimal signed linear risk-sharing rule in this network setting. Subsequently, we consider a special case of this model where all the friends of an agent take on an equal share of the agent's risk, and establish a connection to the graph Laplacian. Our results are illustrated with several examples.

Figures (2)

• Figure 1: Many works on P2P insurance either perform non-olet risk pooling (represented by the star graph) or unrestricted risk-sharing (represented by the complete graph). In this work, we consider networks with general structures such as the Barbell graph.
• Figure 2: A heat map visualization of the optimal $\boldsymbol{A}_*$ for a fully-connected network (left), and $\boldsymbol{A}_*$ for the barbell network (right).

Theorems & Definitions (25)

• Definition 1.1
• Definition 1.2
• Theorem 1.1: Feng, Liu, Taylor feng2023peer
• Definition 2.1: Only friends share risk
• Theorem 2.1: Only friends share risk
• Remark 2.1
• Remark 2.2
• Definition 2.2: Friends take an equal share of risk
• Theorem 2.2: Friends take an equal share of risk
• proof : Proof of Theorem \ref{['thmoptshare']}