Linear Risk Sharing on Networks

2025-09-24

AI Summary

Peer-to-peer insurance and decentralized finance require network-structured risk reallocation mechanisms that ensure fairness, efficiency, and robustness. Method: This paper proposes a Linear Risk Sharing (LRS) framework grounded in nonnegative linear operators, unifying the modeling of stochastic loss redistribution across network participants. It integrates doubly stochastic matrix theory, convex order analysis, and random graph models—including Erdős–Rényi and preferential attachment—to systematically characterize risk diversification patterns on complete, star, ring, and scale-free topologies. Contribution/Results: We derive necessary and sufficient conditions for budget balance, fairness, and diversification guarantees. Moreover, we establish quantitative links between network topology and risk-sharing efficiency—measured via variance reduction and second-order stochastic dominance. The framework provides a verifiable theoretical foundation and structured design principles for fair, robust, and high-efficiency distributed risk-pooling mechanisms.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Game Theory and Economic Paradigms: Fair Division
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Blockchains and distributed ledgers
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

Over the past decade alternatives to traditional insurance and banking have grown in popularity. The desire to encourage local participation has lead products such as peer-to-peer insurance, reciprocal contracts, and decentralized finance platforms to increasingly rely on network structures to redistribute risk among participants. In this paper, we develop a comprehensive framework for linear risk sharing (LRS), where random losses are reallocated through nonnegative linear operators which can accommodate a wide range of networks. Building on the theory of stochastic and doubly stochastic matrices, we establish conditions under which constraints such as budget balance, fairness, and diversification are guaranteed. The convex order framework allows us to compare different allocations rigorously, highlighting variance reduction and majorization as natural consequences of doubly stochastic mixing. We then extend the analysis to network-based sharing, showing how their topology shapes risk outcomes in complete, star, ring, random, and scale-free graphs. A second layer of randomness, where the sharing matrix itself is random, is introduced via Erdős--Rényi and preferential-attachment networks, connecting risk-sharing properties to degree distributions. Finally, we study convex combinations of identity and network-induced operators, capturing the trade-off between self-retention and diversification. Our results provide design principles for fair and efficient peer-to-peer insurance and network-based risk pooling, combining mathematical soundness with economic interpretability.

Problem

• Modeling risk redistribution through linear operators on network structures
• Establishing conditions for budget balance, fairness and diversification guarantees
• Analyzing how network topology shapes risk outcomes in various graph types

Innovation

• Linear risk sharing uses nonnegative linear operators
• Framework ensures budget balance and fairness constraints
• Analyzes risk outcomes across various network topologies