Axiomatic characterizations of some simple risk-sharing rules

TL;DR

This paper provides axiomatic characterizations of simple risk-sharing rules, clarifying their foundational principles and introducing new rules through a formal framework that emphasizes fairness, anonymity, and non-punitiveness.

Contribution

It formalizes key principles of risk sharing and characterizes several simple rules, including the uniform rule, and introduces new classes of risk-sharing rules within a unified axiomatic framework.

Findings

The uniform risk-sharing rule is uniquely characterized by reshuffling and anonymity properties.

Introduces $q$-proportional and $(q_1,q_2)$-linear risk-sharing rules.

Develops a comprehensive axiomatic framework for understanding risk-sharing principles.

Abstract

In this paper, we present axiomatic characterizations of some simple risk-sharing (RS) rules, such as the uniform, the mean-proportional and the covariance-based linear RS rules. These characterizations make it easier to understand the underlying principles when applying these rules. Such principles typically include maintaining some degree of anonymity regarding participants' data and/or incident-specific data, adopting non-punitive processes and ensuring the equitability and fairness of risk sharing. By formalizing key concepts such as the reshuffling property, the source-anonymous contributions property and the strongly aggregate contributions property, along with their generalizations, we develop a comprehensive framework that expresses these principles clearly and defines the relevant rules. To illustrate, we demonstrate that the uniform RS rule, a simple mechanism in which risks are shared equally, is the only RS rule that satisfies both the reshuffling property and the source-anonymous contributions property. This straightforward axiomatic characterization of the uniform RS rule serves as a foundation for exploring similar principles in two broad classes of risk-sharing rules, which we baptize the $q$-proportional RS rules and the $(q_1,q_2)$-based linear RS rules, respectively. This framework also allows us to introduce novel particular RS rules, such as the scenario-based RS rules.