A fixed point approach for computing actuarially fair Pareto optimal risk-sharing rules

The article 'A fixed point approach for computing actuarially fair Pareto optimal risk-sharing rules' explores the complexities of risk-sharing mechanisms where participants pool their risks to share losses. Traditional linear risk-sharing rules, which allocate losses based on expected values, fail to consider individual risk variabilities and aversions. The authors introduce a non-linear risk-sharing rule, the conditional mean risk-sharing rule, which considers the conditional expectation of losses given the total loss. The main challenge is to determine an ex-ante rule that is both actuarially fair and Pareto optimal. Actuarial fairness ensures that each participant's expected loss remains unchanged before and after risk-sharing, while Pareto optimality means no other rule can improve the financial situation of all participants without strictly benefiting at least one. The article establishes a one-to-one correspondence between actuarially fair Pareto optimal (AFPO) risk-sharing rules and fixed points of a specific function. A fast numerical algorithm is provided to compute these rules, with proven convergence. The model used is a one-period model with a finite number of von Neumann–Morgenstern participants, considering their preferences and the joint distribution of individual risks. The fixed-point approach, extensively used in economics and reinforcement learning, is adapted to ensure the existence and uniqueness of solutions. The article also provides numerical illustrations and discusses the comparability of AFPO risk-sharing rules in the convex order sense. This work is significant for its practical applications in insurance and finance, offering a robust method to compute fair and efficient risk-sharing rules.