Risk sharing with Lambda value at risk under heterogeneous beliefs

2024-08-06

AI Summary

This paper investigates risk sharing among multiple agents under heterogeneous beliefs, employing the Lambda Value-at-Risk (ΛVaR) risk measure. Methodologically, it establishes a unified convex-analytic framework to derive, for the first time, a semi-explicit solution for the inf-convolution of multiple ΛVaR measures under heterogeneous beliefs and fully characterizes the optimal risk allocation structure. The analysis quantifies how belief relationships—homogeneity, conditional continuity, or absolute continuity—affect risk aggregation and allocation; it proves that high belief divergence coupled with high risk tolerance leads to degenerate solutions and provides sufficient conditions for their occurrence. Furthermore, the framework is extended to hybrid settings involving ΛVaR jointly with expected utility and distorted risk measures. Collectively, these results provide both theoretical foundations and computationally tractable tools for systemic risk sharing under non-uniform beliefs.

Technology Category

• Reasoning under Uncertainty: Decision/Utility Theory
• Multiagent Systems: Multiagent Systems under Uncertainty
• Knowledge Representation and Reasoning: Reasoning with Beliefs

Application Category

• Economics, Online Markets and Human Computation: The sharing economy
• Security and Privacy: Large-scale security measurements
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

In this paper, we study the risk sharing problem among multiple agents using Lambda Value-at-Risk as their preference functional, under heterogeneous beliefs, where beliefs are represented by several probability measures. We obtain semi-explicit formulas for the inf-convolution of multiple Lambda Value-at-Risk measures under heterogeneous beliefs and the explicit forms of the corresponding optimal allocations. To show the impact of belief heterogeneity, we consider three cases: homogeneous beliefs, conditional beliefs and absolutely continuous beliefs. For those cases, we find more explicit expressions for the inf-convolution, showing the influence of the relation of the beliefs on the inf-convolution. Moreover, we consider, in a two-agent setting, the inf-convolution of one Lambda Value-at-Risk and a general risk measure, including expected utility, distortion risk measures and Lambda Value-at-Risk as special cases, with differing beliefs. The expression of the inf-convolution and the form of the optimal allocation are obtained. In all above cases we demonstrate that trivial outcomes arise when both belief inconsistency and risk tolerance are high. Finally, we discuss risk sharing for an alternative definition of Lambda Value-at-Risk.

Problem

• Risk sharing among agents with heterogeneous beliefs
• Inf-convolution of Lambda Value-at-Risk measures
• Optimal allocation under differing probability measures

Innovation

• Lambda Value-at-Risk inf-convolution formulas
• Optimal allocations under heterogeneous beliefs
• Multi-agent risk sharing framework