Lambda Value-at-Risk under ambiguity and risk sharing

2025-11-01

AI Summary

This paper investigates the Lambda value-at-risk (ΛVaR) and its multi-agent risk-sharing problem under model uncertainty, characterized by a family of probability measures. To address the lack of robustness of conventional ΛVaR under ambiguity, we propose a novel capacity-theoretic extension: first establishing an intrinsic link between ΛVaR and monotone capacities; introducing lower-set families to obtain a tractable analytical representation under monotone Λ functions; and proving closure under inf-convolution. Ambiguity sets are constructed via φ-divergences and likelihood-ratio constraints, unifying ΛVaR and Choquet quantiles. The framework yields explicit formulas for robust ΛVaR and the optimal risk allocation, providing both theoretical foundations and implementable tools for fair and efficient multi-agent risk sharing under model uncertainty.

Technology Category

• Reasoning under Uncertainty: Uncertainty Representations
• Multiagent Systems: Multiagent Systems under Uncertainty
• Planning, Routing, and Scheduling: Planning under Uncertainty

Application Category

• Economics, Online Markets and Human Computation: The sharing economy
• Responsible Web: Machine-in-the-loop, human agency and autonomy
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

In this paper, we investigate the Lambda Value-at-Risk ($Λ$VaR) under ambiguity, where the ambiguity is represented by a family of probability measures. We establish that for increasing Lambda functions, the robust (i.e., worst-case) $Λ$VaR under such an ambiguity set is equivalent to $Λ$VaR computed with respect to a capacity, a novel extension in the literature. This framework unifies and extends both traditional $Λ$VaR and Choquet quantiles (Value-at-Risk under ambiguity). We analyze the fundamental properties of this extended risk measure and establish a novel equivalent representation for $Λ$VaR under capacities with monotone Lambda functions in terms of families of downsets. Moreover, explicit formulas are derived for robust $Λ$VaR when ambiguity sets are characterized by $φ$-divergence and the likelihood ratio constraints, respectively. We further explore the applications in risk sharing among multiple agents. We demonstrate that the family of risk measures induced by families of downsets is closed under inf-convolution. In particular, we prove that the inf-convolution of $Λ$VaR with capacities and monotone Lambda functions is another$Λ$VaR under a capacity. The explicit forms of optimal allocations are also derived. Moreover, we obtain more explicit results for risk sharing under ambiguity sets characterized by $φ$-divergence and likelihood ratio constraints. Finally, we explore comonotonic risk-sharing for $Λ$VaR under ambiguity.

Problem

• Extending Lambda Value-at-Risk to handle ambiguity using probability measures
• Establishing equivalence between robust ΛVaR and capacity-based ΛVaR
• Analyzing risk sharing applications and optimal allocations under ambiguity

Innovation

• Extends ΛVaR to ambiguity using capacity representation
• Derives explicit formulas for φ-divergence ambiguity sets
• Proves inf-convolution closure for risk sharing optimization