Sharing the value‐at‐risk under distributional ambiguity

This paper considers the problem of risk sharing, where a coalition of homogeneous agents, each bearing a random cost, aggregates their costs, and shares the value‐at‐risk of such a risky position. Due to limited distributional information in practice, the joint distribution of agents' random costs is difficult to acquire. The coalition, being aware of the distributional ambiguity, thus evaluates the worst‐case value‐at‐risk within a commonly agreed ambiguity set of the possible joint distributions. Through the lens of cooperative game theory, we show that this coalitional worst‐case value‐at‐risk is subadditive for the popular ambiguity sets in the distributionally robust optimization literature that are based on (i) convex moments or (ii) Wasserstein distance to some reference distributions. In addition, we propose easy‐to‐compute core allocation schemes to share the worst‐case value‐at‐risk. Our results can be readily extended to sharing the worst‐case conditional value‐at‐risk under distributional ambiguity.