Almost Worst Case Distributions in Multiple Priors Models

A worst case distribution is a minimiser of the expectation of some random payoff within a family of plausible risk factor distributions. The plausibility of a risk factor distribution is quantified by a convex integral functional. This includes the special cases of relative entropy, Bregman distance, and $f$-divergence. An ($\epsilon$-$\gamma$)-almost worst case distribution is a risk factor distribution which violates the plausibility constraint at most by the amount $\gamma$ and for which the expected payoff is not better than the worst case by more than $\epsilon$. From a practical point of view the localisation of almost worst case distributions may be useful for efficient hedging against them. We prove that the densities of almost worst case distributions cluster in the Bregman neighbourhood of a specified function, interpreted as worst case localiser. In regular cases, it coincides with the worst case density, but when the latter does not exist, the worst case localiser is perhaps not even a density. We also discuss the calculation of the worst case localiser, and its dependence on the threshold in the plausibility constraint.