Robust Decisions under Risk for Imprecise Probabilities

In this paper we analyze robust approaches to decision making under uncertainty where the expected outcome is maximized but the probabilities are known imprecisely. A conservative robust approach takes into account any probability distribution thus leading to the notion of robustness focusing on the worst case scenario and resulting in the max-min optimization. We consider softer robust models allowing the probabilities to vary only within given intervals. We show that the robust solution for only upper bounded probabilities becomes the tail mean, known also as the conditional value-at-risk (CVaR), with an appropriate tolerance level. For proportional upper and lower probability limits the corresponding robust solution may be expressed by the optimization of appropriately combined the mean and the tail mean criteria. Finally, a general robust solution for any arbitrary intervals of probabilities can be expressed with the optimization problem very similar to the tail mean and thereby easily implementable with auxiliary linear inequalities.