Distributionally Robust Optimization Under Distorted Expectations

Distributionally robust optimization (DRO) has arisen as an important paradigm for addressing the issue of distributional ambiguity in decision optimization. In the case in which a decision maker is not risk neutral, the most common scheme applied in DRO for capturing the risk attitude is to employ an expected utility functional. In this paper, we propose to address a decision maker’s risk attitude in DRO by following an alternative scheme known as dual expected utility. In this scheme, a distortion function is applied to convert physical probabilities into subjective probabilities so that the resulting expectation, called a distorted expectation, captures the decision maker’s risk attitude. Unlike an expected utility functional, which is linear in probability, in the dual scheme, the distorted expectation is generally nonlinear in probability. We distinguish DRO based on distorted expectations by coining the term “distributionally robust distortion risk optimization” (DRDRO) and show that DRDRO problems can be equally, if not more, tractable to solve as DRO problems based on utility functionals. Our tractability results hold for any distortion function, and hence, our scheme provides more flexibility in capturing more realistic forms of risk attitudes. These include, as an important example, the inverse S-shaped distortion functionals in cumulative prospect theory. We demonstrate through a numerical example that a production manager who overly weights “very good” and “very bad” outcomes may act as if the manager is risk averse when distributional ambiguity is considered.