Preference Robust Modified Optimized Certainty Equivalent

Ben-Tal and Teboulle [Management Sci., 32 (1986), pp. 1445--1466] introduce the concept of optimized certainty equivalent (OCE) of an uncertain outcome as the maximum present value of a combination of the cash to be taken out from the uncertain income at present and the expected utility value of the remaining uncertain income. In this paper, we consider two variations of the OCE. First, we introduce a modified OCE by maximizing the combination of the utility of the cash and the expected utility of the remaining uncertain income so that the combined quantity is in a unified utility value. Second, we consider a situation where the true utility function is unknown but it is possible to use partially available information to construct a set of plausible utility functions. To mitigate the risk arising from the ambiguity, we introduce a robust model where the modified OCE is based on the worst-case utility function from the ambiguity set. In the case when the ambiguity set of utility functions is constructed by a Kantorovich ball centered at a nominal utility function, we show how the modified OCE and the corresponding worst-case utility function can be identified by solving two linear programs alternatively. We also show that the robust modified OCE is statistically robust in a data-driven environment where the underlying data are potentially contaminated. Some preliminary numerical results are reported to demonstrate the performance of the modified OCE and the robust modified OCE model.