Preference robust models in multivariate utility-based shortfall risk minimization

Utility-based shortfall risk measure (SR) has received increasing attentions over the past few years. Recently Delage et al. [Shortfall Risk Models When Information of Loss Function Is Incomplete, GERAD HEC, Montreal, 2018] consider a situation where a decision maker's true loss function in the definition of SR is unknown but it is possible to elicit a set of plausible utility functions with partial information and consequently propose a robust formulation of SR based on the worst-case utility function. In this paper, we extend this new stream of research to a multi-attribute prospect space since multi-attribute decision-making problems are ubiquitous in practical applications. Specifically, we introduce a preference robust multivariate utility-based shortfall risk measure (PRMSR) and demonstrate that it is law invariant and convex. We then apply the PRMSR to an optimal decision-making problem where the objective is to minimize the PRMSR of a vector-valued cost function and propose some numerical scheme for solving the resulting optimization problem in the case when the underlying exogenous uncertainty is finitely distributed. Finally, we discuss statistical robustness of the PRMSR based optimization model by examining qualitative stability of the estimator of the optimal value obtained with potentially contaminated data. A case study is carried out to examine the performance of the proposed robust model and numerical scheme.