Preference Robust Optimization with Quasi-Concave Choice Functions in Multi-Attribute Decision-Making: Characterization and Computation

In behavioural economics, a decision maker's (DM's) preferences are often expressed by a preference functional such as expected utility or a distortion risk measure, which assigns a numerical value to a risky prospect. Preference robust optimization (PRO) is about decision making where the DM's preference functional is ambiguous and the optimal decision is based on the worst-case preference functional from a set of plausible ones constructed from available partial information about the DM's true preferences. In this paper, we propose a choice function (a particular class of preference functionals) based PRO model where the DM's preferences over a prospect space satisfy Von Neumann-Morgenstern's (VNM's) axioms of completeness, monotonicity, and continuity. We concentrate on the class of choice functions which are monotonic, quasi-concave, and multi-attribute. The resulting PRO model is broader than the existing expected utility-based PRO models in that: (a) it captures a broader class of DM's preferences; and (b) it can be effectively applied to multi-attribute decision making problems where the DM's preferences over different attributes are related in a nonlinear manner. We propose a cutting plane-type method for evaluating the worst-case choice function and solve the resulting PRO problem by solving a sequence of convex optimization problems. We examine the behavior and scalability of the proposed model and computational schemes numerically on a multi-portfolio optimization problem and a capital allocation problem.