Preference Robust Optimization for Choice Functions on the Space of CDFs

Abstract

In this paper, we consider decision-making problems where the decision maker’s (DM) utility/risk preferences are ambiguous but can be described by a general class of choice functions defined over the space of cumulative distribution functions (CDFs) of random prospects. These choice functions are assumed to satisfy two basic properties: (i) monotonicity w.r.t. the order on CDFs and (ii) quasiconcavity. We propose a maximin preference robust optimization (PRO) model where the optimal decision is based on the robust choice function from a set of choice functions elicited from available information on the DM’s preferences. The current univariate utility PRO models are fundamentally based on Von Neumann-Morgenstein’s (VNM) expected utility theory. Our new robust choice function model effectively generalizes them to one which captures common features of VNM’s theory and Yaari’s dual theory of choice. To evaluate our robust choice functions, we characterize the quasiconcave envelope of L−Lipschitz functions of a set of points. Subsequently, we propose two numerical methods for the DM’s PRO problem: a projected level function method and a level search method. We apply our PRO model and numerical methods to a portfolio optimization problem and report test results.

Keywords: preference elicitation, quasiconcave choice functions, multi-attribute decision-making, prefer- ence robust optimization, level search method