Preference Elicitation and Robust Optimization with Multi-Attribute Quasi-Concave Choice Functions

TL;DR

This paper introduces a robust preference model for multi-attribute prospects using quasi-concave choice functions, enabling decision-making under ambiguity with applications to security budget allocation.

Contribution

It develops a novel robust optimization framework based on quasi-concave choice functions, extending preference elicitation and tractable formulations for ambiguity in multi-attribute decision problems.

Findings

Supports multi-attribute expected utility and S-shaped risk preferences.

Provides mixed integer linear programming and convex risk minimization formulations.

Demonstrates effectiveness through security budget allocation experiments.

Abstract

Decision maker's preferences are often captured by some choice functions which are used to rank prospects. In this paper, we consider ambiguity in choice functions over a multi-attribute prospect space. Our main result is a robust preference model where the optimal decision is based on the worst-case choice function from an ambiguity set constructed through preference elicitation with pairwise comparisons of prospects. Differing from existing works in the area, our focus is on quasi-concave choice functions rather than concave functions and this enables us to cover a wide range of utility/risk preference problems including multi-attribute expected utility and $S$-shaped aspirational risk preferences. The robust choice function is increasing and quasi-concave but not necessarily translation invariant, a key property of monetary risk measures. We propose two approaches based respectively on the support functions and level functions of quasi-concave functions to develop tractable formulations of the maximin preference robust optimization model. The former gives rise to a mixed integer linear programming problem whereas the latter is equivalent to solving a sequence of convex risk minimization problems. To assess the effectiveness of the proposed robust preference optimization model and numerical schemes, we apply them to a security budget allocation problem and report some preliminary results from experiments.