Scale-Invariant Variations of Max Regret

Nic Wilson

Abstract: Max regret is frequently used, in situations when there is uncertainty about a user preference model in a multi-objective optimisation problem, as a measure of how close an alternative is to being necessarily optimal. It is used in a termination condition for a dialogue with a user, and for recommending a compromise solution, and in different ways of generating informative queries. In this paper we consider linear user preference models based on simple weighted sums of the objectives. Unfortunately, max regret lacks a desirable scale-invariance property: changing the units (or the linear scaling) of the objectives can significantly alter the relative values of max regret between alternatives, even though the choice of units is often somewhat arbitrary. In this paper we define variations of max regret in which the regret, of an alternative given a particular user model, is divided by a function expressing a range of utility values. This leads to scale-invariance, and maintains important properties of max regret such as translation-invariance (in contrast with max relative regret). We show how linear programming and extreme points algorithms can be used for computation.