Minimality and comparison of sets of multi-attribute vectors

In a decision-making problem, there is often some uncertainty regarding the user preferences. We assume a parameterised utility model, where in each scenario we have a utility function over alternatives, and where each scenario represents a possible user preference model consistent with the input preference information. With a set A of alternatives available to the decision maker, we can consider the associated utility function, expressing, for each scenario, the maximum utility among the alternatives. We consider two main problems: Firstly, finding a minimal subset of A that is equivalent to it, i.e., that has the same utility function. We show that for important classes of preference models, the set of so-called possibly strictly optimal alternatives is the unique minimal equivalent subset. Secondly, we consider how to compare A to another set of alternatives B, where A and B correspond to different initial decision choices. We derive mathematical results that allow different computational techniques for these two problems, using linear programming, and especially, with a novel approach using the extreme points of the epigraph of the utility function.