Extensions of Expected Utility Theory and some Limitations of Pairwise Comparisons

We contrast three decisions rules that extend Expected Utility to contexts where a convex set of probabilities is used to depict uncertainty: -Maximin, Maximality, and -admissibility. The rules extend Expected Utility theory as they require that an option is inadmissible if there is another that carries greater expected utility for each probability in a (closed) convex set. If the convex set is a singleton, then each rule agrees with maximizing expected utility. We show that, even when the option set is convex, this pairwise comparison between acts may fail to identify those acts which are Bayes for some probability in a convex set that is not closed. This limitation affects two of the decision rules but not -admissibility, which is not a pairwise decision rule. -admissibility can be used to distinguish between two convex sets of probabilities that intersect all the same supporting hyperplanes.