**M. J. Schervish, T. Seidenfeld, J. B. Kadane, and I. Levi**

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.

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