We contrast three decision rules that extend Expected Utility to contexts where a convex set of probabilities is used to depict uncertainty: Gamma-Maximin, Maximality, and E-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 E-admissibility, which is not a pairwise decision rule. E-admissibility can be used to distinguish between two convex sets of probabilities that intersect all the same supporting hyperplanes.
Keywords. Bayes admissible, E-admissible, Gamma-Maximin, Maximality, convex set
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Authors addresses:
Mark Schervish
Department of Statistics
Carnegie Mellon University
Pittsburgh, PA 15213
Teddy Seidenfeld
135J Baker Hall
Carnegie Mellon University
Pgh. PA 15213
Joseph Kadane
Department of Statistics
Carnegie Mellon University
Pittsburgh, PA 15213
Isaac Levi
718 Philosophy Hall
Columbia University
New York
USA
E-mail addresses:
| Mark Schervish | mark@stat.cmu.edu |
| Teddy Seidenfeld | teddy@stat.cmu.edu |
| Joseph Kadane | kadane@stat.cmu.edu |
| Isaac Levi | levi@columbia.edu |