Bayesian consistent prior selection

A subjective expected utility agent is given information about the state of the world in the form of a set of possible priors. She is allowed to condition her prior on this information. A set of priors may be updated according to Bayes' rule, prior-by-prior, upon learning that some state of the world has not obtained. We show that there exists no decision maker who obeys Bayes' rule, conditions her prior on the available information (by selecting a prior in the announced set), and who updates the information prior-by- prior using Bayes' rule. The result implies that at least one of several familiar decision theoretic "paradoxes" is a mathematical necessity.