Evaluating Ambiguous Random Variables and Updating by Proxy

We introduce a new theory of belief revision under ambiguity. It is recursive (random variables are evaluated by backward induction) and consequentialist (the conditional expectation of any random variable depends only on the values the random variable attains on the conditioning event). Agents experience no change in preferences but may not be indifferent to the timing of resolution of uncertainty. We provide two characterization theorems: the first relates our rule to standard Bayesian updating; the second characterizes the dynamic behavior of an agent who adopts our rule.