Bayesian theory specifies how sets of probabilities should merge to a common opinion under the effects of Bayes' rules applied to shared evidence. This pooling provides a robustness in that after the data are obtained, decisions can be taken without knowing which prior opinion was used. Also, it suggests asymptotic theories which use sets of probabilities rather than a single probability to represent a belief can share some of the properties of a strict Bayesian theory. Less familiar, and the focus of this project, is the phenomenon called 'dilation' for sets of probabilities. Dilation occurs when, regardless of the evidence, the set of posterior probabilities for a given even properly contains the set of prior probabilities for that event, as in, for example, sequential decision making. Thus, dilation is the opposite phenomenon to the merging of sets of probabilities. This project raises sever questions about the importance of dilation for sequential decisions, and in particular, for the Harsanyi-Selten theory of sequential games. The project delineates how sequential decisions differ from normal decisions when dilation is present. Also to be examined are issues concerning independence and contaminated models.
