The Bayesian approach to decision making under uncertainty prescribes that a decision maker have a unique prior probability and a utility function such that decisions are made so as to maximize the expected utility. This paradigm was justified by axiomatic treatments but challenged by several classes of evidence: experimental results incompatible with the consistency condition implied by the Bayesian approach, difficulties people have in specifying a prior probability, and economic phenomena contradictory to the theoretical results. Among many models suggested to generalize the Bayesian approach are the nonadditive probabilities model and the multiple priors model. These models have passed an important test: they have been axiomatized. However, these one-shot decision models shed little light on the problem of dynamically updating probabilities as new information is gathered. The natural question then is: what are the reasonable ways for economic agents to update beliefs over time? This study will deal with this problem axiomatically and suggest plausible updating rules which satisfy some basic requirements. A family of pseudo-Bayesian rules, each of which may be considered a generalization of Bayes' rule for a unique additive prior, will be presented and studied. A family of classical updating rules, each of which starts out with a given set of priors, possibly rules some of them out in the face of new information, and continues with the Bayesian updates of the remaining ones, will also be studied. The assumption of a unique given prior probability is clearly unrealistic. New political circumstances, technological advances, new firms, and new products are all counter-examples in which the assumption of a unique prior is unreasonable. Consequently, this study will provide important theoretical foundation upon which we build our understanding of the adjustment of economic agents to new phenomena or new information.
