Axiomatizing the Bayesian Paradigm in Parallel Small Worlds

There is currently much interest in scenario-focused decision analysis (SFDA), a methodology that provides, among other things, supporting analyses in circumstances in which there are deep uncertainties about the future (that is, when experts and decision makers (DMs) cannot come to any agreement on some of the probabilities to use in a Bayesian model). This lack of agreement can mean that sensitivity and robustness analyses show that virtually any strategy may be optimal under the beliefs of one or more participants. Scenario-focused analyses fix the deep uncertainties at interesting values in different scenarios and conduct a (Bayesian) decision analysis within each. The results can be informative to the DMs, helping them understand different possible futures and their reactions to them. However, theoretical axiomatizations of subjective expected utility (SEU), the core of decision analysis, do not immediately extend to the context of SFDA. The purpose of this paper is to provide an axiomatization of SEU that supports SFDA. Scenarios have much in common with Savage’s concept of small worlds . We discuss the parallels and then explore two difficulties in extending his and other writers’ axiomatizations. The development of SEU offered here overcomes these difficulties. Throughout, attention is given to the implications of the theoretical development for the practice of decision analysis.