Subjective and Objective Probabilities in Representation of Preferences under Uncertainty

Dana Scott provides a necessary and sufficient condition for a given probability capacity can be decomposed into a unique additive probability and a unique strictly increasing distortion function. Given this result, we characterize a family of preferences defined over the lottery acts (the Anscombe-Aumann acts), which we call rank-dependent subjective expected utility (RDSEU): The Choquet expected utility (CEU) preferences where a probability capacity is decomposed into an additive probability and a distortion function, both of which are endogenously derived from the preference. Furthermore, we clarify the relationship between this family of preferences and the so-called probabilistically sophisticated (PS) preferences. Specifically, a decision maker’s preference is represented by RDSEU if and only if it is represented by CEU on the Anscombe-Aumann acts and by PS when its domain is restricted to the Savage acts. Moreover, if, in addition, the RDSEU is representable by PS on the entire Anscombe-Aumann acts, it turns out that it must be a subjective expected utility (CEU) on the entire domain. We thus uncover how subjective and objective probabilities are different not only in their concepts but also in their implications for preference representation