Counterfactual Priors: A Bayesian Response to Ellsberg's Paradox

This paper analyzes the root cause of Ellsberg-type choices. This class of problems share the feature that at the time of the decision, t = m, the decision maker (DM) possesses partial information, Im, about the events/propositions of interest F: DM knows the objective probabilities of the sub-class F1, F1 С F only, whereas she is uninformed about the probabilities of the complement F’1 . As a result, DM may slip into the state of "comparative ignorance" (see Heath and Tve rsky 1991 and Fox and Tversky 1995). Under this state, DM is likely to exhibit "ambiguity aversion" (AA) for the events of F’1 relative to those of F1. AA, in turn results in DM having non-coherent beliefs, that is, her prior probability function, PI0m, is not additive. A possible way to mitigate AA is to motivate DM to form her prior in a state of "uniform ignorance". This may be accomplished by inviting DM to bring herself to the hypothetical time t = 0, in the context of which Im was still a contingency, and trace her "counterfactual prior", Pc0, "back then". Under uniform ignorance, DM may adhere to the "Principle of Indiference", thus identifying Pc0 with the uniform distribution. Once Pc0 is elicited, DM can embody the existing information Im into her current, actual set of beliefs Pm by means of Bayesian Conditionalization. In this case, we show that Pm is additive.