Instability of 'ex post' Aggregation in the Bolker-Jeffrey Framework and Related Instability Phenomena

Suppose n Bayesian agents need to make a decision as a group. The group as a whole is also supposed to be a Bayesian agent whose probabilities and utilities are derived or aggregated in reasonable ways from the probabilities and utilities of the group members. The aggregation could be ex ante, i.e., in terms of expected utilities, or it could be ex post, i.e., in terms of utilities only, or in terms of utilities and probabilities separately. This study explores the ex post approach. Using the Bolker/Jeffrey framework, we show that ex post aggregation is subject to an instability phenomenon. That is, it may happen that the group preference between actions "flips back and forth" depending on the level of detail in which the decision problem is described. Structurally very similar phenomena also occur elsewhere in social choice theory, in statistics (Simpson's Paradox), and in voting theory (Ostrogorski's Paradox). 1. EX ANTE AND EX POST AGGREGATION