Opinion pooling on general agendas: linearity or just neutrality?

In "Opinion pooling on general agendas", we characterize linear and neutral opinion pooling functions F : P ! P, where unlike in the classical opinion pooling problem the agenda X of relevant events need not form a -algebra. These characterizations are based on two conditions on the pooling function: independence and implication-preservation. The latter condition is stronger than the standard (Pareto-like) condition of zero-preservation. In the present appendix, (i) we show that our characterizations would not in general hold if instead of implication-preservation we merely require zero-preservation; but (ii) for an interesting class of agendas, zero-preservation (still together with independence) su¢ ces to force pooling to be neutral (Theorem 3) while leaving room for non-linear pooling (Theorem 4). These results suggest that without invoking the requirement of implication-preservation a normative defense of linear pooling becomes di¢ cult. The framework and notation is the same as in the original paper.2 Say that a relevant event A 2 X conditionally entails another one B 2 X (written A ` B) if fAg [ Y entails B (i.e. \C2fAg[YC B) for some countable set Y X that is consistent with A (i.e. \C2fAg[YC 6= ;) and with B (i.e. \C2fBcg[YC 6= ;). The agenda X is pathconnected if for any two events A;B 2 Xnf;; g there exist events A1; :::; Ak 2 X (k 1) such that A = A1 ` A2 ` ::: ` Ak = B. In other words, any two contingent events in the agenda can be connected by a path of conditional entailments.3 For instance, X := fA;A : A R is a bounded intervalg is a pathconnected agenda (a subset of the Borel-algebra over = R).4 One easily shows that