From posteriors to priors via cycles: An addendum

Monotonic models and cycles

“…The distributive property of set theory implies that the collection of all monotonic partitions of the state space is a distributive lattice. 5 The corresponding result is not valid for the set of all partitions of a set, which is a lattice, but not a distributive lattice.…”

Incomplete-Information Models of Large Economies with Anonymity: Existence and Uniqueness of Common Priors

“…The "if" part of the proposition follows from Hellwig's [7] addendum to the result of Rodrigues-Neto [12]. 4 The proposition actually is a special case of the more general Proposition 4 below.…”

“…In contrast, the "if" part of Proposition 1 rests on the …nding in [7] that, for belief systems satisfying (f gjt) > 0 and (ftg) > 0 for all t 2 T and all 2 D; it su¢ ces to consider these equations for "cycles" of length four or less because the equations for longer cycles follow automatically. Without positivity, this is not necessarily true.…”

Incomplete-Information Games in Large Populations with Anonymity

“…This result hinges on the strict positivity of the densities f and g; which in turn is derived from the assumption that the macro belief function is strongly coherent. 17 Strong coherence involves two properties, (i) coherence, i.e., mutual absolute continuity of the measures (t); t 2 T; and (ii) mutual absolute continuity of the measures 2 D; where (Djt) = 1 for all t: Coherence of the measures (t); t 2 T; is restrictive, but, as discussed in Section 4, this property plays a key role in establishing that a given belief function b a admits even an agent-speci…c prior, let alone a common prior.…”