Monotonic models and cycles

Abstract

A partitional model of knowledge is monotonic if there exists a linear order on the state space such that, for every player, each element of her partition contains only a sequence of consecutive states. In monotonic models, the absence of alternating cycles is equivalent to the property that, for every pair of players, the join of their partitions contains only singletons. Under these equivalent conditions any set of posterior beliefs for the players is consistent (i.e., there is a common prior). We describe the lattice properties of monotonic models, develop a test to check if a model is monotonic, propose a simple su¢ cient condition for non-monotonicity, and provide some examples. We also study models having circular orders, a weakening of monotonicity.

Monotonic Knowledge Models, Cycles, Linear Versions and Auctions with Differential, Finite Information

“…Combining Proposition 2 with a previous result in the literature (Rodrigues‐Neto, ; Proposition ) gives the following conclusion.…”

“…This paper makes explicit the links between the connectedness condition found in the literature on common‐value, second‐price auctions with differential information (Einy et al . (), also studied by Forges & Orzach ()) and the concept of monotonic partitions (Rodrigues‐Neto, ) that is associated with Harsanyi Consistency Problem (common prior existence). When bidders’ partitions are not monotonic, it is no longer always the case that the bidder with superior information will do better than others.…”

“…Now suppose that there is a cycle, and thus an alternating cycle, in the reduced model ℜ, and that all joins of two players at a time contain only singletons. By Proposition 1 in Rodrigues‐Neto (), the reduced model ℜ is not monotonic. By Proposition 2, the model is also not monotonic.…”

“…Independently, Rodrigues‐Neto () introduces the concept of a monotonic model; that is, a state space in which states are linearly ordered, together with players’ partitions respecting this linear order in the sense that each cell of every partition is a sequence of consecutive states (according to the linear order). Rodrigues‐Neto () links monotonicity to the cycles approach, and, consequently, to Harsanyi Consistency Problem and games of incomplete information.…”

“…For more details about the meet-join diagram, versions and cycles, seeRodrigues-Neto (2009, 2012, 2014.© 2015 Economic Society of Australia…”