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

In games of incomplete information where each player's information is represented by a partition of the state space, this paper presents a characterisation of monotonic models in terms of connected partitions and another in terms of versions . A model is monotonic if and only if there is a linear version, and this is true if and only if there is a real-valued function on the state space such that every partition of the model is connected with respect to this function. These results help us understand the strength of the connectedness assumption on common-value, second-price auctions with differential, finite information. We offer a simple sufficient condition for non-monotonicity to check if models are monotonic.