Common knowledge with monotone statistics

When individual statistics are aggregated through a strictly monotone function to an aggregate statistic, common knowledge of the value of the aggregate statistic does not imply, in general, that the individual statistics are either equal or constant. This paper discusses circumstances where constancy and equality both hold. The first case arises when partitions are independently drawn, and each individual's information is determined by their own partition and some public signal. In this case common knowledge of the value of the aggregator function implies (with probability one) that the individual statistics are constant, so that in the case where the individual statistics have the same expected value, they must all be equal. The second circumstance is where private statistics are related: affiliation of individual statistics and a lattice condition imply that the individual statistics are equal when the value of the aggregate statistic is common knowledge.