Common knowledge with monotone statistics

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