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, constancy of the individual statistics. This paper discusses two circumstances where it does occur. The first case arises when partitions are independently drawn: in this case common knowledge of the value of the aggregator function implies (with probability one) constancy of the individual statistics. The second case is where private statistics are related: affiliation of individual statistics and a lattice condition imply constancy of the individual statistics when the value of the aggregate statistic is common knowledge.