A note on consensus and common knowledge of an aggregate of decisions.

McKelvey and Page [1986] investigate the effect that common knowledge of an aggregate statistic of individuals' actions has on individual beliefs, assuming that public information can be expressed as a function of the individuals' posterior probabilities of some event. We show that this kind of statistic is not equivalent to a pure function of individuals’ decisions in informational terms. Assuming that all agents follow the same union-consistency decision rule as in Cave [1983], we give a sufficient condition on the aggregate statistic of decisions such that the common knowledge of this statistic implies the equality of all decisions. We also prove that if information partitions are finite, an iterative process of public announcement of the statistic, where agents update their decision on the basis of their private information plus the announced value of the statistic, achieves the situation of common knowledge, and hence converges to an equilibrium characterized by equality of decisions. We then show that all admissible statistics are equivalent in informational terms, which is there's no admissible statistic that leads to a better consensus than an other. Finally, we show that given any decision rule, there exists a refinement of this decision rule that achieves the situation of perfect information.