Smallness of Invisible Dictators

Fishburn (1970) showed that in an infinite society Arrow's axioms for a preference aggregation rule do not necessarily imply a dictator. Kirman and Sondermann (1972) showed that, in this case, nondictatorial rules imply an invisible dictator that, whenever the agent set is an atomless finite measure space, can be viewed as the limit of coalitions of arbitrarily small size. We show first that, when admissible coalitions are restricted to an algebra, there are two sorts of invisible dictators. We next show that, in most cases of interest, we do not need to resort to measures on the agent space to give a precise meaning to the statement that invisible dictators are the limit of arbitrarily small decisive coalitions.