Econometrica: May, 1979, Volume 47, Issue 3

Consistent Majority Rules over Compact Sets of Alternatives

https://doi.org/0012-9682(197905)47:3<627:CMROCS>2.0.CO;2-G
p. 627-636

Joseph Greenberg

Consider a society with n individuals who must choose an alternative from a given non-empty set X. For an integer @< n, a d-majority equilibrium is an alternative x* @? X such that no alternative in X is preferred to x* by at least d individuals. It is proved that when X is a compact and convex set of dimension m, a necessary and sufficient condition that, for every profile of individuals' convex and continuous preferences, there exists a d-majority equilibrium, is that d be greater than (m/(m + 1))n. Using this result for the case when X consists of a finite number, T, of alternatives, it is shown that a necessary and sufficient condition that for every individuals' preference orderings there exists a d-majority equilibrium is that d exceeds ((T - 1)/T)n.