Journal Article Existence and Optimality of Equilibrium in Labour-managed Economies Get access Joseph Greenberg Joseph Greenberg Virginia Polytechnic Institute and State University Search for other works by this author on: Oxford Academic Google Scholar The Review of Economic Studies, Volume 46, Issue 3, July 1979, Pages 419–433, https://doi.org/10.2307/2297011 Published: 01 July 1979 Article history Received: 01 November 1975 Accepted: 01 July 1978 Published: 01 July 1979

[Consider a society with n individuals who must choose an alternative from a given non-empty set X. For an integer d ≤ 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.]