journal of economic theory 75, 271279 (1997) Probabilities of Preferences and Cycles with Super Majority Rules Craig A. Tovey School of ISyE, Georgia Institute of Technology, Atlanta, Georgia 30332 Received September 5, 1996; revised February 26, 1997 Balasko and Cre s [2] introduce a probabilistic model of population preference profiles on n alternatives. For supermajority voting at any level { >.5286... they show that the probability of Condorcet cycles (intransitivity of the aggregate preference) tends quickly to 0 as n increases. This paper offers an alternative analysis that relates voting outcomes to the behavior of sample medians. Condorcet cycles turn out to be asymptotically rare for all { > 1 2 , but for the disappointing reason that preferences are rare. In contrast, for simple majority voting, { = 1 2 , cycles occur with probability converging to 1 as n  . Journal of Economic Literature Classification Number: D71.  1997 Academic Press 1. INTRODUCTION In many models the theoretical properties of simple majority voting are poor in the sense that aggregate preferences can be intransitive unless individual (agent) preferences are highly constrained. One approach to this problem, dating back at least to Greenberg [1], is to consider superma- jority voting, where an aggregate preference of alternative I to alternative II means that some fraction : > 1 2 of the agents prefer I to II. More recently Caplin and Nalebuff [3] show that for a certain structured class of agent spatial preference distributions, aggregate preferences are transitive for any : above 1&1er.64. References to other work along these lines are given in [2]. There is no convincing, generally accepted random model of the preferences of an agent in a population. Any result about a particular dis- tribution is open to doubt about its significance or applicability. Balasko and Cre s [2] seek to get around this difficulty by establishing a result about ``most'' distributions. They propose the following very natural model: Definition of S. Let i =1, 2, ..., n ! index the n ! possible preference orders on n alternatives. Situate a simplex S in n !-dimensional space, whose extreme points are the n ! unit basis vectors e i . Every point in S now denotes a probability distribution on preference orders, because every point is a convex combination of the extreme points. The point  i w i e i article no. ET972310 271 0022-053197 25.00 Copyright  1997 by Academic Press All rights of reproduction in any form reserved.