Quality & Quantity 38: 147–171, 2004.
© 2004 Kluwer Academic Publishers. Printed in the Netherlands.
Consistency in Measures of Social Homogeneity: A
Connection with Proximity to Single Peaked
WILLIAM V. GEHRLEIN
Department of Business Administration, University of Delaware, Newark, DE 19716, U.S.A.
Abstract. A survey of work is presented that is related to the relationship between measures of social
homogeneity and probability that a Pairwise Majority Rule Winner (PMRW), or Condorcet Winner,
exists. Little evidence has been reported to support the notion of a strong relationship between
these two factors. Exact probability representations are derived to show that it is possible to ﬁnd
a parameter that is directly observable from voter proﬁles that can be systematically changed, so that
an expected decrease in social homogeneity will lead to a counterintuitive increase in the expected
probability that a PMRW exists. Using a different observable proﬁle parameter that reﬂects a rough
measure of the proximity of proﬁles to completely single-peaked preferences adds a sufﬁcient degree
of consistency among voters’ preferences to produce the expected outcome.
Consider a situation in which a group of n voters is attempting to select a winner in
an election from a set of m candidates. A Pairwise Majority Rule Winner (PMRW)
in an election would be a candidate who could defeat each of the other candidates
in a series of pairwise elections by majority rule. Condorcet (1785) argued that
the PMRW should be selected as the winner in any election, and the PMRW is
often referred to as the Condorcet Winner in the literature on the topic. However,
it is well known that a PMRW does not necessarily exist (Condorcet, 1785). Kelly
(1974) and Buckley and Westen (1979) produced some interesting results that are
related to the probability that a PMRW exists. They found that different patterns of
probability results are observed as n increases, depending upon whether n is odd
Kuga and Nagatani (1974) and others considered the impact that the presence
of varying degrees of measurable societal factors might have upon the probability
that a PMRW exists. Some intuitively appealing results were found in these studies,
This paper initially evolved from a presentation at a workshop at University of Caen, France.
An earlier version of the paper presented at the Public Choice Society Meeting in Charleston, SC,
March 10–12, 2000. A number of comments from attendees at both presentations were very helpful
in developing the ﬁnal version of the paper.