Quality & Quantity 33: 77–84, 1999.
© 1999 Kluwer Academic Publishers. Printed in the Netherlands.
On the Probability that all Weighted Scoring Rules
Elect the Condorcet Winner
WILLIAM V. GEHRLEIN
Department of Business Administration, University of Delaware, Newark, DE 19711, U.S.A.
Abstract. Many procedures exist to determine the winner in an election, and many studies have been
done to determine conditions under which each of them would work best. Much less attention has
been given to the examination of how frequently these procedures tend to produce the same winner.
The focus of the current study is to examine the likelihood that weighted scoring rules and weighted
scoring elimination rules tend to always elect the same winner in three candidate elections.
Key words: Condorcet winner, weighted scoring rules.
Consider an election on three candidates (A, B,andC). There are six possible
linear preference rankings that voters might have on these alternatives
denotes the number of voters with the associated preference ranking on
candidates. Since all preference rankings are linear, no voter indifference between
candidates is allowed. For an election with n voters, n =
. A weighted
scoring rule, Rule λ, selects a winner by giving relative weights of 1, λ, 0 respec-
tively to each voter’s ﬁrst , second and third ranked candidates. The winner is the
candidate who receives the most total points.
We are interested in the likelihood that all weighted scoring rules will select
the same winner under various conditions. In order to assess probabilities, some
assumption must be made about the likelihood that various combinations of n
occur. A speciﬁc combination of n
’s is referred to as a voter proﬁle, or simply as
a proﬁle. If it is quite certain that one particular candidate has high probability that
it is preferred by a large proportion of the population, then there would intuitively