Quality & Quantity 33: 85–96, 1999.
© 1999 Kluwer Academic Publishers. Printed in the Netherlands.
A Note on the Probability of Having a Strong
and WILLIAM V. GEHRLEIN
University of Caen, France;
University of Delaware, U.S.A.
Abstract. In an election, a strong Condorcet winner is a candidate who is top-ranked by more than
50% of the voters. The purpose of this note is to provide some algebraic representations for the
probability of having a strong Condorcet winner in three-candidate elections. Three alternative pro-
cedures for generating voting situations are considered: the Impartial Culture condition, the Impartial
Anonymous Culture condition and the Maximal Culture condition. It turns out that the conclusions
we obtain strongly depend on the way for generating voting situations.
Key words: social choice, voting, Condorcet winner, probability models.
Consider a group of individuals that is seeking to choose a most preferred candidate
from among several candidates by means of voting. How should they proceed?
Two hundred years ago, Condorcet (1785) proposed the following method: com-
pare candidates in a pairwise fashion with the object of determining who of the
pair receives the majority of the votes. Then the candidate who defeats each other
candidate by this process is selected (such a candidate is called a Condorcet win-
ner). Although this method seems natural and reasonable, it is well known that
its implementation can give rise to serious problems. As a ﬁrst ﬂaw, a Condorcet
winner need not exist, a phenomenon that Condorcet was the ﬁrst to discover: it
is the famous paradox of voting, or Condorcet’s paradox. Secondly, even when
a Condorcet winner exists, examples can be found showing that such a candi-
date does not necessarily represent the most satisfactory social choice (see, e.g.,
Fishburn, 1974: 544). In other terms, asking for the selection of the Condorcet
winner appears to be too strong a requirement in some situations. For this reason,
some authors (Richelson, 1978; Lepelley, 1992, among others) have suggested the
notion of a strong Condorcet winner. A strong Condorcet winner is deﬁned as a
Correspondence on this paper should be addressed to: Dominique Lepelley, Facult
e de sciences
economiques et de gestion, Universit
e de Caen, 14032 Caen Cedex, France.