[EM] Nanson

[Simon Gazeley]

Dear List

For interest.

Simon


 
NANSON'S PROPOSITION
 
Nanson proved that if a series of Borda counts is conducted on the same
set of votes, eliminating each time the candidates whose Borda scores
are below the average, the one candidate who is left at the end is the
Condorcet winner if there is one.
 
BORDA COUNTING
 
I have seen various specifications for a Borda count, many of which are
mutually inconsistent. The specification I use is as follows:
 
a. Voters rank the candidates in order of preference, starting at 1; the
second favourite is 2, the third favourite is 3 and so on. Equal
preferences can be catered for and there is no need to rank all the
candidates.
 
b. Each ballot-paper has a value of r(r-1)/2 points, where r is the
number of candidates.
 
c. At the count, each ballot-paper is examined and each candidate is
awarded the number of points that corresponds with the number of
non-eliminated candidates ranked below him/her on that paper. If the
voter has awarded equal preferences to two or more candidates, those
candidates are awarded the arithmetic mean of the points they would have
had if the preferences had been discrete. Any candidates left unranked
on a ballot-paper are treated as equal preferences and the remaining
points are distributed equally among them.
 
d. A candidate's "Borda score" is the total number of points awarded to
him/her in the count.
 
NANSON'S PROPOSITION PROVED
 
Suppose that v votes have been cast in an election in which r candidates
are standing. In a Borda count involving only two of those candidates a
total of v points are scored of which, barring ties, one of the
candidates has more than v/2, the other has fewer. The Condorcet winner
by definition scores more than v/2 points in such pairwise contests with
each of the r-1 other candidates, making a total of more than v(r-1)/2
points. The total number of points scored in a Borda count is vr(r-1)/2
and the arithmetic mean is v(r-1)/2, less than the Condorcet winner's
total: therefore, if candidates who have less than the arithmetic mean
of points are eliminated, the Condorcet winner, if there is one, will be
among those who survive.