[EM] NANSON - Elections to one seat

[Simon Gazeley]

I was disappointed to see no reference to Nanson on this site.

Nanson proved in effect that if a series of Borda counts is conducted on
the same set of votes, eliminating each time the candidates whose scores
are equal to or below the mean, the one candidate left at the end is the
Condorcet winner, if there is one.  A proof (not Nanson's) is as
follows:

A candidate's Borda score is the sum of the points they score in
Condorcet contests with all the other candidates.  Let there be v voters
and r candidates in the election.  Each Condorcet contest generates v
points; obviously, the mean of the points scored by the two candidates
is v/2.  The Condorcet winner C necessarily scores more than v/2 points
in every contest in which C takes part, therefore C's Borda score will
be above the mean in a Borda count with any one or more of the other
candidates.

The Nanson winner is the candidate who is most preferred by the largest
majority of voters.  If A's Borda score is above B's, then the average
voter ranks more of the currently active candidates below A than below
B.  If A's Borda score is above the mean in a round of a Nanson count,
then the average voter ranks A above a majority of the other candidates
in that round.  If A survives to the end of the count, then the average
voter ranks A above all the other candidates in the election.

Simon


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From:  Simon Gazeley
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