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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