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. ---- Election-methods mailing list - see http://electorama.com/em for list info
