Is there any difference between ways of counting Borda?
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I've heard of three different ways to count Borda-like ballots
- "each one receives n – 1 points for a first preference, n – 2 for a second, and so on"
- "As Borda proposed the system, each candidate received one more point for each ballot cast than in tournament-style counting, eg. 4-3-2-1 instead of 3-2-1-0"
- Sum up the rankings themselves and elect the candidate with the lowest sum
I've always assumed these are exactly equivalent, and will always elect the same candidate with a given set of ballots, but I want to make sure I'm not missing something. Are they the same even in cases where incomplete rankings are allowed, and in cases where equal rankings are allowed?
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Well, it partly depends on what you do with equal ranks or incomplete ballots. If an unranked candidate is scored as 0 then a 4-3-2-1 system would be different from 3-2-1-0. But if it's done in a more sensible way, they would be equivalent.
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https://politicalreform.ie/2023/10/06/my-word-preferendum-et-maintenant-le-preferendum/
But there’s another problem: Jean-Charles de Borda’s voting procedure is not exactly the same as that which today is called the Borda Count BC. He suggested that, in any vote on n options, where the voter casts m preferences, points shall be awarded to (1st, 2nd … last) preferences cast, according to the rule
(m, m-1 … 1).
But some of his contemporaries in l’Académie des Sciences changed this to
(n, n-1 … 1).
or
(n-1, n-2 … 0).
If every voter has submitted a full ballot, the social choice and social ranking of any m- or n-rule analysis will remain the same. If, however, some voters have submitted only a partial ballot, the difference between the m- and n-rule outcomes can be huge. In brief, the m rule gives a voter’s (x)th preference 1 point more than her (x+1)th preference, regardless of whether or not she has cast that (x+1)th preference. The n rules, in contrast, give he who casts only one preference an (n-1) advantage over all the other options. So on really contentious topics, the BC may not be much better, if at all, than a plurality vote. The m rule, however, is unbiased. The n-rules promote division, whereas the m-rule can be the very catalyst of consensus; and the m-rule, Jean-Charles’ original proposal, is today known as the Modified Borda Count MBC, a magnificent Irish contribution to the world’s politics.
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Let's say there are 5 candidates and a voter bullet votes for their favourite. According to the m-rule (the "good" way), the points would be 1, 0, 0, 0, 0. According to the n-rule (the "bad" way) it would be 4, 0, 0, 0, 0 (or equivalently 5, 1, 1, 1, 1).
But 1, 0, 0, 0, 0 would also be equivalent to 5, 4, 4, 4, 4, which doesn't really seem that good either and is biased against voters choosing not to rank all candidates (rather than being unbiased as claimed by the article).
I've always thought the most logical way is to average the ranks. So if 5 ranked candidates would get 5, 4, 3, 2, 1, then bullet voting should give 5, 2.5, 2.5, 2.5, 2.5.
