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.
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"The original Borda count and partial voting" by Emerson, Peter. Social Choice and Welfare; Heidelberg Vol. 40, Iss. 2, (Feb 2013): 353-358. DOI:10.1007/s00355-011-0603-9
explicitly says they are equivalent if everyone submits full rankings, but not equivalent if partial rankings are allowed:
In a Borda count, bc, M. de Borda suggested the last preference cast should receive 1 point, the voter’s penultimate ranking should get 2 points, and so on. Today, however, points are often awarded to (first, second,..., last) preferences cast as per (n, n−1, …, 1) or more frequently, (n−1, n−2,…, 0). If partial voting1 is allowed, and if a first preference is to be given n or n −1 points regardless of how many preferences the voter casts, he/she will be incentivised to rank only one option/candidate. If everyone acts in this way, the bc metamorphoses into a plurality vote… which de Borda criticized at length. If all the voters submit full ballots, the outcome—social choice or ranking—will be the same under any of the above three counting procedures. In the event of one or more persons submitting a partial vote, however, outcomes may vary considerably. This preliminary paper suggests research should consider partial voting. The author examines the consequences of the various rules so far advocated and then purports that M. de Borda, in using his formula, was perhaps more astute than the science has hitherto recognised.
