Donald Saari is an ardent supporter of the Borda count. He has published a number of papers and at least one book on the subject. Reading these is somewhat aggrivating for me, since he frequently cites his own work to back up his propositions, requiring a good deal of background work to verify (as the conscientious scholar should). I was recently struck by one of his quotes:
[F]or any conflict between the BC and Condorcet rankings, all examples support Borda's approach while raising serious doubts about Condorcet's method—the standard of the field. The conflict resides in the failings of the pairwise vote—not the BC. This conclusion contradicts what has been accepted for two centuries.
Explaining all three-alternative voting outcomes, p. 317; italics original
This is typical of his style and claims. The mathematics behind his work is, to the best of my abilty to judge, correct; the underlying philosophy and normative judgments I disagree with. But for this statement, surely a reasonable method presents itself: show instances of the Borda count conflicting with a Condorcet method, and determine which (if either!) is reasonable.
Toward that end, I present this series of examples.
In this example, there is no clear frontrunner. A and B are clearly the best candidates, since both are preferred by a majority to any other candidates. It would only be sensible for the winner to come from one of these two. In fact, A would defeat any other candidate in a pairwise election, so is the Condorset winner. C, however, is the winner under the Borda count, even though C would lose pairwise to either A or B. I believe in this conflict between Condorcet methods and the Borda count, the Condorcet methods are more reasonable.
C is sensibly the Borda count winner, but if Condorset loser D (D loses every pairwise election: A≻D, B≻D, C≻D, in this case with strong majorities) withdraws, the winner changes to A. This of course violates the principle that a weak candidate (not in the Smith set) entering or existing will not change the winner. I believe in this conflict between Condorcet methods and the Borda count, the Condorcet methods are more reasonable.
Here's a simple, reasonable election. The Borda count and any Condorset method agree that B is the winner. Now suppose A', another candidate from the party of A runs, holding the same views but widely considered not as capable as A. A' will appear on all ballots just below A. (This makes A' a clone in the technical sense, because nothing comes between A and A' in preference order.) Suddenly A is the winner with the Borda count. Under Condorcet methods, B would remain the winner. Once again, in a conflict between Condorcet methods and the Borda count, I feel the Condorcet methods are more sensible.
This is shows a problem with the Borda count when there are truncated ballots. Truncated ballots are a major issue in real elections; of the dozens of FEC-registered candidates for election, most presidential voters will only be familliar with perhaps 2–6. A is the Borda and Condorcet winner, but if candidate D was running (without any ballots changing—no one's heard of D), candidate C wins by a large margin. I see no argument that the Borda count is even close to rational in this case.
In this example, candidate A is the first choice of a majority (70%) of voters and as such wins under any Condorcet method. In the Borda count, B wins instead. This is where the Borda count distinguishes itself from not only the Condorcet methods but any majoritarian rule. It is for this reason that I feel the Borda count is not truly democratic.
A worst-case calculation shows that, for c candidates, (c-1)/c of the votes are needed to tie for the win. For c = 2 candidates the Borda count is majoritarian (a simple majority wins), but beyond that it loses this property. My example has c = 4 candidates and a 70% majoriy losing; this is close to the theoretic 75% limit.
Nauru's modified Borda count does not fare so poorly. Only two-thirds of the votes are needed to win any election; in particular, (2c-2)/(3c-2) are needed for the tie.
A theorem of Gibbard (1973) shows that, subject to simple constrants, no voting system is strategyproof. Still, different voting systems are subject to different degrees of manipulability. It is my understanding that the Borda count is ususually susceptible to such schemes. Barbie, Puppe, and Tasnadi write in a 2006 paper:
Our analysis confirms the general view of the literature that the Borda count is highly vulnerable to strategic manipulation. This intuition is made precise here in two ways. First, for any preference ordering, there is only one rich domain that contains the given preference ordering and that renders the Borda count non-manipulable. By contrast, for other choice rules there may exist many different rich and non-manipulable domains that contain a given preference ordering; for instance, there are many rich single-peaked domains that contain a given single-peaked preference ordering. Secondly, any fixed rich domain on which the Borda count is non-manipulable is as small as it could possibly be, since it contains just as many orderings as there are social states. Again, this strong restriction does not apply to single-peaked domains, for instance. Thus, the overall conclusion from our analysis is that the Borda count fares poorly in terms of strategic manipulation, in the sense that there are very few non-manipulable domains all of which are, moreover, very small.
Similarly, Pierre Favardin, Dominique Lepelley, and Jérôme Serais write in a 2002 paper:
Our results (see Table 2) show the great vulnerability of the Borda rule to this manipulation. We can compare it to the classical manipulation attempted by voters misrepresenting their individual preference. In an election contest of three candidates when n is large the vulnerability of the Borda rule to manipulation of coalition of strategic voters is of about 50% (see Favardin, Lepelley, Serais 2001) whereas it is of 62% (resp. 55%) for single (resp. simultaneous) cloning manipulation.
Unlike Dr. Saari, I have not decided which method is best for elections. In fact I harbor serious doubts as to which criteria are of significant value, and which must be discarded to avoid various impossibility theorems. However, alone out of all the methods seriously suggested for use, I find the Borda count unsuitable. It suffers from serious flaws and few advantages. My biggest problems with the Borda count: