Before I argue that the Borda voting system is fatally defective, it may be worth considering what kinds of weaknesses could justify such a verdict. We know from Arrow’s Impossibility Theorem that any nontrivial voting system will encourage strategic/insincere voting in some circumstances and will not always elect the right candidate (unless ‘right’ is defined to coincide with the outcome of the voting system in question). So a fatal defect must be a lot worse than this. I claim that the Borda voting system is so vulnerable to strategic manipulation that it would be completely unworkable, provided only that there are no restrictions on candidacy.

Note: I did a Google before writing this and couldn’t find anything similar, but of course, when I checked again after doing the work, I found this almost perfect anticipation of my counter-example. But having done the work, I thought I’d post it anyway.

My argument can be illustrated with a simple example. Suppose that an election is to be held to fill an office, and that the population is divided into two groups, say Blue and Red. All Blue voters would prefer, of all possible candidates, that B1 fills the job, and all Red voters would prefer, of all possible candidates, that R1 fills the job. There are 60 Red voters and 40 Blue voters. In most systems of voting, R1 is guaranteed to win, no matter what other candidates run and whether or not voters act strategically.

Now consider Borda, and suppose that in addition to R1 and B1, the Blues advance a second candidate, B2[1], who is a little less attractive to all voters than B1. Assuming sincere voting, the Blues will all vote B1, B2, R1 and the Reds will all vote R1,B1, B2. The result will be that B1 gets 240 Borda votes, B2 gets 140 and R1 gets 220, so that B1 is elected.

Even with strategic voting, the Blues still benefit from this strategy provided they can either formally or informally caucus. The Reds’ best strategy is to split their preferences between the two Blue candidates. The Blues’ best response is a mixed strategy, in which, with 50 per cent probability they all vote for B1 and with 50 per cent probability they all vote for B2. With an initial 60-40 split, R1 beats the top Blue candidate 220 to 210. But if the initial split is a bit closer, say 54-46, the Blues win.

The only effective way for the Reds to respond is to run a second candidate of their own, restoring the initial balance in their favor. But then the Blues can put up a third candidate and the process goes on indefinitely. Hence the only way to get a workable election is to restrict the right of candidacy, in which case the restriction procedure effectively amounts to a first round of voting.

fn1. Are you thinking what I’m thinking?