The Borda voting system is fatally defective

by John Quiggin on March 3, 2004

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. Note for Australian readers. Are you thinking what I’m thinking?

{ 11 comments }

1

Brian Weatherson 03.03.04 at 5:40 am

I agree entirely – Borda voting in practice would be a disaster.

Michael Dummett has a book advocating Borda voting where he includes some weak responses to this problem. He suggests (I think, I’m doing this all from memory) that each political party only be allowed to nominate one candidate. There’s an obvious countermove here of just setting up multiple political parties with the same ideals, and I don’t recall Dummett having much good to say at that point.

2

rilkefan 03.03.04 at 6:27 am

“B1″ in the final sentence of para 3 should be “R1″ – or Australians vote very differently than I thought.

3

Maynard Handley 03.03.04 at 6:42 am

“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, B1 is guaranteed to win, no matter what other candidates run and whether or not voters act strategically.”

HUH? Surely if there are 60 Red and 40 Blue R1 is guaranteed to win?
I know nothing about Borda voting, but given your starting example I’m loathe to bother reading the rest of the article. Is this just a typo?

4

John Quiggin 03.03.04 at 7:02 am

Sorry about that. Fixed now.

Note though, that R1 is not guaranteed to win under Borda voting as the rest of the example shows.

5

humeidayer 03.03.04 at 8:50 am

Mr. Quiggin, a quick thank you for your pieces on voting systems which have piqued my curiosity in this area and prompted further investigations.

Of all the systems I’ve studied so far, I’ve yet to find a system in which voters control the weighting of their votes. E.g., I want 75% of my vote to go to Kerry and 25% of it to go to Dean. With such a system, those who really want Kerry can vote 100% Kerry, those who want Dean can vote 100% Dean, those who want either Kerry or Dean but no one else could vote 50% Kerry and 50% Dean. Etc. There probably is such a system, I just haven’t seen it discussed anywhere so far in my investigations.

On a side note, an anecdote that popped into my head which may be amusing or interesting to an economist studying voting systems.

Can you imagine any system in which restricting the vote to lawyers might produce an optimal result?

When the U.S. Constitution was being hammered out and the selection of judges came up, Benjamin Franklin (always an interesting character) noted a Scottish method of selecting judges. According to Convention notes…

“…DOCr. FRANKLIN observed that two modes of chusing the Judges had been mentioned, to wit, by the Legislature and by the Executive. He wished such other modes to be suggested as might occur to other gentlemen; it being a point of great moment. He would mention one which he had understood was practiced in Scotland. He then in a brief and entertaining manner related a Scotch mode, in which the nomination proceeded from the Lawyers, who always selected the ablest of the profession in order to get rid of him, and share his practice among themselves. It was here he said the interest of the electors to make the best choice, which should always be made the case if possible…

Cheers.

6

John Quiggin 03.03.04 at 10:39 am

The Australian Senate and other Upper Houses are elected on various forms of the Hare-Clark system of proportional representation.

In recent modifications of the system, you can either choose an allocation of preferences or opt to follow an allocation chosen by the party of your first preference. In the latter case, the party can choose to divide preferences in a weighted fashion. It doesn’t give you everything you asked for above, but it is yet another instance of the Australian addiction to wild and wonderful voting systems.

In thirty years as an Australian voter I’ve experienced a vast range of systems including preferential (instant runoff), optional preferential, Hare-Clark with and without Robson rotation, modified d’Hondt, approval and of course plurality (what we call first-past-the-post).

7

Jeff Darcy 03.03.04 at 2:21 pm

I thought this rang a bell, but it took me a while to remember why. Check out Dasgupta and Maskin’s The Fairest Vote of All in this month’s Scientific American.

8

LTH 03.03.04 at 2:48 pm

“There are 60 Red voters and 40 Blue voters. In most systems of voting, R1 is guaranteed to win…”

How about when the votes are divided into two constituencies such that there are 30 Reds and 40 Blues in one zone and the remaining 30 reds in the other zone? The best Red can hope for here is a draw.

Hmmm, is this similar to any existing voting systems…?

9

Kerim Friedman 03.03.04 at 4:59 pm

My understanding is that most ranked systems actually being proposed are recursive. See FairVote.org’s web site on Instant Runoff Voting, which is what is already being used in many states.

10

tew 03.04.04 at 8:01 pm

Another good resource on voting, and on the problems with IRV and Borda in particular, is electionmethods.org. I find it a little hard to take them seriously, since their web design is so preposterously bad, but the information seems fine.

11

John Quiggin 03.04.04 at 9:18 pm

I don’t think there’s anything factually wrong, but be aware that electionmethods.org is an advocacy site for Condorcet and tends to gloss over its difficulties while overstating those of other methods. I pointed this out in this earlier post. (Of course, there’s nothing wrong with advocacy – readers will have noted that I’m a fan of IRV, and should apply corresponding scepticism).

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