Letters to the Editor: Lingering Spotlight on Voting Theory

April 9, 2001

To the Editor:

In the article "Making Sense Out of Consenus" (SIAM News, Vol. 33, October 2000), Dana Mackenzie reviews a number of voting systems and concludes with the suggestion that we try voting with the Borda count in order to avoid problems similar to the one he discusses from the California primary during the 2000 U.S. presidential election.

Under the Borda count, each voter ranks the n candidates from 0 (least favorite) to n - 1 (most favorite). These rankings are added, and the candidate with the highest total ranking is elected. Another way to think about the Borda count is to imagine running every possible two-way election, and adding the votes from the various elections.

However, consider an election in a hypothetical country in which there were two well-known candidates, Al and George, and 17 third-party candidates whose views were either unknown or unappreciated by the overwhelming majority of voters. If voters were to honestly vote their minds using a Borda count, they would rank Al and George first and second, according to their individual preferences, and place the third-party candidates in lower positions. Thus, each voter would be giving 19 votes to Al and 18 votes to George, or vice versa. If a number of the voters for George understood the system well, however, they might put Al at the bottom of the list (giving him 0 votes), despite the fact that they prefer Al to any of the third-party candidates. Each such pro-George voter would have to be offset by 19 pro-Al voters in order for Al to win the election.

Now suppose that Al's supporters got the same idea. In this case, virtually all voters would be placing Al and George in the extreme positions (0 and 19 votes) and the universally disliked candidates in the intermediate positions. It is then easy to imagine an unknown third-party candidate (perhaps the first one in alphabetical order!) winning the election even though all voters would have preferred either Al or George.

To conclude, I should mention that after our experiences last November (and on into December), it is no longer California but rather Florida that is representative of the kind of electoral "debacle" that most people are set on avoiding. It is my opinion that, compared with our current system, the Borda count is more difficult for voters to understand and more difficult for counties to implement physically on an election machine.

I would like to give my support to approval voting (also mentioned in the SIAM News article). In approval voting, voters are allowed to cast one vote for as many candidates as they wish. In this case (as in the case of our current voting system in general elections), there is no incentive for voters to misrepresent their actual preferences. Moreover, approval voting is immune to the "overvoting" problems that invalidated a large number of ballots in Palm Beach County and elsewhere in Florida.

Daniel Loeb (loeb@pa.wagner.com), Daniel H. Wagner Associates, Malvern, Pennsylvania, and Laboratoire Bordelais de Recherche en Informatique, Talence, France.

The Author Replies:

Daniel Loeb makes a valid point, one that I did not mention in my article for reasons of space. Indeed, if voters cast their ballots insincerely, you can definitely have surprise third-party winners in the Borda count system. This was, in fact, pointed out to Borda back in the 18th century. His reply was typical of a more optimistic age: "My system is only for honest men."

Even if you take a less optimistic view of human nature, though, you could still argue that voters would recognize that it isn't really in their best interest to rank George number 1 and Al number 17 if, in truth, they would prefer Al to the Socialist candidate, the Green Party candidate, and so forth. Loeb's example shows why. Another way to put it is that, although the Borda count system may not be immune to attempted strategic voting, it is relatively immune to successful strategic voting. (Donald Saari has proved this mathematically, in fact.) So after a few fiascos, voters might stop trying to beat the system. (Of course, after a few fiascos, the voters might also ask for the old system back. . . .)

Last year's experience in the U.S. was, I think, both good and bad news for voting theorists. On the good side, it focused attention on the fact that our election system does not automatically choose the candidate that most people want. On the other hand, it diverted most of this attention to the mechanical question of how the votes are recorded. There still wasn't much debate over the way we determine the winner once we have the votes, or more fundamentally, how we should set up an election so that the winner reflects a true consensus of the electorate. One thing this election proved, beyond a doubt, is that the plurality system leads to polarization rather than consensus. Unfortunately, few commentators seem to have grasped that point.

If Loeb prefers the approval voting system, I have no complaint with that, although Saari might!

Dana Mackenzie


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