When Math Warps Elections

It is a little disturbing for democracy. One candidate could win with some rules and lose with others.

Even if you do not live in an early-primary state, it's almost impossible to avoid online polls and "elections." How much their results square with reality remains to be seen, but one online poll is intriguing less for any predictive power than for what it says about the interaction of math and elections (and I don't mean the funny way they count votes in Florida). The American Mathematical Society and other scholarly groups have launched a site where you pick your favorite presidential candidate—as well as choose any of eight you deem acceptable and rank them from one to eight. (To play, go to www.amstat.org/mathandvoting.) Now the fun begins. The three different methods produce, when I tried it, at least two different winners.

For anyone who believes in democracy, this is a little disturbing. What it means is that "election outcomes can more accurately reflect the choice of an election rule than the voters' wishes," writes mathematician Donald Saari of the University of California, Irvine. One candidate could win with some rules and lose with others. In fact, as mathematicians analyze voting systems, they are turning up other oddities that can yield a "winner" who does not reflect the will of even a plurality, much less a majority. The discoveries are especially relevant this year. "The severity of the problem escalates with the number of candidates," notes Saari, and one thing this primary season has is a lot of still-viable candidates.

One of the most surprising aberrations mathematicians have found comes in a four-way race. There, of course, one candidate wins a plurality and another comes in last. Saari examines what happens if the third-place candidate drops out and, in the next round of voting, people have the same ordered preference as before (A is the first choice of the most, followed by B, then D). Consider an election with 30 voters, who mentally rank the candidates this way:

In our system, McCain wins, with nine first-place votes, trailed by Giuliani (eight), Huckabee (seven) and Romney (six). Now let's say Huckabee drops out. Cross out his name where he came in first, and notice who is now the first choice of his former supporters: two go with Giuliani and five with Romney. That pushes Romney, formerly in last place, to the top, with 11 first-place votes. As the GOP field prunes itself, don't be surprised if the new leader comes from the back of the pack.

If Super Tuesday produces a clear GOP front runner, he could be one whom many and perhaps most Republicans will have to hold their nose to vote for in November. Our pick-your-favorite system, known as plurality voting, "may produce a winner who is the least acceptable to the majority of [GOP] voters," says Steven Brams of New York University, a pioneer in the application of math to voting. That happened in the 2000 presidential election, when Ralph Nader got about 95,000 votes in Florida. George W. Bush's winning margin was about 500. "Since a significant majority of Nader voters preferred Al Gore to Bush," says Brams, "the winner was the candidate least preferred by most voters."

One fix for that is approval voting, in which voters choose any number of candidates they deem acceptable. This not only would avert the distortions of 2000, but would let candidates regarded as unelectable draw their true share of supporters. "You don't have to desert your preferred candidate for fear of 'wasting' your vote," says Brams. Hypothetically, if supporters of Joe Biden, who dropped out of the Democratic race after Iowa, didn't have to worry that a vote for him might benefit, say, Barack Obama, whom they like less than Hillary Clinton, they could have shown support for their man and his foreign-policy expertise by voting for both him and Clinton. That might have clarified somewhat how much voters value experience. "Election returns would better reflect the overall acceptability of candidates, rather than being distorted by considerations of electability or about wasting your vote," says Brams. The best-known race decided by approval voting is for secretary-general of the United Nations. That, more than plurality voting, tends to ensure that an extremist candidate cannot best two centrists who split the majority's vote and let the fringe candidate in.

With three viable Democrats remaining, it's unlikely that the nominee will be someone whom most Dems rank their least favorite. But with four viable Republicans, that is a real possibility. If Florida 2000 wasn't enough to get us to re-examine plurality voting, though, grumbling in the GOP ranks probably won't be either. It is said that a nation gets the leaders it deserves. Maybe we also get the voting system we deserve.