Wednesday, March 19, 2008

The Spoil-Proof Election

A Lesson in Election Mathematics

When Ralph Nader recently announced he was entering the 2008 presidential race, many Democrats groaned. It was his fault, they say, that George Bush defeated Al Gore in 2000. But Nader retorted that the Democratic Party has only itself to blame for the loss in 2000.

Mathematicians offer a different perspective. The problem, they say, doesn't lie with Nader or with the Democrats. It lies with our voting system.

Complaints about the obscure Electoral College system are common, but the mathematicians' objection is even more basic. Presidential elections in the United States are decided using a variation of a method known as plurality voting: each person votes for one candidate, and the candidate with the most votes wins.

Seems like the obvious approach—but obvious doesn't always mean effective. "The plurality vote is pretty much the worst voting system there is," says Donald Saari, a mathematician at the University of California, Irvine.

The 2000 election gave a vivid demonstration of plurality voting's limitations. Polls indicated that most people who voted for Nader would have preferred Gore to Bush. The votes for Nader and Gore combined in Florida would have beat Bush. But with the votes divided between them, Bush emerged the winner.

Though this example is especially dramatic, Saari has found that determining voters' preferences from their ballots is often tricky. For example, suppose three candidates, A, B, and C, are competing. The preferences of the voters are as follows:

  • 3 people rank A first, B second, and C third;
  • 2 people rank A first, C second, and B third;
  • 2 people rank B first, C second, and A third; and
  • 4 people rank C first, B second, and A third.

Plurality voting would name A the winner, with 5 votes.

On the other hand, suppose one wanted the candidate that was least disliked. Six people rank A last, two people rank B last, and three people rank C last, so in that case, B should win.

Yet another method would be to assign 2 points for a first place vote, 1 point for second place and none for third. In this method, known as the Borda count, C walks away the winner with 12 points, beating out B's 11 points and A's 10.

So who should win the election?

Examples like this lead Saari to conclude that "election outcomes can more accurately reflect the choice of an election rule rather than the voters' wishes." He even jokes to colleagues that for a price, he could rig the election of their next department chair to guarantee that their preferred candidate would win. He would interview everyone in the department to determine their preferences and then choose an election method—one that could be argued to be fair—that would produce the desired outcome.

Indeed, he has found that 69 percent of the time, an election result can be changed by changing the voting rules.

But that doesn't mean there's no basis for choosing the best rule. Although mathematicians haven't settled on a single choice, they've done a lot of work to explore the consequences of choosing one method over another.

Saari's preferred method is the Borda count, because he believes it reflects the voters' wishes most accurately. Suppose, he argues, that voters prefer candidate A to candidate B to candidate C and candidate B then drops out. The voters should still prefer A to C, right? Saari found that for three-candidate elections, the Borda count is the method most likely to ensure that.

Plurality voting, on the other hand, often does terribly at this test. For example, in the contest above, suppose candidate B withdraws. Although A should be the winner just like before, in fact, five people would prefer A to C and six would prefer C to A, making C the winner. Similarly, if C withdraws, A should still win, but in fact, B would.

Saari says this property explains much of the horse-race jockeying between candidates during the presidential election. The media, for example, often speculate on the impact it will have on other candidates if one drops out of the race. In a system like the Borda count, a candidate dropping out wouldn't change the rankings of the other candidates.

But Steven Brams of New York University, another researcher in the area, prefers a different voting method, one known as approval voting. In that method, voters vote yes for each candidate they find acceptable.

"The major problem with the Borda count is its manipulability," Brams says. "If you have a favorite candidate, and your second choice is his or her fiercest competitor, you have no reason to vote sincerely." Ranking the competitor second, after all, will give away a point, weakening your effective support for your favorite. As long as you're confident that your third pick isn't likely to win, you're better off putting your least-favorite candidate second.

Brams acknowledges that approval voting is subject to paradoxes that the Borda count isn't. But, he says, "I think these paradoxes in many cases are artificial, constructed, contrived. They're rare events."

The two are in agreement, however, that the problems with the current system are not rare. "The real examples are everywhere," Saari says. "Just look at Nader."


References and sources for this post::

Brams, S. 2008. Mathematics and Democracy: Designing Better Voting and Fair-Division Procedures. Princeton, N.J.: Princeton University Press. See

Saari, D. 2001. Decisions and Elections: Explaining the Unexpected. Cambridge, England: Cambridge University Press. See

Chaotic Elections! A Mathematician Looks at Voting. Providence, R.I.: 2000 American Mathematical Society. See

Further Readings:

Donald Saari has a website at It includes videos on the Mathematics of Voting, Creating Voting Paradoxes, and an explanation of What Causes Voting Paradoxes.

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