Nov 1, 2004 12:00 PM

The Creative Math Behind Elections

In a democracy, every vote is supposed to count. But the outcome of an election really depends on how you do the math. Different balloting systems – from cumulative voting to instant runoffs – can lead to different results. Some systems encourage citizens to vote tactically, others can be gamed. All can lead to results […]

In a democracy, every vote is supposed to count. But the outcome of an election really depends on how you do the math. Different balloting systems - from cumulative voting to instant runoffs - can lead to different results. Some systems encourage citizens to vote tactically, others can be gamed. All can lead to results that don't reflect voters' intentions (proof of which won economist Kenneth Arrow a Nobel in 1972.) Experts who study the math agree on one thing: The electoral college system used to decide US presidential elections is one of the worst. Here's a look at the promises of - and the problems with - a half dozen voting methods.

Plurality Voting
Promise: One person, one vote.
Precinct: Democracies worldwide
Problem: It's perfect when there are only two candidates, but that's rarely the case - especially in primaries. So voters may abandon their first choice for fear of "wasting" their vote. In general elections, minority blocs can hand a victory to a wild-card candidate who doesn't have majority support. A classic example: the 1998 gubernatorial race in Minnesota. The Republican and the Democrat got only 35 percent and 28 percent, respectively. An Independent, former professional wrestler Jesse Ventura, pinned down the election with just 37 percent of the vote.

Cumulative Voting
Promise: You get a bunch of votes - say, 10 - to distribute however you'd like: 10 votes to your favorite candidate, or maybe 3 to one and 7 to another.
Precinct: Numerous jurisdictions in Alabama and Texas, and many corporate boards (Toys "R" Us, Hewlett-Packard)
Problem: Results can vary substantially, because no single strategy will put a candidate in office. Giving all 10 votes to your top choice might handicap your second, who then turns out to be the one with a chance to beat the ultimate winner (who you hate). But 5 votes each to your first and second choices might hurt your favorite.

Instant Runoff
Promise: Voters rank the candidates, and their top picks are tallied. If that doesn't yield a majority, the candidate with the fewest votes is dropped, and his supporters' ballots are recounted and allocated to the second choice. The process continues until there's a winner.
Precinct: Australia, Ireland, and San Francisco
Problem: This is a strong challenger to plurality, but it isn't flawless. It fails the monotonicity criterion - the principle that voting for your favorite candidate should always benefit that candidate. Mathematical models prove that ranking a candidate lower can cause that person to climb in the overall rankings.

Approval Voting
Promise: Voters give one point to every candidate they consider acceptable.
Precinct: Many professional associations, notably the Mathematical Association of America and the American Statistical Association
Problem: The system doesn't distinguish between a great candidate and an adequate one. Most people vote for only one candidate, their favorite, while some select several. The results can be wildly erratic.

Borda Count
Promise: Point values are assigned to voter rankings of each candidate. If there are eight contenders, first place is worth eight points, second place is worth seven, and so on. High score wins. The system's inventor, Jean-Charles de Borda, helped popularize the metric system - and that worked out pretty well.
Precinct: The Associated Press polls for college football and basketball
Problem: This one is a favorite of voting-math nerds; it's theoretically immune to outcomes opposite of voter intentions. But voters may not know or care enough about each candidate to decide, say, who should be sixth versus seventh. With millions of voters, millions of votes would be arbitrary.

Electoral College
Promise: The winner of the presidential election in each state gets all of its pre-assigned electoral votes (equal to its number of seats in Congress), regardless of the margin of victory - except in Maine and Nebraska, which allocate electoral votes proportionately, and in Colorado, which will decide this year on whether to divide them.
Precinct: Only in America
Problem: "This is an embarrassing relic," says George Washington University math professor Daniel Ullman. Everyone agrees it warps the final count, but critics are split over whether it empowers or disenfranchises voters in big states. Maybe someday the Supreme Court will rule on that.

- Douglas McGray

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