Link to What's New This Week How Beef-Hungry Voters Can Get Tofu for President

Dear Habermas Logo and Link to Site Index A Justice Site

Plurality Voting

Mirror Sites:
CSUDH - Habermas - UWP - Archives

California State University, Dominguez Hills
University of Wisconsin, Parkside
Soka University Japan - Transcend Art and Peace
Created: March 14, 2003
Latest Update: March 14, 2003

E-Mail Icon

Site Teaching Modules Backup of How Beef-Hungry Voters Can Get Tofu for President
By Sharon Begley
SOURCE: Wall Street Journal
Copyright: Source Copyright.
Included here under Fair Use Doctrine for teaching purposes.


How Beef-Hungry Voters Can Get Tofu for President

With the Democratic presidential primaries so front-loaded in 2004, by this date next year it should all be over but the hotel arrangements for the convention in Boston.

Thanks to the accelerated schedule, those odd ducks who scrutinize returns, calculate how each additional candidate affects the others' chances and analyze strategic voting are already hard at work. I refer, of course, to mathematicians.

Yes, there is a mathematics of elections, and it has nothing to do with how they count votes in Florida. The research has identified various voting systems world-wide in which, paradoxically, becoming more popular can make a candidate lose, abstaining gives your preferred candidate a better chance, and picking a winner means accepting someone a majority of voters don't want.

This last paradox characterizes the U.S. system of plurality voting (vote for one; the top vote-getter wins). It works fine when there are two candidates, but with three or more, plurality voting can come up short.

A year from now, in other words, there is a good chance that President Bush's challenger may be a candidate few of the Democratic faithful like.

For a democracy, the mathematicians' most robust result is chilling. "It's surprisingly difficult to identify a voting system that accurately captures the will of the people," says Donald Saari of the University of California, Irvine.

So as not to inflame passions with examples such as Bush-Gore-Nader, I'll illustrate his point with food. You and two colleagues are planning an office party, and the caterer offers chicken, steak or tofu. You poll 17 invitees:

5 people prefer chicken to steak to tofu.

2 people prefer chicken to tofu to steak.

4 people prefer steak to tofu to chicken.

4 people prefer tofu to steak to chicken.

2 people prefer tofu to chicken to steak.

One organizer tallies the ballots by the plurality method, counting only first-place votes. Chicken wins (7 votes), while steak is last (4 votes). A second organizer uses "approval voting," in which voters mark all acceptable choices (everyone's top two choices are acceptable). Now steak wins with 13, tofu gets 12 and chicken is last with 9. The third organizer asks people to give their first choice 2 points, their second 1 and their last 0. Now tofu wins with 18, steak gets 17, chicken 16.

"The 'winner' changes with the choice of election procedure," Prof. Saari writes in the current issue of the journal Ethics. "An 'election winner' could reflect the choice of an election procedure" rather than the will of the people.

It gets better. Thanks to a mathematical property called nonmonotonicity, in some voting systems, ranking a choice higher can defeat it, explains Steven Brams of New York University. In a plurality-with-runoff system, the two candidates with the most first-place votes face one another in round two, as in France's presidential races. This time, we invite other departments to our office party, and get this first-round result:

27 prefer chicken to steak to tofu.

42 prefer tofu to chicken to steak.

24 prefer steak to tofu to chicken.

Chicken (27 votes) and tofu (42) reach the runoff. Assuming steak fans maintain their preference and give their second-round votes to tofu, tofu wins the runoff.

That seems fair. But what if four people in the group of 27 chicken lovers are last-minute converts to vegetarianism and, in round one, prefer tofu to chicken to steak, like the group of 42? Now steak (24 first-place votes) and tofu (46) make the runoff, in which steak beats tofu 47 to 46. Tofu's late surge turned its win into a loss.

Such paradoxes tend to occur under specific but far from unusual circumstances. With plurality voting, the most common is when two centrists face an extremist. The majority splits its vote between the centrists, allowing the fringe candidate to squeak in. In Minnesota's 1998 governor's race, Hubert Humphrey got 28% of the vote, Norm Coleman 34% and Jesse Ventura won with 37%, even though most voters ranked him last.

Thanks to such outcomes, scientists say what's most needed is "a way for voters to register their second and third choices," Prof. Saari says, "especially in primaries, where there tends to be a large field." Both a ranking system (give candidates 4, 3, 2 or 1 point) and approval voting accomplish that.

The U.N. chooses a secretary-general by approval voting. "It is particularly appealing in elections with many candidates," says Prof. Brams. "If your favorite candidate is a long shot, you can vote for both him and a candidate with a better chance without wasting your vote on the long shot. Approval voting would do a lot to address the problem of presidential-primary victors not being the choice of most voters."

The best system is one that brings to the polls more than the lamentable 51% of eligible voters who turned out in the 2000 presidential election. Approval voting could well make more people (especially supporters of long shots) feel their ballot matters. Still, no system is perfect. As Nobel-winning economist Kenneth Arrow proved mathematically in 1951, no voting system is guaranteed to be free of paradoxes in a race with three or more candidates, except one -- a dictatorship.

Write to Sharon Begley at

Updated March 14, 2003

Site Copyright: Jeanne Curran and Susan R. Takata and Individual Authors, March 2003.
"Fair use" encouraged.