Lila Guterman: When Votes Don't Add Up, Chronicle of Higher Education 47,10 (3 Nov. 2000) page A18

Mathematical theory reveals problems in election procedures

WHEN they approach the polls next week, supporters of Ralph Nader and Pat Buchanan will face a quandary because of their candidates' slim chances of winning the presidential election. Believing that a vote for either is wasted, the voters could support their second choice to try to influence the outcome of the tight race between Vice President Al Gore and Texas Gov. George W. Bush.

The frustration of voters who feel forced to choose between conviction and strategy after they enter the voting booth reveals a flaw in our voting system, say several mathematicians and political scientists, who are using the close 2000 presidential election to emphasize the benefits of alternative scoring methods. Under the current system, the candidate with the most votes wins in each state whether or not that person has a majority, thereby allowing a third-party candidate to upset an election, even with just a small fraction of the votes.

It also means that in races with more than two candidates, the candidate considered acceptable by the largest percentage of the electorate can actually lose.


That happened in 1991 in the Louisiana gubernatorial race, says Donald G. Saari, a mathematician at the University of California at Irvine. In a 12-way general election, Republican David Duke, a former Ku Klux Klan grand wizard, received 32 percent of the vote, while Democrat Edwin W. Edwards, a former governor who bragged about his gambling and had been indicted twice on federal racketeering charges, got 34 percent, both eking out more votes than the incumbent Republican governor Charles E. (Buddy) Roemer, who received 27 percent. "It was reasonable to suspect that incumbent governor Roemer would have beaten either of them in a head-to-head race," says Mr. Saari.

The result was a widely disparaged "Krook-or-Klan" runoff. Bumper stickers supporting Mr. Edwards read "Better the lizard than the wizard." Mr. Edwards won the runoff with 61 percent of the vote. A poll found that almost half of the voters who chose Mr. Edwards said their main motive was to defeat Mr. Duke.

Examples of this phenomenon abound in less visible races. In New York in 1970, a moderate incumbent candidate for the Senate, Republican Charles E. Goodell, received only 24 percent of the vote after losing votes to candidates on both the left and the fight: to Democrat Richard L. Ottinger, who received 37 percent of the vote, and to Conservative James L. Buckley, who won with 39 percent. Like Mr. Roemer in Louisiana, Mr. Goodell could have beaten each of the other two in one-on-one contests, making him the strongest candidate, says Steven J. Brams, a professor of politics at New York University. "I argue that I got the wrong senator for six years."


There are other, better ways to choose single winners from groups of candidates, argue Mr. Saari and Mr. Brams. Voters could assign ranks to the candidates or select as many as they deem acceptable.

Sometimes the way the election is carried out can influence the results as much as the voters' choices do. Mr. Saari says that for three-way races, in about three-quarters of the possible scenarios based on different voters' preferences and several voting systems, election outcomes will vary according to which scoring procedure is used.

Mathematical theory can help sort out which methods will cause fewer problems. It has been proven, however, that no voting procedure is perfect.

In 1952, Kenneth J. Arrow, an economist at Stanford University, showed that no voting procedure--apart from a dictatorship, where only one person votes--can satisfy five basic conditions of fair elections when there are three or more candidates. The conditions seem straightforward. For example, one condition says that if all voters prefer candidate A to candidate B, then A should rank higher in the election results. Another states that how voters feel about candidate C should not affect whether A ranks higher than B.

A dramatic flaunting of the latter condition took place in the women's World Figure Skating Championships in 1995. The U. S. champion, Nicole Bobek, had skated into second place behind Chen Lu of China. In third place after her final performance was Surya Bonaly from France. Then, a relatively unknown skater, 14-year-old Michelle Kwan, took the audience and judges by storm with a performance that catapulted her into fourth place.

Ms. Kwan's skate did not alter any of the judges' scores for Ms. Bobek or Ms. Bonaly. But after the votes were tallied, their positions flipped. Ms. Bonaly went home with the silver, and Ms. Bobek won the bronze.

The scoring rules called for each judge to rank the skaters and then for the group to determine winners by using a modified plurality vote based on those rankings. Ms. Bobek had had more second-place rankings than Ms. Bonaly until Ms. Kwan skated, but got bumped out of second in enough judges' rankings to give the silver medal to Ms. Bonaly.

[Editor's note: Actually, calling the horrific voting system they used a "modified plurality vote" is highly oversimplified, kind of like calling the New York City subway system a "modified rickshaw." The system they actually used acquired numerical scores on an 0-to-6 scale for each skate from each judge (such as "5.9"); but after the skates were all done the voting system would (foolishly!) discard those numbers by converting them into rank orderings; then the "lower median ranking" of each skater would be computed (e.g. if you had 10 scores which were 1st, 2nd, 2nd, 3rd, 3rd, 4th, 4th, 4th, 4th, 5th your "lower median" would be "4th"; actually more precisely it was the worst ranking awarded by the smallest majority of judges ranking you the best); then skaters were compared based on those lower medians; except that this would almost always yield a vast number of ties; those ties then were broken using Borda score. I learned this from the paper by Carroll, Rykken, Sorensen in MAA Math Horizons 10 (Feb 2003). The fact it was all based on rankings and not on the numerical scores (which really were only an optical illusion) was the reason Arrow's theorem was applicable to predict this problem would happen.]

The skaters did not question the methods that caused the flip-flop in standings, instead acknowledging that the rules were very complex and that strange things can happen as a result. Like the general public's attitudes toward the plurality vote in elections, Mr. Saari says he thinks skaters put up with a procedure that causes such paradoxes "because they don't know how to question it." Mr. Brams agrees. "By and large, the ordinary voter doesn't realize there are alternatives," he says.


Although no voting method is perfect, fairer procedures exist, researchers say. Mr. Brams's favorite is approval voting, in which each person votes for as many of the candidates as he or she approves of. "It violates the principle of one person, one vote," Mr. Brams says. "We have a different slogan. We say, one candidate, one vote."

Mr. Brams says the method would tend to elect centrist candidates who have the support of a wide swath of the electorate and would reduce negative campaigning because candidates might not want to alienate their opponents' supporters. His analysis of the 1970 New York senatorial race shows that Mr. Goodell would have received about 55 percent of an approval vote, while his two opponents would have garnered only about 50 percent each.

Supporters of minority-party candidates like Mr. Nader could vote for both their favorite candidate overall and their preferred major-party candidate. "Under approval voting, you can have your cake and eat it too," Mr. Brams says.

Mr. Brams's research into approval voting inspired him to become "something of an activist pushing for its adoption," he says. He has testified before congressional committees in New York and New Hampshire and appeared on Good Morning America to try to raise awareness of this alternative voting procedure. Although a bill in North Dakota to adopt approval voting passed in the state Senate, it lost in the House of Representatives.

"In a way, I secretly hope that the 2000 presidential election will produce a result that makes a lot of people unhappy," such as Mr. Nader or Mr. Buchanan gaining enough votes to shift the election results, he says. "It would show the fragility of our system."

He's not the only one who wouldn't mind an electoral upset. "There's some real chance this year [that] Gore could win the presidency but lose the popular vote nationwide," says Robert D. Richie, the executive director of the Center for Voting and Democracy, a nonprofit organization in Takoma Park, Md. That hasn't happened since 1888, when Benjamin Harrison won in the Electoral College even though Grover Cleveland won the popular vote. If it happens again, Mr. Richie predicts "a great movement" would arise to get rid of the Electoral College. That could focus attention on alternative voting procedures.

His organization has been promoting a system called instant runoff or single transferable vote. Under that system, voters rank the candidates from first to last place. If a candidate gains a majority of first-place votes, that person wins. But if no one does, the votes are re-tallied after the person with the fewest first-place votes is eliminated. The votes of people who chose that person first are shifted to their second choice. The process continues until one candidate gains a majority.

Mr. Richie says an advantage of the instant runoff system is that minority-party candidates do not spoil elections. The procedure, already used in some other countries, may be making headway in the United States: Alaska will vote on a ballot measure in 2002 to employ instant runoff voting in its elections, and Vermont is debating a similar measure, according to Mr. Richie. Both measures would include the presidential election, since the states have the power to decide how to run such elections.


By contrast, Mr. Saari claims that a procedure called the Borda count is preferable to either approval voting or instant runoff in representing the views of the voters. In the Borda count, voters rank the candidates and total points are added. In a five-person race, for instance, a voter's top choice would receive five points, the next choice four, and so on.

In fact, Mr. Saari argues, the Borda count is the best procedure available--in theory, at least. Last winter, he published a proof in the journal Economic Theory that the Borda count is the only voting method that fairly treats preferences that are symmetrical. That means that in a three-candidate race among A, B, and C, a voter who preferred A to B to C would have his or her ballot canceled, in essence, by one who preferred C to B to A; similarly if one person liked A better than B and B better than C, while another preferred B to C to A, and a third voter chose C over A and A over B, these votes would result in a three-way tie. All other voting procedures have "hidden, subtle inadequacies" that can result in one candidate gaining support despite the fact that such scenarios should be ties, he says.

The Borda count results in "by far" the lowest likelihood of paradoxes such as the most popular candidate losing or a skater flip-flopping the ranking of two others, according to Mr. Saari. In a six-candidate election, he has estimated, plurality voting can cause 1050 times as many paradoxes as the Borda count.

Mr. Saari says that since Mr. Roemer was probably the second choice of many voters in the Louisiana gubernatorial race, he likely would have beaten both Mr. Duke and Mr. Edwards under the Borda count


Many social issues concerning the adoption of new voting procedures remain. How would voters adapt to choosing more than one candidate under approval voting? Could voters understand the Borda count? But even treated strictly at the level of theory, experts cannot agree on which procedure is the most fair. Mr. Saari says that in approval voting, decisions about how many borderline candidates to vote for can cause enormous changes in the results. Mr. Brams says that instant runoff can create bizarre situations where ranking a candidate higher can actually hurt that person's chances of winning. He also argues that the Borda count encourages strategic voting, rather than honest rankings, in races like next week's. For example, voters who prefer Mr. Bush to Mr. Gore and Mr. Gore to Mr. Nader would have an incentive to place Mr. Nader ahead of Mr. Gore in their rankings to lower the points given to Mr. Bush's stronger opponent.

"It's my view that there is no universally best system," says Arnold B. Urken, a professor of political science at Stevens Institute of Technology in Hoboken, N.J. He says one of the problems with coming up with workable voting systems that stand a chance of being adopted is that the gap between pure analytical theory and hardball politics has not been bridged. "The gatekeepers are the political leaders in the status quo," he says. "They're going to be unlikely to want to consider any alternatives."

How to Choose a Winner

In the United States, the most common method in elections is to have each voter choose one candidate, and the person who garners the most votes wins, regardless of whether that person has achieved a majority. But many alternative methods exist for picking one winner out of several candidates.

Runoffs or sequential voting. Each voter chooses one candidate, but to win, a candidate must gain a specific fraction of the votes, often a majority. If no candidate wins that fraction, a second election is held between the top vote-getters. This procedure is used in many other countries, as well as in several U.S. cities and states.

The Condorcet method. In the late 1700s, the Marquis de Condorcet, a French mathematician and revolutionary, suggested that a candidate who can beat all the others in head-to-head matchups should win an election. The procedure is not in widespread use perhaps because it requires multiple elections and it often results in so-called cyclic ties, where A beats B, then B beats C, and then C beats A, producing no clear winner.

The Borda count. Named for Jean-Charles de Borda, a French physicist and a contemporary of Condorcet, the Borda count requires each voter to rank the candidates and assign points. For instance, in a three-way vote, a voter's first choice receives three points, the second choice receives two, and the third choice receives one. The candidate who gains the most points wins the election. The Associated Press and the Coaches Poll, run by USA Today and ESPN, use the Borda count to rank teams in certain college sports, including basketball and football.

Instant runoff or single transferable vote. In this method, voters rank the candidates but only the first-place votes get counted. If no candidate achieves a majority, the candidate with the smallest number of first-place votes gets eliminated and the top choice on each ballot among the candidates remaining is counted. That means that a vote then goes to the second-choice candidate if a voter had ranked the eliminated candidate first. The procedure is repeated until one candidate gains a majority. This method is used in Australia and was recently adopted for London's mayoral race.

Approval voting. Each voter gives one vote to each candidate whom he or she approves of. The candidate with the most approval votes wins. Approval voting is used in several professional societies, including two mathematical societies and the 350,000-member Institute of Electrical and Electronics Engineers.

Dictatorship. Only one person's vote counts in making decisions. As well as being used in many countries, this voting procedure also crops up in corporations where one investor owns more than half of the stock.

PHOTO (COLOR): Donald G. Saari of Irvine says Louisiana governor Buddy Roemer was rooked out of re-election in 1991 by our plurality voting system.

PHOTO (COLOR): Steven J. Brams of New York University argues that a flawed voting system resulted in New York getting "the wrong senator for six years" in the 1970 election.


By Lila Guterman