COLUMN ONE : Vetoing the Way We Vote : Blame America’s electoral woes on how we pick our leaders, mathematicians say. They claim dumping winner-take-all pluralities for other methods could reduce apathy, extremism and mudslinging.

Negative campaigning, low voter turnout, elections polarized by such issues as race and abortion--such all-too-familiar ills appear more and more frequently as blotches on the face of American democracy. It does not appear that politics alone has much of a chance to make changes for the better.

But a widely overlooked factor in the election equation--the mathematics of voting itself--could have the power to at least alleviate, if not cure, these ills, according to political scientists and mathematicians who have studied the intricacies of voting systems.

In a country where people vote on just about everything--from presidents to prom queens--remarkably little attention is paid to the underlying rules of voting. Most Americans take for granted that the “winner takes all” system used in most U.S. elections is sacred. To the extent that they think about the system at all, people assume that as long as the process of voting is fair, the outcome will represent the wishes of most people, most of the time.

Mathematicians know differently. In fact, for more than 200 years, they have been studying the flaws of voting systems and arguing about which is the least harmful.

The subject is well known in academic circles. In 1951, Stanford economist Kenneth Arrow proved mathematically that no democratic voting system can be completely fair (and won the Nobel prize for his efforts). This notion is known as “Arrow’s impossibility theorem” because it proves that perfect democracy is impossible.

“Every system has something wrong with it,” said Temple University mathematician John Allen Paulos, “but some work better more often.” Mathematicians do not agree on the best system. But they have no problem pointing their fingers at the worst: the plurality system used in most U.S. elections.

The issue is of more than just academic interest. Plurality systems, used in most U.S. elections, hand victory to the candidate with the most votes even if that candidate falls far short of a majority and even if the candidate is the person the majority likes least. The current systems can encourage extremism, reward name-calling, alienate voters and fail to reflect the wishes of most of the people much of the time.

Until recently, the flaws of traditional voting systems received little public attention. But today, the issues that once were the province of mathematicians have begun to emerge into the broader political debate.

The Supreme Court recently provided one strong incentive. The court ruled in June that states may not consider race when they draw congressional districts. The decision has prompted lawmakers to think more seriously about other ways of ensuring adequate representation of minority groups. Georgia’s Legislature this week began considering how to redraw its district lines--the first steps in a process that could raise racial tensions across the South.

Another spur has been the continuing apathy of so many potential voters--in the 1992 presidential election, for instance, only 55% of the eligible adults cast ballots. Such turnout figures have led leading advocates of reform to look for changes that will make more people feel like players in the election process.

Although the tremors have not yet been felt by most lawmakers, they have the potential to shake the very foundations of U.S.-style democracy. How fair are voting systems? Fair to whom? How do people decide?

It is easy to show that election outcomes often depend quite directly on how voting takes place. In fact, outcomes are frequently determined by choice of voting system alone, according to political scientists and mathematicians who study the issue.

The Paradox

Often, those outcomes can be paradoxical, particularly when the traditional plurality system is used. In 1980, for example, a plurality system gave conservative Republican Alfonse M. D’Amato a U.S. Senate seat even though two liberal candidates had 55% of the vote to his 45%. The liberals split their votes between Republican incumbent Jacob Javits and Rep. Elizabeth Holtzman, a Democrat. Almost certainly, Holtzman would have won in a runoff, or under another voting system.

“Plurality is the worst because it has the highest likelihood of paradoxes,” said Northwestern University mathematician Donald Saari. “Of all the methods, it’s the one where you can have most people preferring A over B, yet B is the winner.”

Plurality--meaning the winner must get more votes than anyone else, not necessarily more than 50%--also means that a candidate need not be all that popular to win.

In the 1992 presidential election, Bill Clinton received 43% of the vote. Many Republicans have argued that George Bush would have won but for the presence of Ross Perot in the race, although exit polls indicated that Perot’s voters would have split--leaving Clinton in the lead.

In presidential primaries, where multiple candidacies are the rule, the plurality system frequently gives victory to candidates who win with a relatively small share of the vote. In 1976, plurality allowed Jimmy Carter to win the New Hampshire Democratic primary with 28% of the vote, setting him on the road to winning his party’s nomination and the presidency.

The same thing could well happen to Republicans this time around.

“I think the Republicans are likely to cut themselves to pieces [in the presidential primaries],” said New York University political scientist and mathematician Steven Brams. “They could get a winner with less than 20% of the vote.” If that scenario develops, said Brams, the winner is likely to be an extremist--”the one with the most vociferous support”--not the one with the broadest appeal to the general electorate.

Appropriately, mathematicians first started taking the problems with plurality seriously in the years leading up to the French Revolution. In the 1770s, Jean-Charles Borda showed that pluralities pick the most popular candidate only in two-person races. In elections involving three or more candidates, the one who has a plurality might easily lose when matched up against each of the other candidates one on one.

One Solution

One solution to the problem of pluralities is a runoff, which several states require when no one receives more than 50% of the vote. (The requirement does not apply in California for state or federal offices, but a runoff is used for many local offices.) Still, a simple runoff can result in a winner whom most voters would not have wanted. For example, a candidate who might have been broadly acceptable as a second choice may not receive enough votes on the first round to get into the runoff, and the current system gives no opportunity to indicate second choices, or other acceptable candidates. Brams and others are drawn to systems that allow a broader spectrum of choices.

The experts disagree, sometimes heatedly, about which alternative system is best.

Some, for example, argue for approval voting, which allows everyone to cast one vote per candidate--changing the “one person, one vote” dictum to “one candidate, one vote.” A liberal voter in the 1980 New York Senate race could have voted for Holtzman and Javits. The candidate thus “approved of” by the most voters wins.

While plurality systems encourage candidates to take extreme positions that develop a hard core of support, approval voting requires candidates to appeal for broad support. The system, experts believe, might do a lot to eliminate negative campaigning. “You would want to get at least partial approval from supporters of your opposition,” Brams said.

When he and his colleagues recalculated the 1992 presidential election under approval voting, they found that while Clinton still would have received the most votes, “the nature of the election would have been a lot different.” Relying on surveys that measured how warmly voters rated their second choices, Brams calculated that Perot’s support would have increased from about 19% to 42%, somewhat behind Bush, who would have gotten 50%. Clinton would have received about 55%, they calculated.

Possibilities for ’96

Looking ahead to 1996, approval voting would also help minority candidates such as the Rev. Jesse Jackson (or Patrick J. Buchanan), said Brams. “Even if Jackson couldn’t win under approval voting, more majority candidates would ape his views. Minorities can’t be ignored because majority candidates need their support to win.”

Approval voting is not without its detractors, however. Paulos argues that it tends to produce bland, mediocre winners. “Someone who doesn’t have any sharp edges won’t turn people off,” he said. “But sometimes you want someone who polarizes people, because one of the poles is right.”

The system Paulos likes much better is cumulative voting, in which voters can pile up several votes for a single candidate (or issue) they feel strongly about. As in approval voting, each voter has as many votes as candidates and can distribute those votes among the candidates or give them all to one candidate.

“It’s a good way to give minorities a voice without carving out race-based districts,” said Paulos. “Any group that has sufficient cohesion to vote as a bloc can win. There need be no reference to race. They could be wiry-haired mathematicians.”

Cumulative voting is widely used in the United States to elect corporate boards of directors. It’s the system championed by C. Lani Guinier, who in 1993 was Clinton’s nominee to head the Justice Department’s civil rights branch. Guinier’s nomination was withdrawn after Clinton decided her legal writings were “radical” views--despite the fact that cumulative voting has been around for more than a hundred years, and is used in city council elections even in all-American Peoria, Ill.

Neither approval nor cumulative voting is “fair” enough to satisfy Saari. Approval voting tends to be indecisive, he said, because it does not discriminate between first and second choices, and cumulative voting is too complicated, in that it requires people to develop group strategies to elect winners. The system he favors was developed by Borda himself, and is appropriately known as the Borda count. Under it, each voter ranks each candidate as first choice, second, third and so on.

“You still need to have broad support,” Saari said, “but it’s better than approval because you designate first and second choices.” Borda has worked out mathematically (in his book, “The Geometry of Voting”) that Borda counts produce the fewest contradictions and paradoxes--for example, situations where the least favored candidate can win.

Others support different alternatives. A Washington-based activist group, the Center for Voting and Democracy, has helped draft legislation recently introduced in Congress by Democratic Rep. Cynthia McKinney of Georgia, an African American whose district prompted the Supreme Court’s decision rejecting the use of race in drawing district lines. Her bill would permit states to experiment with alternative voting schemes. All proposed options would replace one-member districts with larger, multi-member districts.

Currently, federal law requires states to use single-member congressional districts. The law, passed in 1967, was designed to prevent minority votes from being diluted in at-large elections in Southern states, but, ironically, scholars like Guinier now argue the rule impedes minority representation.

“Single-winner races can open up a lot of permutations,” said Robert Richie, the director of the Center for Voting and Democracy. Richie’s choice is a system called preference voting--also known as the transferable ballot--a method already used to elect both houses of the Australian Parliament, the Irish president and the Cambridge, Mass., City Council.

Better Turnouts

Under preference voting, each voter ranks each candidate first, second, third and so forth. But if after an initial count, someone’s first-place choice seems doomed to defeat then that voter’s second-place vote is counted instead. While preference voting may sound complicated, Richie said it is used in other countries without problems and generates far better voter turnouts than in the United States. It is used to choose the five nominees in each Academy Award category--a fact that should vouch for its simplicity, he argued.

“It’s not any harder than learning the rules of baseball or basketball,” Richie said. Despite often-archaic rules, basketball remains “a pretty good game,” he said, although it can be difficult for short people to succeed at it. “The difference is, politics is the kind of game we want everyone to have a chance to play,” he said.

The U.S. government already runs on a fairly complex system of rules, observers point out. Indeed, the Constitution provides for a hodgepodge of voting schemes specifically designed to protect minority interests: Small states are protected by the Senate’s two-member-per-state rules, while populous states can wield more power under proportional representation in the House; the President can rein in an out-of-control majority with a veto; amendments to the Constitution require two-thirds approval of the states.

In the end, deciding on the fairest voting system will probably come down to, well, voting. “Before you decide on the substance, you have to decide what voting method you’re going to use,” said Paulos. “And then, what method are we going to use to choose the method? It can get all tangled and awful.”

‘Disaster for Voters’

The one thing that people who study voting do agree on is that America’s current system is the worst of all possible worlds. Indeed, the United States is virtually the only democracy that gives the plurality winner all of the power. “The current system is just a disaster for most voters,” said Richie, “and it’s getting worse and worse.”

Still, reform is not likely to come easily. Many people still feel that there is something vaguely un-American about choosing voting methods based on predicted outcomes--even though the experts say that is inevitable.

Then there is the troubling fact that the people who currently have the power to change the voting systems were elected by the systems now in place. Whatever system keeps people in power, proponents like to “wrap . . . in the mantle of democracy,” says Paulos. But that mantle “can come in many different styles, all of them with patches.”

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The Math of Voting

Different ways of voting can produce different winners--even when the voters don’t change their mind about their choices. Consider the hypothetical case of a Southland family trying to decide which movie to see on Saturday night. Sam and Sally, the 8-year-old twins, want to see “Pocahantas.” Tiffany, age 13, has been dying to see “Clueless,” while Tom, 18, is dead set on “Waterworld.” Dad, Mom and Stepmom, all 40-something, want to wax nostalgic at “Apollo 13,” but Grandma wants to try out her new pacemaker on “Species.” Grandpa and his sweetheart vote for “Pocahantas” along with the twins.

WINNER #1: On the face of it, “Pocahantas” wins with four votes. Using the plurality system common to most U.S. elections, the candidate with the most votes gets to make the decision for everyone else--even if, as in this case, “Pocahantas” is a minority candidate.

WINNER #2: Dad, Mom and Stepmom protest that the plurality system isn’t fair--because more people would rather see “Apollo 13” than “Pocahantas” if those were the only two choices (certainly Tiffany and Tom wouldn’t be caught dead at a Disney cartoon.) So the family votes again, pitting “Pocahantas” against “Apollo 13”; “Apollo 13” wins.

WINNER #3: But wait, says Tiffany, who’s beginning to see how this voting thing works and who suspects that both Grandma and Tom would rather see “Clueless” than even “Apollo 13.” This time around, the family rigs its election like a tennis tournament. Pitting “Pocahantas” against “Waterworld” eliminates “Waterworld.” “Pocahantas” against “Apollo 13” eliminates “Pocahantas.” But “Apollo 13” loses to “Clueless.”

WINNER #4: Now grandma has an idea. What if she arranges the tournament so at the last vote, only “Pocahantas” and “Species” are left? She’s pretty sure that under those circumstances, “Species” would win. So she asks for a revote, and this time, in the first round, she strategically votes for “Pocahantas,” even though it’s her least favorite choice. Now “Pocahantas” has enough votes to eliminate Apollo 13. Grandma votes for “Pocahantas” again against “Clueless”--effectively eliminating all her serious competition. When the dust settles, “Species” wins.