Remember John McCain? Oh, sure you do now, but will you remember him 20 years from now? If history is any guide, McCain's 2000 campaign will end up on the curio shelf next to John Anderson's presidential run in 1980 or Paul Simon's campaign in 1988. And yet, as Al Gore and George W. Bush vie for the presidency this month, it's easy to imagine a very different election— one that might better reflect the true wishes of the people.As late as February of this year, McCain's "Straight Talk Express" was on a roll. He won the New Hampshire primary, hit a bump in South Carolina, then won a highly publicized Michigan primary, in spite of the Michigan governor's promise to deliver the state to George W. Bush. The news media loved McCain, and so did the voters. In a February 7 CNN/USA Today/Gallup poll, more Americans thought favorably of him than of any other candidate. When the poll was repeated on February 28, his lead had widened: 66 percent had a favorable opinion of him compared with 59 percent for Al Gore, 57 percent for George W. Bush, and 54 percent for Bill Bradley. Yet nine days later, McCain was out of the race.

What happened?

The short answer is Super Tuesday. McCain was wiped out in the delegate-rich states of California and New York, while winning only a few small states in New England. The loss in California, a winner-take-all primary, hurt most. An exit poll by the Sacramento Bee suggested that McCain had been too critical of right-wing religious leaders. The poll also showed that voters believed Bush was more likely to beat Gore in November than McCain was. Yet the same voters, in the same exit poll, said they would vote for Gore over Bush (51 to 43 percent), and they would vote for McCain over Gore (48 to 43 percent).

While campaign strategists and political pundits comb through the wreckage to see where McCain went wrong, ordinary voters would do well to raise a different set of questions. All the signs suggest that the most popular candidate— a candidate who could have drawn voters from both parties— is not on the ballot this November. If a candidate like McCain can't win, is something wrong with our election system?

The answer, say voting theorists Donald Saari of the University of California at Irvine and Steven Brams of New York University, is a resounding yes. America's presidential election system is fundamentally flawed, they argue, because both the primaries and the election are based on the plurality vote. McCain is just the most recent candidate done in by the paradoxes of election mathematics.

In a democratic election between two candidates, the winner is the person with the majority of the votes. But when three or more candidates run, things are seldom so simple. The winner often amasses only a plurality, not a majority, of the votes. (Bill Clinton, for example, won the presidency with 43 percent of the vote; Jesse Ventura won the Minnesota governorship with 37 percent.) The plurality winner could be everybody else's least favorite candidate and could even lose to each of the other candidates in a head-to-head battle. As Saari puts it: "The plurality vote is the only procedure that will elect someone who's despised by almost two thirds of the voters."

Voting theorists have recognized the weakness of the plurality system for centuries. "The apparent will of the plurality may in fact be the complete opposite of their true will," wrote the Marquis de Condorcet, a close friend of Thomas Jefferson and author of a proposed French constitution, in 1793. That is why runoff elections are often used in races that are expected to draw a lot of candidates.

What students don't learn in high-school civics class, and certainly won't hear from the two major parties, is that there are many voting systems besides the plurality vote and the runoff election. Some systems are even older than that of the United States. In recent years, there has been a steady increase of research on voting theory (also called "consensus theory" or "social choice theory"). For the first time ever, Saari says, "We have the tools to systematically analyze all the procedures."

So what is the best procedure?

That's the catch. Saari favors a method called the Borda count; Brams advocates a method called approval voting. Many would argue that there can be no definitive answer until other systems are tested. Alas, consensus is as elusive as ever— even on the theory of consensus.

Approval voting, the simplest of the alternative methods, dates back to at least the 13th century, when Venetians used it to help elect their magistrates. It was subsequently reinvented many times, although it didn't acquire a name until 1976. In an approval vote, a person casts one vote for every candidate he or she considers qualified for the office. It's like an opinion poll, only the results are added up to determine a winner.

Brams argues that all presidential elections should be decided this way. "With approval voting, you can eat your cake and have it, too," he says. Voters who like a dark horse don't have to feel as if they are wasting their votes: "You can vote for all the out-of-the-running candidates you want to, and a safe choice as well." Even if the safe candidate wins, the support for other candidates will be noted. At the same time, Brams believes, the quality of debate would also improve: "Campaign strategies would change. You would have to be more expansive in your appeal." Third-party candidates, like Ralph Nader or Ross Perot, would get more votes and get their ideas out to a wider audience.

Can approval voting truly change the outcome of an election? Absolutely, as the CNN/USA Today/Gallup poll demonstrated. If this year's election had been decided by an approval vote in February, McCain would have won. True, by the November election his rivals would have had more time to dig up dirt against him, but the polls showed that three weeks of intense campaigning by Bush didn't make a dent in McCain's approval ratings. Besides, Brams argues, negative campaigning wouldn't be as widespread if we used approval voting, because it would be more likely to backfire against its users.

The advantages of an approval vote— and the perils of plurality voting— are most apparent in contests like the Louisiana governor's race of 1991. The primary that year was dominated by three candidates: Edwin Edwards, the often-indicted former governor; David Duke, a former grand wizard of the Ku Klux Klan; and incumbent governor Buddy Roemer. Edwards won the primary with 34 percent of the vote compared with 32 percent for Duke and 27 percent for Roemer. But it was Duke's surprisingly strong showing, despite his overtly racist stance, that won national headlines. Time and Newsweek ran long articles about the politics of hate in America. Bumper stickers, anticipating an Edwards-Duke runoff election, urged Louisianans to "Vote for the crook: It's important."

In the end, Edwards walloped Duke by a 61 to 39 percent margin. But the result was hardly a triumph for the runoff system. Say what you will about Louisiana voters, it's unlikely that anyone other than Edwards's core supporters really wanted to put a "crook" in the governor's office. And the election returns from November show beyond a doubt that very few people approved of Duke, outside of the 32 percent who originally voted for him. Roemer, on the other hand, had no strikes against him except that he had recently switched parties. In an approval vote, he might well have finished first, sparing Louisianans the choice between racketeering and racism. By the same token, approval voting might have spared Minnesotans from electing a professional wrestler to the governor's seat two years ago, or New Hampshirites from handing Pat Buchanan a triumph in the 1996 presidential primary.

Some theorists object that approval voting violates the "one person, one vote" principle. But that is partly a misconception. The Supreme Court decision that enshrined that principle— Baker v. Carr, in 1962— was meant to ensure that each voting district in a state had the same population. Approval voting wouldn't change that. More important, Brams argues, approval voting gives each voter equal "sovereignty" over the way his or her vote is counted. "Voters are more equal if they have an equal opportunity to express themselves," he argues. "If I prefer one candidate above everyone else, I can better express myself with a bullet vote for him. Another voter who hates one candidate can express his preferences by voting for the other four."

Approval voting has some strong advocates in the voting science community, but it's hardly the clear front runner (see "Mathematicians in the Voting Booth," page 82). Its chief competitor is the Borda count, the method championed by Donald Saari. The Borda count was named after a French physicist— and later a hero in the American Revolution— named Jean-Charles de Borda, who proposed it in 1770. But it was used in the Roman senate as long ago as A.D. 105. Although it sounds obscure, sports fans should recognize it as the method used to rank college football and basketball teams.

In a Borda count election, each voter ranks all of the candidates from top to bottom. If there are, say, five candidates, then a voter's top-ranked candidate gets 5 points, his second-ranked candidate gets 4, and so on. Finally, the points from all the voters are added up to determine the winner.

Though more complicated than approval or plurality votes, Borda counts sidestep certain pitfalls. Suppose, for instance, that three voters are trying to decide whom to vote for in a primary. Alice ranks McCain first, Bush second, and Gore third. Betty ranks Bush first, Gore second, and McCain third. Cheryl ranks Gore first, McCain second, and Bush third. The situation is completely symmetrical; each candidate has one first-place, one second-place, and one third-place ranking. "I have yet to find anyone who says this should be anything else but a tie," Saari says, and in a Borda count, it is.

Consider a two-stage election, however, consisting first of a Republican primary followed by a runoff between Gore and the Republican winner. In stage one, Cheryl and Alice would vote for McCain, and so he would win the primary, 2 to 1. In stage two, Betty and Cheryl would vote for Gore over McCain, and so Gore would win the runoff, 2 to 1. In this format, the result isn't a tie: Gore wins. In fact, in this year's Michigan primary, Bush's supporters complained that some Democrats, like Cheryl, crossed party lines to help McCain beat Bush. But it wasn't their fault— the system encouraged it.

Now consider a four-person election in which two voters have exactly the opposite preferences: Andrew's preferences are, in descending order, Gore, Bradley, McCain, and Bush. Bob prefers Bush first, McCain second, Bradley third, and Gore fourth. Once again, most people would call this a four-way tie, and a Borda count would agree with them. But in a plurality vote, Gore and Bush would tie for first and Bradley and McCain would be out of luck.

"The reason voting procedures give us paradoxes and unwanted outcomes is that they do not respect the symmetries of data that give us ties," Saari says. Is there any system that would consistently declare the vote of Alice, Betty, and Cheryl a tie, and would do the same for Andrew and Bob? "Only one procedure does that," Saari says. "The Borda count." To prove his point, he published two 50-page papers on the Borda count in the January issue of the journal Economic Theory.

How would McCain, Bush, and Gore have fared in a general election under the Borda count? The Sacramento Bee poll, plus the official tallies for California's open primary, offers a clue. According to the Bee's results, McCain would have beaten Gore 48 to 43, and Gore would have beaten Bush 51 to 43. But could McCain have beaten Bush among all voters, not just Republicans? Here's one way to figure it out. In the California primary, 60 percent of Republicans voted for Bush and 35 percent voted for McCain. We can assume that those percentages would have stayed the same had they been slated to run against each other in November. On the Democratic side, nearly 800,000 voters broke party ranks in March's primary to vote for Republicans. Of these, 64 percent voted for McCain and 31 percent voted for Bush. Again, the simplest thing to do is assume that the entire Democratic party would have split the same way if they were forced to choose between those two candidates in November.

So in our hypothetical Bush-McCain race, Bush would win by 60 to 35 among the Republican voters, and McCain would win by 64 to 31 among the Democratic voters. Overall, McCain would beat Bush, 50 percent to 45 percent. (To arrive at those figures I first added together each candidate's percentage of the Democratic and Republican vote— 60 and 31 for Bush, 64 and 35 for McCain— and divided them in two. That gave 45.5 for Bush and 49.5 for McCain. I then converted those totals to percentages and rounded them off.) To see how all three candidates would do under the Borda count, we add up the results of the head-to-head matchups: