Imagine that in the 1992 presidential election President Bush receives 40 percent of the vote, Sen. Lloyd Bentsen (Tex.) as the Democratic nominee gets 35 percent and Jesse L. Jackson, running as an independent, receives the remaining 25 percent.
Bush wins. But would this necessarily be a fair election?
What if all the Bentsen voters liked Jackson second and Bush least. And let's say all the Jackson voters also liked Bush least. Then Bush would have been elected even though 60 percent of the electorate considered him the least acceptable candidate. Under these circumstances, in a two-way race with either candidate, Bush would be beaten handily.
To avoid electing candidates who do not have majority support, some jurisdictions use a two-stage process that includes a runoff between the two top vote-getters. Were the voting preferences the same, then, Jackson would lose in the first round. And in the runoff, Bensten would add Jackson supporters and defeat Bush, reversing the outcome of the one-stage system.
But wait a minute: Suppose everyone who voted for Bush liked Jackson second best and Bentsen least. That would mean Bentsen becomes president even though in a two-man race with Jackson, supposedly the least popular candidate, Bentsen would have lost in a landslide.
Which of the three men really deserves to be president?
Many Ways to Pick a Winner
This is a hypothetical example, of course. In real life the workings of the electoral college could easily nullify this theoretical three-way contest. But the confusing and paradoxical picture it paints of elections is real. In any contest where there are more than two alternatives -- from choosing a president to picking Miss America -- even the most democratic and common-sense voting methods can sometimes produce bizarre results that seem to have little to do with what people actually want.
In the 300 years since these voting paradoxes were first identified, some of the most brilliant minds in mathematics and political science have looked for ways around the problem. Jeremy Bentham, the 18th century British political philospher, and Lewis Carroll, the English mathematician and writer, along with countless others have proposed ingenious variations on simple plurality voting to ensure that in situations like the imaginary Bush-Bentsen-Jackson election the right candidate wins.
But in recent years, voting theorists have reached a potentially chilling conclusion: Devising a voting system that unfailingly interprets what people want is impossible.
Consider the accompanying illustration. In it, five voters list their order of preference for five candidates. Under a simple plurality election, candidate Jones would win because he has two first-place votes.
Using the method devised in the 18th century by the Marquis de Condorcet, a well-known scheme in which each candidate is matched up in head-to-head contests with every other candidate, Smith wins.
Using the Bentham method, in which voters rank the candidates using numbers from 1 to 5 and the rankings are tabulated, Cook wins. In fact, almost every time a different and seemingly reasonable voting scheme is used, a different candidate wins.
Political scientists who have studied the question consider this extraordinary. What it says is that the results of elections depend as much on how the votes are counted as on what people actually want.
Far from being objective reflections of popular will, democratic voting systems are actually powerful and independent forces, capable of shaping or mangling public preferences in the process of producing a result.
It gets worse.
In the early 1950s, in an acclaimed series of theorems that later won him the Nobel Prize, American economist Kenneth Arrow proved that all conceivable voting schemes have some potential for being unfair or producing a paradoxical result.
In other words, the fact that in the example in the illustration candidates Jones, Smith, Cook or Phillips could each win the same election, depending on how the preferences are tallied, means the electoral system problem cannot be solved.
There is no one "correct" winner. And if all systems are flawed, we have no way of knowing which candidate is the people's choice.
Arrow's theorem, in the 40 years since it was proposed, has led to an avalanche of research in economics, political science and philosophy. Today "social choice theory," as it is known, ranks as one of the hottest intellectual topics.
After all, consider what it says about democracy. Could it be that the choice of our leaders and the shape of our political history is sometimes simply an accident? Did the country choose Abraham Lincoln over Stephen Douglas, John Breckenridge and John Bell in 1860 or Richard M. Nixon over Hubert H. Humphrey and George C. Wallace in 1968 because it wanted those men as president or because that was the idiosyncratic interpretation the U.S. political system put on the votes cast in that year?
And if we have no idea whether an electoral result actually represents the will of the people, how can any democratic government be considered legitimate?
Creating Artificial Majorities
Most experts do not go that far. But their work has has led to certain amendments to classical liberal democratic theory. For example, Arrow's disciples no longer talk about the will of the people because they are convinced by the multiple failures of voting systems that such collective preferences are impossible to calculate or may not even exist at all.
"It's a classic case of anthropomorphism," said Charles Plott, a poltiical scientist at the California Institute of Technology. "We ascribe to groups a property that we see in ourselves, namely of having a single mind, and they don't."
Arrow disciples stress that the majorities created by democratic systems are actually artificial creations made up of many different and sometimes contradictory groups. Democracies, they say, should simply be sensitive to this fact when they make decisions.
They are also keenly aware that voting, far from being a sacred democratic rite, is a process that can be manipulated.
Imagine a political philosopher, for example, asked to decide whether to increase or cut the salary of his university president.
The only other professor asked to vote on the issue, a sociologist, thinks the president should be paid more. The philosopher thinks the whole idea of professors voting on salaries is silly. He abstains.
In a straight vote, then, the philosopher's preference would lose because he casts no ballot and the sociologist votes yes. But if the philosopher votes against the proposal, his vote will cancel out the sociologist and the deadlocked outcome will be exactly what he wanted: Someone else would decide on the president's pay.
This is the final paradox. In order to get the voting system to reflect what he wants, the philosopher has to cast his vote for something in which he does not believe.
He has to lie.