Tuesday, August 08, 2006

A System for Every Taste

Factoring Perot: The Last May Be First; In a Three-Way Race, It's Tough to Figure Out the Will of the People

BY Barry Nalebuff
June 21, 1992
The Washington Post

IN A MATCH race like the America's Cup, the fastest boat wins. In the same way, in most presidential races two candidates compete in a match for the most votes (with the occasional third-party candidate sailing along like a dinghy trying to be one of the big boats). But this year, there are three big, fast entries in the race, and it turns out that in an election, as opposed to a sailboat race, there's more to winning than coming in first. In fact, determining the people's choice can not only get incredibly complicated, it can be impossible.

One of the deep paradoxes of democracy is that there is no right way of adding up the votes to determine the winner in a three-way race -- Stanford professor Kenneth Arrow won a Nobel prize for showing that any system must be flawed.

One problem is that the winner of a three-way race may not be the person voters would have chosen in a two-way race.

Some flaws are worse than others. The warts of our system seem particularly bad as we begin to consider the possible outcomes of a split vote among George Bush, Bill Clinton and Ross Perot. Instead of trying to predict what will happen if no candidate secures a majority of the electoral votes, let us consider what should happen: How can we determine the will of the people? If the election is sent to the House, what information can we provide to help our representatives make a better decision? It helps to think of an election as if it were a scientific investigation. There is some best candidate but we are unsure who it is. To find out the truth, we do an experiment. We ask each voter to examine the candidates and give us an opinion. With enough observations, enough votes, perhaps we can discover this truth. The problem is how to add up all the opinions, taking into account not only the first preferences of the voters, but their second and third choices. The numbers are outdated now, but just for argument's sake, consider the results of an April ABC/Washington Post survey. Among those who expressed a preference, the first choices were split:

Bush: 40 percent

Clinton: 32 percent

Perot: 28 percent

Does that mean that Bush is the most representative choice? Not necessarily. We know that the numbers would have been different if Perot had not run. Did Perot spoil Clinton's chance or amplify Bush's lead? Or is Clinton the spoiler, denying Perot the chance to beat Bush? It seems unfair to rank candidates according to first-choice votes since that allows someone to win by splitting the opposition. In fact, in a fairer system it is possible (maybe even likely) the Perot could come in third in a three-way race and still be able to beat both other candidates in one on-one competition.

To find out the truth, we need to examine the voters' second and third choices. There are exactly six possibilities. For the sake of argument, imagine that the population is split the following way:

Bush, Clinton, Perot: 21 percent

Bush, Perot, Clinton: 19 percent

Clinton, Bush, Perot: 5 percent

Clinton, Perot, Bush: 27 percent

Perot, Clinton, Bush: 16 percent

Perot, Bush, Clinton: 12 percent

This means that 21 percent of the population has Bush first, Clinton second, and Perot third. As in the April poll, the first place choices are split: Bush 40 percent, Clinton 32 percent and Perot 28 percent. With this information, we can predict what would happen if Bush ran against Clinton alone or against Perot alone or if Clinton ran against Perot. The results of these pair wise contests offers a better insight into whatever is meant by the will of the people. (Candidates might use this insight to focus their strategy: Who should they attack and with whom should they align?)

To predict how people would vote in each of the three pair-wise contests, we follow their stated preferences. For example, a person with preferences "Bush, Clinton, Perot" would vote for Bush over either Clinton or Perot, and for Clinton over Perot. The three election results would be:

Bush beats Clinton: 52 to 48

Clinton beats Perot: 53 to 47

Perot beats Bush: 55 to 45.

The paradox is that no one candidate can beat the other two. How did this happen? Bush had sufficiently more first-place supporters to beat Clinton (in spite of Perot supporters favoring Clinton). Clinton beat Perot in their pair-wise contest with help from the Bush supporters. Perot rebounds to beat Bush because Clinton supporters overwhelmingly prefer Perot to Bush. This paradox is not unique to voting. Which would you say is the best basketball team if the Trailblazers beat the Bulls, the Bulls beat the Cavaliers, but the Cavaliers could beat the Trailblazers? Your answer might depend on the size of the winning margins. Same thing in voting. How much did they win by when they won and how much did they lose by when they lost?

Basketball avoids the problem of a three-way competition by awarding the championship through a playoff series. Similarly, we can imagine a runoff election between the top two vote getters. Voters need not even return to the ballot box. By employing what is called a "single transferable vote," we could re-allocate the third-place finisher's votes. Perot's 28 percent would be split according to these voters' designated second choices. Bush picks up 12 percent and Clinton 16 percent, and that's enough for Bush to beat Clinton 52-48. But it seems quite unfair to eliminate Perot because he came in third and then choose someone Perot can beat in a one-on-one competition.

Although Perot came in last, he ends up first under a method proposed just over 200 years ago by Jean-Charles de Borda, a member of French Academy of Science. Borda suggested that we count up the number of votes each candidate gets in his two pair-wise contests. For example, Bush gets 52 percent against Clinton and 45 percent against Perot for a total of 97. The results for all three are:

Bush: 97

Clinton: 101

Perot: 102

Perot's decisive win against Bush more than offsets his mild loss against Clinton, and thus he should be chosen.

Borda's rival in the French Academy was Marie J.A.N. Caritat, the Marquis de Condorcet. Condorcet suggested the following rule. For each candidate, look at the largest vote against that candidate. Choose the person whose largest vote against them is smallest. The largest vote against Bush is 55, the largest against Perot is 53, and the largest vote against Clinton is 52. In this case, Clinton wins.

The motivation for this method is that a large majority against you is much more damning than a small majority against you. To put this another way, we look for the candidate who comes closest to beating all of his rivals. No candidate is able to get at least 50 percent of the vote against all comers. Nor can any candidate get even 49 percent of the vote against all challengers. But Clinton is able to get at least 48 percent of the vote against any challenger. This is better than either Perot or Bush can do.

2 Comments:

Blogger Duck said...

It amazes me in this day of round the clock investigative journalism that politicians continue to involve themselves with crooked money men, or to take the case of Duke Cunningham, continue to abuse their public offices for the sake of luxury vacations and cash. Is there a chance in a million that their schemes can continue very long without discovery? And yet when they are finally caught they are all weepy and contrite and blame their behavior on alcohol or painkiller addiction. How is it that people who are bright enough to advance to high office are dumb enough to think that they can get away with these scams?

August 08, 2006 7:36 AM  
Blogger Susan's Husband said...

I actually just posted about that in a different situation. My view, after years of experience, is that such people simply do not visualize the future in sufficient detail to notice the problem. They take much of their situation to be axiomatic rather than conditional and therefore do not ponder at all what are, to them, counter-factuals.

August 08, 2006 9:40 AM  

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