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An unexpected thing happened in the California recall election: the winner got more votes than the loser.

Few people thought that would happen. Under California’s unique law, the incumbent Gray Davis had to get 50 percent of the vote on the recall portion of the ballot, or he was out. He got 45 percent. In the separate gubernatorial vote, his replacement needed only a plurality. With 135 candidates, that meant thatin principlethe new governor could be elected by less than 1 percent of the voters, even though 45 percent liked the incumbent!

But it didn’t turn out that way. Arnold Schwarzenegger won with 49 percent. Even some opponents were grateful that, at least, the results were decisive. Californians had grown tired of the pre-election ridicule of their ridiculous law. Yet peculiar election results aren’t new. In the last presidential election, the winner, George W. Bush, received 47.9 percent of the votes, while the loser, Al Gore, got more: 48.4 percent. Bill Clinton won the presidency in 1992 with an even smaller percentage: 43 percent.

Election math is screwy. Why don’t we fix it? Well, the problem may not be as simple as changing California’s law, or abolishing the Electoral College. Election math is fundamentally unfixable. That is a celebrated result of a mathematical theorem proven in 1952 by Kenneth Arrow, who won the Nobel Prize in Economics for this and other work.

Consider, for example, the “instant runoff” system, in which every voter ranks every candidate. This method is already used in several municipalities, including Cambridge, MA, and it has been proposed as a replacement for the Electoral College system. From Arrow’s theorem, we expect to be able to find cases in which instant runoffs are unfair-and indeed, such examples are not hard to find.

Here’s how instant runoffs work. Imagine three candidates  who represent the left, middle, and right of the political spectrum-L, M and R for short. Voters rank the three candidates in order of preference. After the vote, the two candidates with the most first-place votes are retained. Each ballot that indicated a first-place preference for the losing candidate is then allocated to the candidate chosen on that ballot as the second preference. The ultimate winner always has a majority, although it may be a combination of first place, second, and lower ranking votes. Isn’t that the best way to decide an election?

Not necessarily. Consider this plausible situation: L gets 34 percent, M gets 32 percent, and R gets 34 percent of the first place votes. We can assume that supporters of both of the extreme candidates put M in their second slot. Yet the moderate M, who represents the center-of-mass of the voters, is eliminated because he narrowly lost the first-round plurality. The decision will be determined by second place votes. The country will get one of the extremists, either L or R, despite the fact that the overwhelming majority of voters put M as either their first or second choice. 

Moreover, there is good reason to think that candidates cannot, in principle, be put in an order of preference. This fact is so counterintuitive that it requires a specific example. Please bear with me through the following math. It will be worth it. And just to make the case even more convincing, let’s begin not with politics, but with chess.

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