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.

Imagine three chess teams, A, B, and C, each with three players. A match between two teams, say A and B, consists of each player of A playing each player of B. The team with the most victories wins the match.

Let’s assume that luck is not involved, so that the stronger chess player always beats the weaker. I’ll set up the teams in such a way that A always beats B, and B always beats C. In math notation we say A > B > C. A seems to be the strongest. Now here’s the paradox: if A plays C, then A will lose. In symbols, C > A. How can that be?

Here is one way to do it. Assume that the names of the players on team A are A2, A6, and A7. (The higher the number, the more skilled the player.) For team B, the names are B1, B5, and B9, and for C they are C3, C4, and C8. First, consider the match between teams A and B:

A2 beats B1 and loses against B5 and B9

A6 beats B1 and B5 and loses against B9

A7 beats B1 and B5 and loses against B9

So team A wins 5 of the 9 games, and A > B.

When team B plays team C, it works out as follows:

B1 wins none and loses against C3, C4, and C8

B5 beats C3 and C4 and loses against C8

B9 beats C3, C4, C8 and loses none

Team B wins 5 of the 9 games. So B > C. You would expect that A > C. But look what happens when they play:

A2 wins zero games and loses to C3, C4, and C8

A6 beats C3 and C4 and loses to C8

A7 beats C3 and C4 and loses to C8

C wins 5 of the 9 games. We have A > B, B > C, and C > A. If all three teams compete, the winner will be decided by the order in which they play. If A and C play in the first round, A is eliminated. If B and C play in the first round, then C is eliminated.

Such peculiar behavior is not actually peculiar at all. In math, we say the chess competition is “not transitive.” That means that A > B and B > C does not necessarily imply that A > C. We can also say the objects (in this case, the teams) cannot be “ordered” under the operation of round-robin competition. Non-orderable objects abound in math-and in the real world.

Does this example also relate to baseball, football, tennis, and soccer? Yes, the order of the playoffs can determine the winners, regardless of the real strengths. If you are a sports fan, you probably already know that. (That’s why your team lost.) The same principle applies in election campaigns. After all, we have playoffs in politics too. They’re called primaries.

The paradox even works for your individual choice of candidate. Suppose you are a middle-of-the-road voter and you rank each candidate on three issues, such as their stands on human rights, on use of military forces, and on taxes. Your evaluation of candidate L on these issues is (2, 7, 6), where higher numbers indicate higher affinity for the candidate’s positions. For M it is (9, 5, 1), and for R it is (4, 3, 8). When you compare L to M, you’ll prefer L on military and taxes, but not on human rights; since he is better on two of three issues, you decide L is better than M, i.e. that L > M. When you compare M to R, you’ll find M is better on two issues, human rights and military, so M > R. Thus far we have L > M and M > R, so you would think L > R, right? Wrong. Compare L directly to R. You prefer R on human rights, L on use of military, and R on taxes. R wins on two of three issues, so R > L. Political preferences can be intransitive.

If the playoff system doesn’t work, can we come up with a system that will? That brings us back to Arrow’s notorious theorem. Arrow made a series of postulates that were so reasonable that every voting system should obey them, and then he proceeded to show that they were incompatible with each other. In other words, there is no voting system that will always satisfy fundamental criteria of fairness.

Does that mean all hope is lost?

No. Arrow’s theorem only guarantees that you can find a situation in which the election is unfair; it doesn’t guarantee bad results in all cases. Moreover, the theorem is true only when there are three or more candidates. Let’s consider the two-candidate case: L and R. Everyone votes for the one who is closest to their preference. In real elections, the positions of the candidates on issues may not be absolutely immutable. Both L and R realize this, so to maximize their chance of winning, they both start shifting toward the middle. They both know that whoever best takes possession of the middle will get the most votes.

By the time the voting takes place, the candidates’ positions are almost indistinguishable. Voters complain that they have no real choice. And in some sense this is true. The center has been found-the position where M would have been, had he (or she) run. Both candidates, to get elected, have moved to the position where they best represent the average of all voters.

Notice how the existence of a primary election can interfere with this center-finding process. A candidate who wins in a primary is often the one who represent the center of his party-a position that tends to be pretty far toward one side or the other of the political spectrum.

If, however, people vote in the primaries for a candidate who “can win” in the general election rather than one who is closest to their own preference, then the two-party system works well, and results in candidates who are close to the center. If there is sufficient time during the subsequent campaign, the candidates can move even closer to the middle position. Democrats can co-opt Republican issues, and Republicans can co-opt Democratic stands. This leads to a surprising irony. With both candidates moving toward the center, many people ultimately see such little difference that they lose interest in the election. Turnout is low-but for this example, that actually reflects the fact that the election process is working well.

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