The Bizarre Math of Elections


Low voter turnout can be a healthy sign

that the electoral system is working well


Richard A. Muller

Technology for Presidents

Technology Review Online

October 2003


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 that -- in principle -- the 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, Massachusetts, and it has been proposed as a replacement for the current national 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 named Left, Middle, and Right -- L, M and R for short. After the vote, the top candidates are retained, and the eliminated candidate has his votes allocated to them. 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 first place votes. Everyone who voted for L or R puts 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 100 percent of the 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 counter-intuitive 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 numbers indicate their skill.)  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 (the teams) cannot be 'ordered' under the operation of round-robin competition. Non-orderable objects abound in math. And also 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 knew that. (That's why your team lost.) Does it relate to politics? Yes, in the same way. 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). 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 Kenneth 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.


So any system we might choose to replace current local, state, or national election procedures would have its own frustrating flaws.  It wouldn't cover all possible situations.  But 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 towards 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 that 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 extreme one who represent the center of his party rather than the center of the whole population.


If, however, people vote in the primaries for a candidate who 'can win' 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 towards 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.