October 17, 2003

The Bizarre Math of Elections

Continued from page 1

By Richard A. Muller

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.