Is Voting Really Fair?

Voting---whether for a presidential election or for which kind of cake is the favorite of a kindergarten class---is considered by most people to be the fairest way to come to decisions. But when analyzed mathematically, voting can look a bit shady.

According to mathematician Donald Saari, who recently proved an important result in the mathematics of voting theory, it is possible to create through voting any misrepresentation one likes. Once one has some notion of what the voters think, it is possible to set up packages where a majority of the the voters keep approving of one package over the other until you have them agreeing on the desired outcome. The key here is that who is in the "majority" can shift each stage.

Here is a simple example. Suppose there are 30 voters, and suppose that the A, B, and C are the choices one has to vote on. To say that a voter has A > B, means that voter prefers choice A to choice B. Now let's suppose we have this configuration:

10 have A > B > C,

10 have B > C > A,

10 have C > A > B.

Now, we could have an election comparing two candidates and then having the winner against the remaining candidate. Who do you like? Whoever it is, an election can be arranged to ensure she wins. For instance, if you like C, have the first election between A and B. (Here A wins.) Then, have the winner run against C; C wins.

If you prefer A, have the first election between B and C where B wins. We already know that A beats B, so A is the winner. For B, the same idea: just have the first election between A and C where C wins. Then B beats C.

Notice, in each election, the winner wins with over 66% of the vote. "What a landslide!" says Saari. "Nobody would or should question the outcome, and that is the delight of the scam. It gets much worse; with more candidates I can invent scenarios where *everyone* prefers A to B, yet B is the overall winner."

Saari's work appeared in the article, "A Chaotic Exploration of Aggregation Paradoxes," in the March 1995 issue of SIAM Review, published by the Society for Industrial and Applied Mathematics.

-Allyn Jackson