There should be an image here!The best thing about elections is when they are over, the political ads stop for a while.

Take a moment to consider what elections do. Do they decide the best candidate, or at least the candidate preferred by the most people? Not necessarily-particularly in races with more than two candidates and almost all races start out that way.

To see how this happens, first consider dice with non-standard markings. The faces can have any value in the range 1-6. Assume a set consists of four dice labeled A, B, C, D. One die is thrown at a time. The highest value showing wins. If we are told that on the average A beats B, B beats C, and C beats D, then we automatically assume A beats D. This is called transitivity. (An example of transitivity is “heavier than”: if A is heavier than B, and B heavier than C, then A is heavier than C.) Is it possible to number a set of four dice such that, on the average die A beats B, B beats C, C beats D, and D beats A. Is that possible?

The surprising answer is yes. I first saw this in the book by Martin Gardner, Wheels, Life, and Other Mathematical Amusements more than twenty years ago. This is a startling phenomenon. Chapter 4 of the referenced book discusses this in detail, and several other people have published discussions of it.

The essential error made when considering transitivity in dice is assuming that throwing many times to get an average is equivalent to weighing objects to see which is heavier. This is not the case. Averaging decreases information, and that information is critical. To see how it works, I reproduce here a figure from Martin Gardner (he gives references to how it was invented.). The figure shows four dice that have been unfolded to show the markings on all sides. Each arrow points toward a losing die. If you were allowed to pick one of these dice to throw in a bet against me, I could always pick another one which has a 2/3 probability of winning on each toss. Try going through each of the combinations to see if you agree with that statement.

Other than changing how you might bet in rare circumstances, transitivity also has ramifications in voting when the ballot has more than two candidates. The majority of voters can easily prefer Amy to Bert; Bert to Charley, and Charley to Amy. Who should be elected? Whoever gets the most votes generally wins unless a runoff is required. Is a runoff guaranteed to result in an election that displeases the minimum number of voters? In a mayoral election in San Diego, three candidates split the vote almost evenly. No matter who was declared winner between the three contenders, 2/3 of the voters disapprove. The spread in the total votes cast was small enough to consider statistically insignificant. If the top two vote getters were subjected to a runoff, would that improve matters?

I’ve mentioned the logical problems with voting in other posts. Perhaps these issues seem remote, but in many elections, even without fraud, technical maneuvering, or help from the courts, there is no rational reason to believe the favorite candidate is always elected. The example non-transitive dice demonstrates how our intuition can mislead us.

Political scientists might disagree with me, but I think our system could be greatly improved. Often we do not ask the right questions: like what do we want to accomplish by voting? Define what you mean by “will of the people” and how you measure it.

For those who wish to learn more about this fascinating subject, I recommend Basic Geometry of Voting by Donald G. Saari.

[Photo above by John C. Abell / CC BY-ND 2.0]