There are a set of special dice, called Efron's Dice. Each die will win over the last. And it will keep going until there's an ultimate winner - which will lose to the original loser. There's a lesson, there, that frustrates many sports fans. Here's how to snatch victory from the jaws (or even the stomach) of defeat.

Efron's Dice

Let us play a little game with a set of Efron's Dice. The game consists of a series of throws, one die against another, with the pot compounding the whole time. Efron's Dice aren't quite like others. They are a standard six-sided shape, but the first match with be between one die with faces that read 5, 5, 5, 1, 1, 1, and one die that reads, 6, 6, 2, 2, 2, 2. Despite the seeming advantage of that extra five, the second die has a probability of winning of two-thirds.


The winner moves on! The die reading 6, 6, 2, 2, 2, 2, is now up against a die reading 3, 3, 3, 3, 3, 3. This new die has a probability of winning of two-thirds, so it will, in most matches, overthrow the previous champion and move on.

Next up! The all-threes die will go up against a die reading 4, 4, 4, 4, 0, 0. This new die has a probability of winning of two-thirds.


Now for the final match. This winning die, this die that came out on top, will go up against the die that got beaten in the first match. The champion 4, 4, 4, 4, 0, 0, will go up against old, unlucky 5, 5, 5, 1, 1, 1. A quick look at the comparative sides will tell you what's likely to happen. That disgraced losing die now has a probability of winning of two-thirds.

That's the trick of Efron's Dice. Put in the right sequence of steps, and every die has a two-thirds chance of winning out over the last die, until you get to the end of the sequence and find out that the highest "step" is suddenly lower than the lowest one.

The Problem With Play-Offs

The trick is to stop seeing the odds as steps, or as terms of absolute odds, or anything else, because those ideas are transitive. Objects have a transitive relationship if there is a kind of linear progression between them. A dog is bigger than a mouse. Because size is (for the most part) a transitive relation, we can confidently say that anything bigger than a dog is also bigger than a mouse. Likewise, anything smaller than a mouse is also smaller than a dog.

We think of odds, especially in games, as getting progressively higher in absolute terms. Each step is higher than the last (a transitive property). Each risk or benefit is greater than the last (another transitive property). Efron's Dice, and other nontransitive dice or nontransitive games, throw us into confusion not because they're impossible to understand, but because they circumvent the easy assumption that if A is "better" than B, and B is "better" than C, then A has to beat C, because it's better.

Sports fans probably have bitter experience with nontransitive odds during play-offs that pit random teams or players against each other. Given the right sequence of games, and the right odds, a relatively weak team can hang on while better teams are eliminated. The game is run as if there were a transitive relation between winners and losers, and if the Rabid Squirrels beat the Plague Fleas, the Squirrels have to be better than every team that the Fleas have beaten.

But it's not that simple. Even aside from random chance, every team (and for that matter every individual player) is a complicated grouping of strengths and weaknesses. Although there's always a case to be made that whoever won must be the better team, everyone has seen a strong team taken out by a relatively weaker one that manages to hit the stronger team's one weakness. (For those who wish to stay entirely within the realm of geekdom, every Death Star has at least one exhaust port that can be taken out by a whiny backwoods kid with three days of training at using The Force.) That weak team goes on to get crushed during their next time around (Empire Strikes Back), leaving the fans with a thoroughly unentertaining show (ewoks). The point is, treating things like they are transitive when they are not can lead to big problems.

Then again, it's not all bad. When you face a disappointment in your life, losing out on something, look at Efron's Dice and take heart. You might not be a loser. You might be a nontransitive winner. (And if you win, don't get cocky, kid. You're just a nontransitive loser.)

Image: Ana

[Via Wolfram Mathworld, Plus Maths.]