## Non-Transitive Dice

Here are four interesting and famous dice. They are known as non-transitive dice, or Efron's dice. Here is the game we play with them. You can choose any die (singular of dice) that you want, and then I will choose my die; we roll our dice and the higher number wins. Which die would you choose?

It turns out that it doesn't matter which one you choose, I will win 2/3 of the time, you will only win 1/3 of the time (in other words, I have a 2:1 advantage). Does that make sense to you? Whichever die you choose, I will choose the one immediately to its left. If you choose the left one, I will choose the right one.

The logical way to see if one die beats another is to list the 36 possible outcomes, like this:

 4 4 4 4 0 0 3 * * * * 3 * * * * 3 * * * * 3 * * * * 3 * * * * 3 * * * *

These are the first two dice, and we see that 4-4-4-4-0-0 beats 3-3-3-3-3-3 2/3 of the time. The asterisks show which rolls are won by the 4-4-4-4-0-0 die. Similar tables show that 3-3-3-3-3-3 beats 6-6-2-2-2-2 2/3 of the time, 6-6-2-2-2-2 beats 5-5-5-1-1-1 2/3 of the time, and 5-5-5-1-1-1 beats 4-4-4-4-0-0 2/3 of the time. You might want to verify all of that.

We are used to transitive situations. If a=b and b=c then a=c, the famous transitive law. In fact, if a>b and b>c then a>c, another transitive law. In games, we like to think that if player a beats player b and b beats c then a will beat c, but that doesn't always work. Above we see that not only are people sometimes non-transitive, but dice and other randomizing tools can also be non-transitive.

There are other possible non-transitive dice.