A Trick With Dice

Ten is bigger than five, and five is bigger than one. And so ten is bigger than one. Basic stuff. In symbols (we mathematicians like symbols, more than we like numbers):

If A > B and B > C then A > C.             (*)

This is common sense. Wait, not so fast! (There is no such as thing as common sense to mathematicians.) This is not true for everything.

Do you play Rock, Paper, Scissors? Rock beats Scissors, Scissors beats Paper. In symbols

R > S > P

But it is not true that R > P. Because Paper beats Rock.

R > S > P > R > S > …

When something obeys (*), as numbers do, then we call that Transitive. But like Rock, Paper, Scissors there are many fun things that are not transitive, and some of them even involve numbers, in a way. And can even be used to win drinks at a bar. (This could turn into a theme.) Let’s see an example.

You play games with dice, then you are used to the standard dice with six sides, numbered 1, 2, 3, 4, 5, 6. Each side equally likely to be rolled. We are going to roll dice, you and I, to see who pays for the drinks. But we aren’t going to use regular dice. They will be six sided and they will be unbiased. But we are going to use six-sided dice with slightly different numbers on them. Here are three such dice:

A: 1, 1, 3, 5, 5, 6

B: 2, 3, 3, 4, 4, 5

C: 1, 2, 2, 4, 6, 6

We each choose a die, we roll. If we draw we roll again. Loser pays for the drink. All the dice are very similar, no? The pips on all the dice add up to 21, so the mean roll is 21/6 = 3 1/2, just like for a regular die.

I am feeling generous, I will let you choose your die first. You choose A? Ok, I choose C. Who wins most often? You’d think that would be equally likely to win because the means are the same, no? No! Look at table:

Along the top is your roll, die A. Down the left-hand side is my roll, die C. Thirty-six possibilities, out of which I win 17, you win 15, and there are four ties and we roll again. And this is true whichever die you choose. Choose B and I choose A, choose C and I choose B. Every time you pay for more drinks than me! Ok, so not a big advantage but it adds up over a typical lock-in session.

Warren Buffett is big fan of non-transitive dice. I don’t know if he is a big drinker.

Mathemagical Thinking Lesson

What can we learn from this? Other than don’t go to a bar with me. The most important lesson is that what happens on average does not give us the whole picture. All these dice have the same mean roll. That is irrelevant. What about other types of average? The median is the middle number in an ordered list. Die A has a median of 4, die B 3.5 and die C 3. Now there are differences, perhaps you’d choose die A. Doesn’t matter. Similarly the modes are different. The mode is the most frequent number, here all dice are bimodal, and are different. But again that form of average is irrelevant.

Normally we are great fans of common sense. But sometimes common sense can be highly misleading. Sometimes you have to ask questions that defy common sense.

See also the excellent essay by Stephen Jay Gould, “The Median Isn’t the Message,” https://journalofethics.ama-assn.org/article/median-isnt-message/2013-01