Nontransitive Dice

I’d always thought of dice as being fair. Like flipping a coin, I’d assumed that a throw of the dice was random. Then I learned about nontransitive dice and it blew my mind.

Nontransistive dice are a set of dice that don’t behave in the normal way. Even though each die has an expected value of 3, some dice are better than others in 1:1 matchups. Just to make things even stranger, there’s no set of best dice, just like in the way that the game of “Rock, Paper, Scissors” has no best move.

The most common set of nontransitive dice is Efron’s dice.

With Efron’s dice, there are four different colored dice. They have the sides with the numbers above. The yellow die is particularly weird as it always comes up 3. When you roll the red die vs. a yellow, the yellow will win 2/3 of the time (3 against 2) and lose 1/3 of the time (3 against 6), but the yellow will lose against the blue 2/3 of the time and win 1/3 of the time for similar reasons. If you do the math you’ll see that on average blue beats yellow, yellow beats red, and red beats green. So you would think that red is the “best” and green is the “worst.” But surprisingly, yellow beats blue. That’s because these dice aren’t transitive. If you want to dig deeper, Microsoft has a nice writeup of the dice on their website.⁽¹⁾They gave them away during an event. I grabbed the image above from their site.

Things that are transitive can be chained together. Like if Adam weighs more than Bob, and Bob weighs more than Charlie, Adam weights more than Charlie. But these dice are nontransitive, which means there’s no “best” move. Just like in the game rock, paper, scissors, the best move depends on what the other player has chosen.

I spent many years trying to find a set of these dice. The best place to buy Efron’s Dice in the US is the Museum of Math. They are $5 and come with instructions. You might want to get 2 sets because if you roll two dice at once, the winning order changes. If you’re looking for something fancier, you can get a nice set of Grime Dice from the UK.⁽²⁾Grime dice were created by James Grime. Here’s a video of him describing how they work. If you have a 3D printer, you can 3D print your own Grime dice at home. Or, considering there’s so much time during the coronavirus pandemic, you can print the dice out on regular paper and tape them together.

One of my friends uses them in an introductory college class to teach probability.⁽³⁾Here’s what my friend says about using nontransitive dice in class:

I’ve always liked nontransitive dice.  It’s both really neat, and also a great pedagogical example.

When I use this in class, I always begin with the game. First I’ll start with some “normal” dice where one has a higher expected outcome than the other.  You can then ask questions like, “Is it always better to take the die with the higher expected value?”  Then you can give examples of non-transitive dice (where the expected values are the same).  The class is now distracted by expected value…. and have to work out the probabilities.  At which point they are surprised to discover the non-transitive property.  (If you just ask directly, a few will guess what’s going on too quickly… a little bit of diversion always helps the magician look like a magician 🙂 ).

Then (if it’s a smaller class), I’ll ask them to think of what types of “research” questions they might want to ask about said dice.  (I stole this from Michael Mitzenmacher’s intro probability class.). For example, you might want to ask about larger sets of dice, dice with unique values, dice with more sides, dice with arbitrary “edge” relations (i.e., which beats which) not just cycles, etc.   Helps to get students thinking about the “right” type of questions… As students play around with these dice they notice a pattern with some dice beating out others. Once they start working out the probabilities, they discover the nontransitive effect. Then he gets them to ask more questions about the dice (e.g., what happens if you throw more than one die at a time) to get them really thinking.

Other people, most famously Warren Buffett, use them to unfairly win a game of dice with friends. Buffet tried to do this with Bill Gates and Ed Thorpe, the creator of card counting. In this game, you have the other person choose their dice before you do. It’s like getting to choose rock or scissors once your opponent has chosen paper.