# Non-Transitive Dice Follow up

Yesterday I asked you to pick the best die. If you guessed Blue, you would be wrong. Red is all wrong. Likewise, Green is incorrect. The correct answer is that…

There is no best die!

As crazy as this might sound, you have all experience this type of situation before. Consider rock-scissors-paper:

In the above game, there is no best choice. Each symbol beats one symbol and loses to the other. The dice are similar. Consider the following diagram:

This is a chart displaying the possible outcomes between for Blue vs Red. It is clear that he Blue die will win more often, 21 times out of 36. The red die will only win 15 times out of 36. You can construct similar tables for Red vs Green and Green vs Blue. The results are summarized in the following diagram:

These dice are called “non-transitive” because the ability to win does not transfer. Red is better than Green, and Green is better than Blue, but Red is not better than Blue.

Warren Buffett is known to be a fan of non-transitive dice. Buffett once attempted to win a game of dice with Bill Gates using non-transitive dice. Buffett suggested that each of them choose one of the dice, then discard the other. They would bet on who would roll the highest number most often. Buffett offered to let Gates pick his die first. This suggestion instantly aroused Gates’s curiosity. He asked to examine the dice, after which he demanded that Buffett choose first.