Efron's dice

Introduction

Commonly relations are transitiv, e.g. if \(a < b\) and \(b < c\) then \(a < c\). Also the probability that one ordinary 6-sided dice beats another ordinary 6-sided dice is transitive, because they are equal.
$$P(A>B) = P(B>C) = P(A>C)$$

Enter Bradley Effron, American statistician, best know for the bootsrap resampling technique, which I applied in frequently in my research. He came up with a set of non-transitive dice. Best know example for a non-transitive relation might be the game rock-paper-scissors. Rock beats scissors, scissors beats paper and papaer in turn beats rock, or \( r>s>p>r \).

Efron’s set of dice

Efron came up with a set of four 6-sided dice with mutual beating probabilities
$$P(A>B) = P(B>C) = P(C>D) = P(D>A) = \frac{2}{3}$$.

The sides of this set of dice are as follows:

Dice Numbers

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A 0, 0, 4, 4, 4, 4
B 3, 3, 3, 3, 3, 3
C 2, 2, 2, 2, 6, 6
D 1, 1, 1, 5, 5, 5

Read more on a generalization of Efron’s dice in this note I wrote some time ago.

Games with Efron’s dice