# Intransitive cubes

Intransitive dice (the opposite sides of each cube are labeled with the same number)

Intransitive dice are called a set of special game dice in which there is a different dice for each of the dice, against which it loses in the long run, i.e. compared to which it shows a smaller than a larger number more often. An example are the three intransitive cubes A, B and C shown on the right: A against B, B against C and C against A wins with probability 5/9. The example of the intransitive cubes shows that the relation “is more likely to be greater “ Does not have to be transitive for random variables . A similar example of an intransitive relation is the game of scissors, stone, paperin which each symbol wins against one and loses against another.

The outcome of the game contradicts the intuition that an advantage must be transitive. This idea would be correct if the result were the sum of the numbers rolled in a large number of game rounds, rather than the number of rounds won. The Condorcet paradox shows a similar error .

## Efron's Cube

Efron's cubes are four intransitive cubes invented by the American statistician Bradley Efron .

The four dice A, B, C and D have the following numbers on their six sides:

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

For each of the dice there is another one who has a 2/3 chance of defeating it:

• P (A> B) = P (B> C) = P (C> D) = P (D> A) = 2/3.

The probabilities of comparing A to C and B to D are

• P (A> C) = 4/9 and P (B> D) = 1/2.

## Miwin's Cube

Miwin's Cube

The Miwin'schen dice were invented by the Austrian physicist Michael Winkelmann 1975th They are labeled as follows:

Sentence 1

• III: 1, 2, 5, 6, 7, 9
• IV: 1, 3, 4, 5, 8, 9
• V: 2, 3, 4, 6, 7, 8

Sentence 2

• IX: 1, 3, 5, 6, 7, 8
• X: 1, 2, 4, 6, 8, 9
• XI: 2, 3, 4, 5, 7, 9

Against each of the dice, one of the other two has the following odds: win 17/36, loss 16/36 and tie 3/36. Winkelmann has also constructed intransitive dodecahedron- shaped cubes .

## literature

• Hugo Steinhaus , Stanisław Trybuła: On a paradox in applied probabilities , Bulletin de l'Académie Polonaise des Sciences. Série des sciences mathématiques, astronomiques et physiques 7, 1959, pp. 67-69 (English with a Russian summary; Zentralblatt review )
• Stanisław Trybuła: On the paradox of three random variables , Zastosowania Matematyki 5, 1961, pp. 321–332 (English; Zentralblatt review )
• Li-chien Chang: On the maximin probability of cyclic random inequalities , Scientia Sinica 10, 1961, pp. 499–504 (English; Zentralblatt review )
• Zalman Usiskin: Max – min probabilities in the voting paradox , The Annals of Mathematical Statistics 35, June 1964, pp. 857–862 (English; Zentralblatt review )
• Stanisław Trybuła: On the paradox of n random variables , Zastosowania Matematyki 8, 1965, pp. 143–156 (English; Zentralblatt review )
• Martin Gardner : Nontransitive dice and other probability paradoxes , Scientific American 223, December 1970, pp. 110–114 (English)
• Richard P. Savage: The paradox of nontransitive dice , American Mathematical Monthly 101, No. 5, 1994, pp. 429-436 (English; Zentralblatt review )
• Noga Alon , Graham Brightwell, HA Kierstead, AV Kostochka, Peter Winkler: Dominating sets in k-majority tournaments , Journal of Combinatorial Theory Series B 96, No 3, May 2006, pp. 374–387 (English)