Intransitive dice

A set of dice is intransitive (or nontransitive) if it contains X>2 dice, X1, X2, and X3... with the property that X1 rolls higher than X2 more than half the time, and X2 rolls higher than X3 etc... more than half the time, but where it is not true that X1 rolls higher than Xn more than half the time.

In other words, a set of dice is intransitive if the binary relation – X rolls a higher number than Y more than half the time – on its elements is not transitive.

It is possible to find sets of dice with the even stronger property that, for each die in the set, there is another die that rolls a higher number than it more than half the time.

Using such a set of dice, one can invent games which are biased in ways that people unused to intransitive dice might not expect (see Example).

The probability that A rolls a higher number than B, the probability that B rolls higher than C, and the probability that C rolls higher than A are all ⁠5/9⁠, so this set of dice is intransitive.

In fact, it has the even stronger property that, for each die in the set, there is another die that rolls a higher number than it more than half the time.

Now, consider the following game, which is played with a set of dice.

If this game is played with a transitive set of dice, it is either fair or biased in favor of the first player, because the first player can always find a die that will not be beaten by any other dice more than half the time.

If it is played with the set of dice described above, however, the game is biased in favor of the second player, because the second player can always find a die that will beat the first player's die with probability ⁠5/9⁠.

The following tables show all possible outcomes for all three pairs of dice.

If one allows weighted dice, i.e., with unequal probability weights for each side, then alternative sets of three dice can achieve even larger probabilities than

The largest possible probability is one over the golden ratio,

[5] Efron's dice are a set of four intransitive dice invented by Bradley Efron.

[4] The four dice A, B, C, D have the following numbers on their six faces: Each die is beaten by the previous die in the list with wraparound, with probability ⁠2/3⁠.

[4] If each player has one set of Efron's dice, there is a continuum of optimal strategies for one player, in which they choose their die with the following probabilities, where 0 ≤ x ≤ ⁠3/7⁠:[4] Miwin's Dice were invented in 1975 by the physicist Michael Winkelmann.

Consider a set of three dice, III, IV and V such that Then: A number of people have introduced variations of intransitive dice where one can compete against more than one opponent.

Consequently, for arbitrarily chosen two dice there is a third one that beats both of them.

Namely, Whatever the two opponents choose, the third player will find one of the remaining dice that beats both opponents' dice.

Dr. James Grime discovered a set of five dice as follows:[7][8] One can verify that, when the game is played with one set of Grime dice: However, when the game is played with two such sets, then the first chain remains the same, except that D beats C, but the second chain is reversed (i.e. A beats D beats B beats E beats C beats A).

Consequently, whatever dice the two opponents choose, the third player can always find one of the remaining dice that beats them both (as long as the player is then allowed to choose between the one-die option and the two-die option): A four-player set has not yet been discovered, but it was proved that such a set would require at least 19 dice.

The following tables show all possible outcomes: In "A versus B", A wins in 9 out of 16 cases.

In analogy to the intransitive six-sided dice, there are also dodecahedra which serve as intransitive twelve-sided dice.

The points on each of the dice result in the sum of 114.

Miwin's dodecahedra (set 1) win cyclically against each other in a ratio of 35:34.

The miwin's dodecahedra (set 2) win cyclically against each other in a ratio of 71:67.

Miwin's intransitive prime-numbered dodecahedra win cyclically against each other in a ratio of 35:34.

A generalization of sets of intransitive dice with

, we define the set of dice

as the random variables taking values each in the set

The set of dice obtained in tis case is equivalent to the first example on this page, but removing repeated faces.