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).

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.

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,

[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.

In the book Fortune's Formula: The Untold Story of the Scientific Betting System that Beat the Casinos and Wall Street, a discussion between him and Edward Thorp is described.

Buffett and Thorp discussed their shared interest in intransitive dice.

"These are a mathematical curiosity, a type of 'trick' dice that confound most people's ideas about probability."

"Buffett suggested that each of them choose one of the dice, then discard the other two.

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.

"[6] In 2010, Wall Street Journal magazine quoted Sharon Osberg, Buffett's bridge partner, saying that when she first visited his office 20 years earlier, he tricked her into playing a game with intransitive dice that could not be won and "thought it was hilarious".

[7] 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.

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.

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

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.