Nontransitive dice
Encyclopedia
A set of nontransitive dice is a set of dice
Dice
A die is a small throwable object with multiple resting positions, used for generating random numbers...

 for which the relation "is more likely to roll a higher number" is not transitive
Transitive relation
In mathematics, a binary relation R over a set X is transitive if whenever an element a is related to an element b, and b is in turn related to an element c, then a is also related to c....

. See also intransitivity
Intransitivity
In mathematics, the term intransitivity is used for related, but different, properties of binary relations:- Intransitivity :A relation is transitive if, whenever it relates some A to some B, and that B to some C, it also relates that A to that C...

.

This situation is similar to that in the game Rock, Paper, Scissors
Rock, Paper, Scissors
Rock-paper-scissors is a hand game played by two people. The game is also known as roshambo, or another ordering of the three items ....

, in which each element has an advantage over one choice and a disadvantage to the other.

Example

Consider a set of three dice, A, B and C such that
  • die A has sides {2,2,4,4,9,9},
  • die B has sides {1,1,6,6,8,8}, and
  • die C has sides {3,3,5,5,7,7}.

Then:
  • the probability
    Probability
    Probability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The...

     that A rolls a higher number than B is 5/9 (55.55 %),
  • the probability that B rolls a higher number than C is 5/9, and
  • the probability that C rolls a higher number than A is 5/9.


Thus A is more likely to roll a higher number than B, B is more likely
to roll a higher number than C, and C is more likely to roll a higher
number than A. This shows that the relation "is more likely to roll a
higher number" is not transitive with these dice, and so we say this
is a set of nontransitive dice.

Efron's dice

Efron's dice are a set of four nontransitive dice invented by Bradley Efron
Bradley Efron
Bradley Efron is an American statistician best known for proposing the bootstrap resampling technique, which has had a major impact in the field of statistics and virtually every area of statistical application...

.

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

Probabilities

Each die is beaten by the previous die in the list, with a probability of 2/3:

B's value is constant; A beats it on 2/3 rolls because four of its six faces are higher.

Similarly, B beats C with a 2/3 probability because only two of C's faces are higher.

P(C>D) can be calculated by summing conditional probabilities for two events:
  • C rolls 6 (probability 1/3); wins regardless of D (probability 1)
  • C rolls 2 (probability 2/3); wins only if D rolls 1 (probability 1/2)

The total probability of win for C is therefore

With a similar calculation, the probability of D winning over A is

Best overall die

The four dice have unequal probabilities of beating a die chosen at random from the remaining three:

As proven above, die A beats B two thirds of the time but beats D only one third of the time. The probability of die A beating C is 4/9 (A must roll 4 and C must roll 2). So the likelihood of A beating any other randomly selected die is:

Similarly, die B beats C two thirds of the time but beats A only one third of the time. The probability of die B beating D is 1/2 (only when D rolls 1). So the likelihood of B beating any other randomly selected die is:

Die C beats D two thirds of the time but beats B only one third of the time. The probability of die C beating A is 5/9. So the likelihood of C beating any other randomly selected die is:

Finally, die D beats A two thirds of the time but beats C only one third of the time. The probability of die D beating B is 1/2 (only when D rolls 5). So the likelihood of D beating any other randomly selected die is:

Therefore the best overall die is C with a probability of winning of 0.5185.

Variants with equal averages

Note that Efron's dice have different average rolls: the average roll of A is 8/3, while B and D each average 9/3, and C averages 10/3. The non-transitive property depends on which faces are larger or smaller, but does not depend on the absolute magnitude of the faces. Hence one can find variants of Efron's dice where the odds of winning are unchanged, but all the dice have the same average roll. For example,
  • A: 6, 6, 6, 6, 0, 0
  • B: 4, 4, 4, 4, 4, 4
  • C: 8, 8, 2, 2, 2, 2
  • D: 7, 7, 7, 1, 1, 1

or
  • A: 7, 7, 7, 7, 1, 1
  • B: 5, 5, 5, 5, 5, 5
  • C: 9, 9, 3, 3, 3, 3
  • D: 8, 8, 8, 2, 2, 2

These variant dice are useful, e.g., to introduce students to different ways of comparing random variables (and how only comparing averages may overlook essential details).

Numbered 1 through 24 dice

A set of four dice using all of the numbers 1 through 24 can be made to be nontransitive.
With adjacent pairs, one die will win approximately 2 out of 3 times.

For rolling high number, B beats A, C beats B, D beats C, A beats D.

  • A: 1, 2, 16, 17, 18, 19
  • B: 3, 4, 5, 20, 21, 22
  • C: 6, 7, 8, 9, 23, 24
  • D: 10, 11, 12, 13, 14, 15


Relation to Efron's dice

These dice are basically the same as Efron's dice, as each number of a series of successive numbers on a single die can all be replaced by the lowest number of the series and afterwards renumbering them.
  • A: 1, 2, 16, 17, 18, 19 -> 1, 1, 16, 16, 16, 16 -> 0, 0, 4, 4, 4, 4
  • B: 3, 4, 5, 20, 21, 22 -> 3, 3, 3, 20, 20, 20 -> 1, 1, 1, 5, 5, 5
  • C: 6, 7, 8, 9, 23, 24 -> 6, 6, 6, 6, 23, 23 -> 2, 2, 2, 2, 6, 6
  • D: 10, 11, 12, 13, 14, 15 -> 10, 10, 10, 10, 10, 10 -> 3, 3, 3, 3, 3, 3


Miwin's dice


Miwin's Dice were invented in 1975 by the physicist Michael Winkelmann.

Consider a set of three dice, III, IV and V such that
  • die III has sides 1, 2, 5, 6, 7, 9
  • die IV has sides 1, 3, 4, 5, 8, 9
  • die V has sides 2, 3, 4, 6, 7, 8

Then:
  • the probability
    Probability
    Probability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The...

     that III rolls a higher number than IV is 17/36
  • the probability that IV rolls a higher number than V is 17/36
  • the probability that V rolls a higher number than III is 17/36

Three-dice set with minimal alterations to standard dice

The following intransitive dice have only a few differences compared to 1 through 6 standard dice:
  • as with standard dice, the total number of pips is always 21
  • as with standard dice, the sides only carry pip numbers between 1 and 6
  • faces with the same number of pips occur a maximum of twice per die.
  • only two sides on each die have numbers different from standard dice:
    • A: 1, 1, 3, 5, 5, 6
    • B: 3, 2, 3, 4, 5, 4
    • C: 1, 2, 2, 4, 6, 6


Like Miwin’s set, the probability of A winning versus B (or B vs. C, C vs. A) is 17/36. The probability of a draw, however, is 4/36, so that only 15 out of 36 rolls lose. So the overall winning expectation is higher.

Freivalds's investigation

The set of nontransitive dice were investigated by the Latvian computer scientist and mathematician Rusins Freivalds. He showed that if there is a set of n dice, and each die beats the next with probability p, then p can be arbitrary close (but not equal) to 3/4 = 0.75 when n goes to infinity.

Warren Buffett

Warren Buffett
Warren Buffett
Warren Edward Buffett is an American business magnate, investor, and philanthropist. He is widely regarded as one of the most successful investors in the world. Often introduced as "legendary investor, Warren Buffett", he is the primary shareholder, chairman and CEO of Berkshire Hathaway. He is...

 is known to be a fan of nontransitive dice. 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 nontransitive dice. "These are a mathematical curiosity, a type of 'trick' dice that confound most people's ideas about probability."

Buffett once attempted to win a game of dice with Bill Gates
Bill Gates
William Henry "Bill" Gates III is an American business magnate, investor, philanthropist, and author. Gates is the former CEO and current chairman of Microsoft, the software company he founded with Paul Allen...

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

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 nontransitive dice that could not be won and "thought it was hilarious".

External links

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