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. More simply, X1 normally beats X2, X2 normally beats X3, but X1 does not normally beat Xn.

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. This is different in that instead of only "A does not normally beat C" it is now "C normally beats A". 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).^[1]^[2]^[3]^[4]

Example

Consider the following set of dice.

Die A has sides 2, 2, 4, 4, 9, 9.
Die B has sides 1, 1, 6, 6, 8, 8.
Die C has sides 3, 3, 5, 5, 7, 7.

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.

The first player chooses a die from the set.
The second player chooses one die from the remaining dice.
Both players roll their die; the player who rolls the higher number wins.

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.

A C	2	4	9	B A	1	6	8	C B	3	5	7
Player 1 chooses die A Player 2 chooses die C				Player 1 chooses die B Player 2 chooses die A				Player 1 chooses die C Player 2 chooses die B
3	C	A	A	2	A	B	B	1	C	C	C
5	C	C	A	4	A	B	B	6	B	B	C
7	C	C	A	9	A	A	A	8	B	B	B

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 ${\frac {5}{9}}\approx 0.56$ that each die beats the next one in the cycle. The largest possible probability is one over the golden ratio, ${\frac {1}{\varphi }}\approx 0.62$ .^[5]

Variations

Efron's dice

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:

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

Each die is beaten by the previous die in the list with wraparound, with probability ⁠2/3⁠. C beats A with probability ⁠5/9⁠, and B and D have equal chances of beating the other.^[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]

P(choose A) = x

P(choose B) = ⁠12⁠ - ⁠56⁠x

P(choose C) = x

P(choose D) = ⁠12⁠ - ⁠76⁠x

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 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

Warren Buffett

Warren Buffett is known to be a fan of intransitive 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 intransitive 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 using intransitive 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."^[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]

Intransitive dice set for more than two players

A number of people have introduced variations of intransitive dice where one can compete against more than one opponent.

Three players

Oskar dice

Oskar van Deventer introduced a set of seven dice (all faces with probability ⁠1/6⁠) as follows:^[8]

A: 2, 02, 14, 14, 17, 17
B: 7, 07, 10, 10, 16, 16
C: 5, 05, 13, 13, 15, 15
D: 3, 03, 09, 09, 21, 21
E: 1, 01, 12, 12, 20, 20
F: 6, 06, 08, 08, 19, 19
G: 4, 04, 11, 11, 18, 18

One can verify that A beats {B,C,E}; B beats {C,D,F}; C beats {D,E,G}; D beats {A,E,F}; E beats {B,F,G}; F beats {A,C,G}; G beats {A,B,D}. Consequently, for arbitrarily chosen two dice there is a third one that beats both of them. Namely,

G beats {A,B}; F beats {A,C}; G beats {A,D}; D beats {A,E}; D beats {A,F}; F beats {A,G};
A beats {B,C}; G beats {B,D}; A beats {B,E}; E beats {B,F}; E beats {B,G};
B beats {C,D}; A beats {C,E}; B beats {C,F}; F beats {C,G};
C beats {D,E}; B beats {D,F}; C beats {D,G};
D beats {E,F}; C beats {E,G};
E beats {F,G}.

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

Grime dice

Dr. James Grime discovered a set of five dice as follows:^[9]^[10]

A: 2, 2, 2, 7, 7, 7
B: 1, 1, 6, 6, 6, 6
C: 0, 5, 5, 5, 5, 5
D: 4, 4, 4, 4, 4, 9
E: 3, 3, 3, 3, 8, 8

One can verify that, when the game is played with one set of Grime dice:

A beats B beats C beats D beats E beats A (first chain);
A beats C beats E beats B beats D beats A (second chain).

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):

Sets chosen by opponents		Winning set of dice
Sets chosen by opponents		Type	Number
A	B	E	1
A	C	E	2
A	D	C	2
A	E	D	1
B	C	A	1
B	D	A	2
B	E	D	2
C	D	B	1
C	E	B	2
D	E	C	1

Four players

A four-player set has not yet been discovered, but it was proved that such a set would require at least 19 dice.^[9]^[11]

Intransitive 4-sided dice

Tetrahedra can be used as dice with four possible results.

Set 1

A: 1, 4, 7, 7
B: 2, 6, 6, 6
C: 3, 5, 5 ,8

P(A > B) = P(B > C) = P(C > A) = ⁠9/16⁠

The following tables show all possible outcomes:

B A	2	6	6	6
1	B	B	B	B
4	A	B	B	B
7	A	A	A	A
7	A	A	A	A

In "A versus B", A wins in 9 out of 16 cases.

C B	3	5	5	8
2	C	C	C	C
6	B	B	B	C
6	B	B	B	C
6	B	B	B	C

In "B versus C", B wins in 9 out of 16 cases.

A C	1	4	7	7
3	C	A	A	A
5	C	C	A	A
5	C	C	A	A
8	C	C	C	C

In "C versus A", C wins in 9 out of 16 cases.

Set 2

A: 3, 3, 3, 6
B: 2, 2, 5, 5
C: 1, 4, 4, 4

P(A > B) = P(B > C) = ⁠10/16⁠, P(C > A) = 9/16

Intransitive 12-sided dice

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. There are no repetitive numbers on each of the dodecahedra.

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.

Set 1:

D III	purple	1	2			5	6	7		9	10	11			14	15	16		18
D IV	red	1		3	4	5			8	9	10		12	13	14			17	18
D V	dark grey		2	3	4		6	7	8			11	12	13		15	16	17

D III
D IV
D V

Set 2:

D VI	cyan	1	2	3	4					9	10	11	12	13	14			17	18
D VII	pear green	1	2			5	6	7	8	9	10					15	16	17	18
D VIII	light grey			3	4	5	6	7	8			11	12	13	14	15	16

D VI
D VII
D VIII

Intransitive prime-numbered 12-sided dice

It is also possible to construct sets of intransitive dodecahedra such that there are no repeated numbers and all numbers are primes. Miwin's intransitive prime-numbered dodecahedra win cyclically against each other in a ratio of 35:34.

Set 1: The numbers add up to 564.

PD 11	grey to blue	13	17	29	31	37	43	47	53	67	71	73	83
PD 12	grey to red	13	19	23	29	41	43	47	59	61	67	79	83
PD 13	grey to green	17	19	23	31	37	41	53	59	61	71	73	79

PD 11
PD 12
PD 13

Set 2: The numbers add up to 468.

PD 1	olive to blue	7	11	19	23	29	37	43	47	53	61	67	71
PD 2	teal to red	7	13	17	19	31	37	41	43	59	61	67	73
PD 3	purple to green	11	13	17	23	29	31	41	47	53	59	71	73

PD 1
PD 2
PD 3

Generalized Muñoz-Perera's intransitive dice

A generalization of sets of intransitive dice with $N$ faces is possible^[12]. Given $N\geq 3$ , we define the set of dice $\{D_{n}\}_{n=1}^{N}$ as the random variables taking values each in the set $\{v_{n,j}\}_{j=1}^{J}$ with

$\mathbb {P} \left[D_{n}=v_{n,j}\right]={\frac {1}{J}}$ ,

so we have $N$ fair dice of $J$ faces.

To obtain a set of intransitive dice is enough to set the values $v_{n,j}$ for $n,j=1,2,\ldots ,N$ with the expression

$v_{n,j}=(j-1)N+(n-j){\text{mod}}(N)+1$ ,

obtaining a set of $N$ fair dice of $N$ faces

Using this expression, it can be verified that

$\mathbb {P} \left[D_{m}<D_{n}\right]={\frac {1}{2}}+{\frac {1}{2N}}-{\frac {(n-m){\text{mod}}(N)}{N^{2}}}$ ,

So each die beats $\lfloor N/2-1\rfloor$ dice in the set.

Examples

3 faces

$D_{1}$	1	6	8
$D_{2}$	2	4	9
$D_{3}$	3	5	7

The set of dice obtained in tis case is equivalent to the first example on this page, but removing repeated faces. It can be verified that $D3>D_{2},D_{2}>D_{1}\ {\text{and}}\ D_{1}>D_{3}$ .

4 faces

$D_{1}$	1	8	11	14
$D_{2}$	2	5	12	15
$D_{3}$	3	6	9	16
$D_{4}$	4	7	10	13

Again it can be verified that $D_{4}>D_{3},D3>D_{2},D_{2}>D_{1}\ {\text{and}}\ D_{1}>D_{4}$ .

6 faces

$D_{1}$	1	12	17	22	27	32
$D_{2}$	2	7	18	23	28	33
$D_{3}$	3	8	13	24	29	34
$D_{3}$	4	9	14	19	30	35
$D_{3}$	5	10	15	20	25	36
$D_{4}$	6	11	16	21	26	31

Again $D_{6}>D_{5},D_{5}>D_{4},D_{4}>D_{3},D_{3}>D_{2},D_{2}>D_{1}\ {\text{and}}\ D_{1}>D_{6}$ . Moreover $D_{6}>\{D_{5},D_{4}\},D_{5}>\{D_{4},D_{3}\},D_{4}>\{D_{3},D_{2}\},D_{3}>\{D_{2},D_{1}\},D_{2}>\{D_{1},D_{6}\}\ {\text{and}}\ D_{1}>\{D_{6},D_{5}\}$ .

References

^ Weisstein, Eric W. "Efron's Dice". Wolfram MathWorld. Retrieved 12 January 2021.
^ Bogomolny, Alexander. "Non-transitive Dice". Cut the Knot. Archived from the original on 2016-01-12.
^ Savage, Richard P. (May 1994). "The Paradox of Nontransitive Dice". The American Mathematical Monthly. 101 (5): 429–436. doi:10.2307/2974903. JSTOR 2974903.
^ ^a ^b ^c ^d Rump, Christopher M. (June 2001). "Strategies for Rolling the Efron Dice". Mathematics Magazine. 74 (3): 212–216. doi:10.2307/2690722. JSTOR 2690722. Retrieved 12 January 2021.
^ Trybuła, Stanisław (1961). "On the paradox of three random variables". Applicationes Mathematicae. 4 (5): 321–332.
^ Bill Gates; Janet Lowe (1998-10-14). Bill Gates speaks: insight from the world's greatest entrepreneur. New York: Wiley. ISBN 9780471293538. Retrieved 2011-11-29.
^ "Like a Marriage, Only More Enduring". Yahoo! Finance. The Wall Street Journal. 2010-12-06. Archived from the original on 2010-12-10. Retrieved 2011-11-29.
^ Pegg, Ed Jr. (2005-07-11). "Tournament Dice". Math Games. Mathematical Association of America. Archived from the original on 2005-08-04. Retrieved 2012-07-06.
^ ^a ^b Grime, James. "Non-transitive Dice". Archived from the original on 2016-05-14.
^ Pasciuto, Nicholas (2016). "The Mystery of the Non-Transitive Grime Dice". Undergraduate Review. 12 (1): 107–115 – via Bridgewater State University.
^ Reid, Kenneth; McRae, A.A.; Hedetniemi, S.M.; Hedetniemi, Stephen (2004-01-01). "Domination and irredundance in tournaments". The Australasian Journal of Combinatorics [electronic only]. 29.
^ Muñoz Perera, Adrián. "A generalization of intransitive dice". December 2024

Sources

Gardner, Martin (2001). The Colossal Book of Mathematics: Classic Puzzles, Paradoxes, and Problems: Number Theory, Algebra, Geometry, Probability, Topology, Game Theory, Infinity, and Other Topics of Recreational Mathematics (1st ed.). New York: W. W. Norton & Company. p. 286–311.^{[ISBN missing]}
Spielerische Mathematik mit Miwin'schen Würfeln (in German). Bildungsverlag Lemberger. ISBN 978-3-85221-531-0.

External links

[MathWorld-1] Weisstein, Eric W. "Efron's Dice". Wolfram MathWorld. Retrieved 12 January 2021.

[2] Bogomolny, Alexander. "Non-transitive Dice". Cut the Knot. Archived from the original on 2016-01-12.

[3] Savage, Richard P. (May 1994). "The Paradox of Nontransitive Dice". The American Mathematical Monthly. 101 (5): 429–436. doi:10.2307/2974903. JSTOR 2974903.

[efron-4] Rump, Christopher M. (June 2001). "Strategies for Rolling the Efron Dice". Mathematics Magazine. 74 (3): 212–216. doi:10.2307/2690722. JSTOR 2690722. Retrieved 12 January 2021.

[5] Trybuła, Stanisław (1961). "On the paradox of three random variables". Applicationes Mathematicae. 4 (5): 321–332.

[6] Bill Gates; Janet Lowe (1998-10-14). Bill Gates speaks: insight from the world's greatest entrepreneur. New York: Wiley. ISBN 9780471293538. Retrieved 2011-11-29.

[7] "Like a Marriage, Only More Enduring". Yahoo! Finance. The Wall Street Journal. 2010-12-06. Archived from the original on 2010-12-10. Retrieved 2011-11-29.

[8] Pegg, Ed Jr. (2005-07-11). "Tournament Dice". Math Games. Mathematical Association of America. Archived from the original on 2005-08-04. Retrieved 2012-07-06.

[A-9] Grime, James. "Non-transitive Dice". Archived from the original on 2016-05-14.

[10] Pasciuto, Nicholas (2016). "The Mystery of the Non-Transitive Grime Dice". Undergraduate Review. 12 (1): 107–115 – via Bridgewater State University.

[11] Reid, Kenneth; McRae, A.A.; Hedetniemi, S.M.; Hedetniemi, Stephen (2004-01-01). "Domination and irredundance in tournaments". The Australasian Journal of Combinatorics [electronic only]. 29.

[12] Muñoz Perera, Adrián. "A generalization of intransitive dice". December 2024

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