Intransitive Dice

Description

This repository demonstrates some cool properties of intransitive dice, which are dice wherein each die in a set beats the next in line in a loop. This repository at the moment contains two such strategies. The first is a modified form of Efron's dice, wherein die 0 beats die 1 with 2/3 probability, same for 1 beating 2, 2 beating 3, and 3 beating 0. A second, more straightforward infinite-player strategy with 5/9 probability to beat the next was invented by yours truly.

This repo also contains one comparison function and one function to compare all dice in rotation, as documented.

Efron's Dice

Efron created the following dice list:

(6, 6, 2, 2, 2, 2)
(5, 5, 5, 1, 1, 1)
(4, 4, 4, 4, 0, 0)
(3, 3, 3, 3, 3, 3)

Which can be demonstrated to be a more specific case (n=4) of some n dice which each beats the next 2/3 of the time. By splitting the final die into two dice, we get the insertion of a penultimate die with some arbitrarily small adjustment ε value:

(6, 6, 2, 2, 2, 2)
(5, 5, 5, 1, 1, 1)
(4, 4, 4, 4, 0, 0)
(3+ε, 3+ε, 3+ε, 3+ε, 3-ε, 3-ε)
(3, 3, 3, 3, 3, 3)

Thus 5 and 1 are unchanged, so have the same win ratio. The relation between 3 and 4 is the same as the relationship between what is now 3 and 5, so result in the same 2/3 win for 3. and 4 beats 5 with 2/3s ratio. The only trick is now to adjust the values such that ε can be inflated to a whole integer value. Hence, all values greater than the split die value are incremented by 2 and the list becomes:

(8, 8, 2, 2, 2, 2)
(7, 7, 7, 1, 1, 1)
(6, 6, 6, 6, 0, 0)
(5, 5, 5, 5, 3, 3)
(4, 4, 4, 4, 4, 4)

This process can be repeated an infinite number of times, resulting in dice with faces:

3 "turnaround dice", for i-th dice (starting index at 1), with the following faces:
- i+1 faces of 2*n-1-i
- 5-i faces of 3-i
n-3 "split dice" where the i-th dice (starting index at 4; continuing the index from the previous dice) has faces:
- 2 faces of i-1
- 4 faces of 2*n-i-1

These dice are so named because the 'split dice' is a potentially infinitely long list of dice where each die beats the next, and this is nested within a set of 3 dice that serve as almost like a turnaround such that the bottom of the list of split dice can wrap around to beat the top one once more.

One can see that when n=4, the 4th dice has 2 faces of 4-1 = 3 and 4 faces of 8-5 = 3, hence it is a die of all 3s. This is true for the final die since the face i-1 is equal to 2*n-(i+1) when i equals n.

Multiplayer Dice

Multiplayer dice are dice which reduce the efficiency from the Infinite Efron series of 6/9 to 5/9 and can only be generated for odd n. However, they are a more straightforward way to generate dice where every i-th die beats the next n//3 dice, with wrap around, where each die is uniquely numbered with all equal numbers. Also this generates only 3 numbers for the dice, so these can be simply doubled for writing on 6-sided dice. An example is:

(7, 11, 15)
(6, 10, 17)
(5, 9, 19)
(4, 8, 21)
(3, 14, 16)
(2, 13, 18)
(1, 12, 20)

Notice that the first column is all the numbers from 1 to n, the second n+1 to 2n, and the third is 2n+1 to 3n. These three columns represent the low, medium, and high values for each die. Each die beats the one below it

This works because every die beats the next n//2 (three, in this case) dice on two each of the low, medium, and high values. Therefore 2/3s of cases are just decided by a low vs medium vs high roll. However, on the remaining 1/3 of cases, 2/3 of them are decided in this fashion. Hence, the winning bias is 5/9 under this system.

The Multiplayer Dice class has a "winning" class, wherein you can input the dice your competitors pulled in order to see which dice beats all of them. If you have m competitors, you can guarantee a winning die with n = 2m+1 dice.

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