Nontransitive Dice Cycle

Nontransitive Dice Cycle Theory

Concept Overview

In probability theory, a set of random variables is nontransitive if the relation "is more likely to be strictly greater than" is not transitive. For dice, this means we can construct a set of dice (such as A, B, and C) where die A beats die B most of the time, die B beats die C most of the time, and yet die C beats die A most of the time.

The mean face value of a die does not necessarily determine its winning probability against another die. Nontransitive sets illustrate that in pairwise competitions, no universally "best" die exists; the optimal choice depends strictly on what your opponent chooses, forming a cyclic advantage analogous to Rock-Paper-Scissors.

Worked Example

Consider the independently constructed six-sided fair dice A = [3, 4, 5, 7, 17, 18] and B = [1, 2, 13, 14, 15, 16]. Let's calculate the probability that a single roll of die A strictly beats a single roll of die B.

Pairwise Comparison Calculation

Both dice have 6 faces, so there are exactly 6 × 6 = 36 equally likely ordered pairs (face A, face B).
Let's count how many times A strictly beats B by looking at A's lower faces:
- For A = 3, it beats B = 1, 2 (2 wins).
- For A = 4, it beats B = 1, 2 (2 wins).
- For A = 5, it beats B = 1, 2 (2 wins).
- For A = 7, it beats B = 1, 2 (2 wins).
Now consider A's higher faces:
- For A = 17, it beats all faces of B (6 wins).
- For A = 18, it beats all faces of B (6 wins).
Summing these wins gives a total of 2 + 2 + 2 + 2 + 6 + 6 = 20 strict wins for A out of 36 possibilities.

Expected Result:

Strict win probability = 20 / 36 = 0.5555555555555556
Tie probability = 0
Strict loss probability = 16 / 36 = 0.4444444444444444

Interactive Experiment

You can verify the cyclical relationship using the Nontransitive Dice Cycle interactive module.

Set First die to A and Second die to B. Observe that A's strict win probability is roughly 0.555 (20/36), making it the favorite.
Change First die to B and Second die to C. B is now the favorite with a 0.666 (24/36) strict win probability.
Change First die to C and Second die to A. Notice that C beats A with a 0.638 (23/36) strict win probability, completing the cycle.
Change Tie credit to Half and compare the same die against itself (e.g., A vs A) to see how ties factor into the selected win score.

Limitations

This module considers only independent single-roll comparisons between finite six-sided dice. It does not calculate combinations like rolling two dice against two dice, or multi-die transitive tournaments where different cyclic dependencies might form. Additionally, this implementation uses a specific fixed fixture of three dice (A, B, C) selected via finite search and does not provide an unbounded solver to generate completely arbitrary nontransitive sets.

References

The cyclical nature of these dice allows us to explore deeper mathematical properties of comparisons. For further reading, consider the following primary sources:

NRICH: Non-transitive Dice - A detailed teacher resource explaining nontransitive mathematical relationships and cyclical dice sets.

Nontransitive Dice Cycle