Nontransitive Dice Cycle

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.

Interactive Experiment

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

1. 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.
2. Change First die to B and Second die to C. B is now the favorite with a 0.666 (24/36) strict win probability.
3. 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.
4. 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.