Nontransitive Dice, explained.
Nontransitive dice form a cycle of pairwise winning probabilities, so no single die is favored against all of the others.
Why it happens
A die’s average face value does not specify how often it exceeds another die’s roll. Counting all face pairings makes the opponent-dependent comparison explicit and can reveal a cycle.
Pairwise superiority need not produce an overall ranking: one die can beat another more often, yet lose to a third.
Read the result
Inspect the six equally likely faces of each die and roll repeatedly. The counter-die is deliberately chosen against your selection. The displayed exact chance comes from all 36 pairings, while your win count is a finite sample.
A worked example
A cycle without a best die
A beats B in 20 of 36 pairs; B beats C in 20; C beats A in 20.
After choosing A, the opponent chooses C, leaving your chance at 16 of 36.
Changing to another die changes the opponent too, so there is no universally favored first choice.
OPTIONAL DEEPER DETAILGo deeper: inside the model
Inside this model
Dice A=[2,2,4,4,9,9], B=[1,1,6,6,8,8], C=[3,3,5,5,7,7] form a cycle with winning probability 5/9 in each favored pairing. Rolls sample faces uniformly.
Where this idea is useful
A practical use
A comparison that depends on the opponent cannot always be summarized by a single best option.
A common misconception
“The highest average face must win most often.”
Winning probability counts comparisons, whereas the mean also weights the size of unusually high faces.
What this explanation leaves out
- These deliberately constructed dice are not ordinary six-sided dice. Finite roll counts fluctuate around their exact probabilities.
Are these ordinary dice?
No. Their repeated face values were constructed to illustrate a nontransitive comparison.
Does a ranking you use remain valid when the comparison partner changes?
Further reading
Explore the original research or the teaching reference behind this experiment.