nontransitive

Nontransitive describes a relation, order, or system in which the transitive property does not hold universally. In a transitive relation, if a is related to b and b is related to c, then a is related to c. Nontransitive systems show deviations from this pattern, often producing cycles or probabilistic dominance without a simple linear ranking.

Formally, a relation R on a set is transitive if, for all elements a, b, and c, aRb and bRc imply aRc. A system is nontransitive when there exist elements a, b, and c such that aRb, bRc, but not aRc. In practice, nontransitivity is often observed as cycles or probabilistic dominance: A can beat B, B can beat C, and C can beat A, each with a favorable likelihood though no single element dominates all others.

A well-known informal example is the game rock-paper-scissors. Rock beats Scissors, Scissors beats Paper, and Paper beats Rock, forming a nontransitive cycle with no single winning option. In probability and games, nontransitive dice illustrate a similar idea. Sets of three or more dice can be arranged so that die A tends to beat die B, die B tends to beat die C, and die C tends to beat die A, with each matchup showing a probability greater than 0.5 in the corresponding direction. Famous examples include Efron’s dice and other published sets.

Nontransitivity is not the same as noncomparability, where neither item is comparable with another under the relation. It is also distinct from certain logical notions of intransitivity. Applications appear in game design, decision theory, and probability puzzles, where cyclic dominance can create counterintuitive or strategic dynamics. See also transitivity and nontransitive dice.