Intransitivity

Intransitivity refers to a property of a relation in mathematics and logic. A relation R on a set S is transitive if, whenever a is related to b and b is related to c, then a is related to c (for all a, b, c in S). A relation is intransitive if there exist elements a, b, and c such that a is related to b and b is related to c, but a is not related to c. Intransitivity thus describes a breakdown of the transitive condition rather than a universal negation of the relation.

Common illustrations include cyclic dominance games. The classic example is rock-paper-scissors: rock defeats scissors, scissors defeats paper, and paper defeats rock. Here the “defeats” relation is intransitive, since the chain rock → scissors and scissors → paper does not guarantee rock defeats paper. Another well-known example is a set of nontransitive dice, where die A tends to roll higher than die B, die B higher than die C, and die C higher than die A, with probabilistic dominance forming a cycle.

In social choice and game theory, intransitive or cyclic preferences can arise when collective judgments fail to produce a linear order, a phenomenon connected to the Condorcet paradox. In biology and ecology, cyclic or intransitive interactions can occur among competing species or strategies, influencing population dynamics and evolutionary stability.

See also: transitivity, nontransitive relations, cyclic dominance.