nontransitively

Nontransitive refers to a characteristic of a relation, game, or process in which the transitivity property does not hold. Transitivity is the logical principle that if an element A is related to a second element B, and B is related to a third element C, then A must be related to C. A nontransitive relation violates this implication; thus, A may relate to B and B to C without A relating to C. The most common illustration is the game of rock–paper–scissors, where rock beats scissors, scissors beats paper, and paper beats rock, creating a cycle of dominance that is nontransitive.

In mathematics, nontransitive relations appear in order theory, preference relations, and social choice theory. A frequently studied example is nontransitive dice. Three sets of dice can be constructed so that die A beats die B with probability greater than one half, die B beats die C with probability greater than one half, and die C beats die A with probability greater than one half. This phenomenon demonstrates that pairwise comparison can lead to circular defeat relationships, challenging intuitive notions of fairness and ranking.

Nontransitivity also arises in economics, where certain market preferences can lead to cyclical behavior, and in cryptography, where nontransitive protocols can provide security against specific types of attacks. In computer science, nontransitive dependencies may appear in package management or version control systems, requiring careful handling to avoid inconsistency. The concept underscores the importance of recognizing that many real-world systems exhibit cycles and that transitive reasoning may not always be applicable.