A relation is transitive if, whenever it relates some A to some B, and that B to some C, it also relates that A to that C. Some authors call a relation intransitive if it is not transitive, i.e.
For instance, in food chains: wolves eat deer, and deer eat grass, but wolves do not eat grass. Thus, the eat relation among life forms is intransitive, in this sense.
Often the term intransitive is used for a stronger property.
We just saw that the eat relation is not transitive, but it still contains some transitivity: for instance: humans eat rabbits, rabbits eat carrots, and human also eat carrots.
A relation is antitransitive if this never occurs at all, i.e.,
Many authors use the term intransitivity to mean antitransitivity.
An example of an antitransitive relation: the defeated relation in knockout tournaments. If player A defeated player B and player B defeated player C, A can never have played C, and therefore, A has not defeated C.
In practice, the term intransitivity is mostly used when speaking of scenarios in which a relation describes the relative preferences between pairs of options, and weighing several options produces a "loop" of preference:
A well-known example is the game rock, paper, scissors.
Assuming no option is preferred to itself, a preference relation with a loop is not transitive. For if it is, each option in the loop is preferred to each option, including itself. This can be illustrated for this example of a loop among A, B, and C. Assume the relation is transitive. Then, since A is preferred to B and B is preferred to C, also A is preferred to C. But then, since C is preferred to A, also A is preferred to A.
Therefore such a preference loop (or "cycle") is known as an intransitivity.
This is not the same property as the relation not being transitive, nor is it the same as the relation being antitransitive.
It has been suggested that Condorcet voting tends to eliminate "intransitive loops" when large numbers of voters participate because the overall assessment criteria for voters balances out. For instance, voters may prefer candidates on several different units of measure such as by order of social consciousness or by order of most fiscally conservative.
In such cases intransitivity reduces to a broader equation of numbers of people and the weights of their units of measure in assessing candidates.