1. Intransitivity

2. Antitransitivity

3. Cycles

4. Occurrences in preferences

5. Likelihood

6. References

In mathematics, intransitivity (sometimes called nontransitivity) is a property of binary relations that are not transitive relations. This may include any relation that is not transitive, or the stronger property of antitransitivity, which describes a relation that is never transitive.

Intransitivity

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. (if the relation in question is named )

This statement is equivalent to

For instance, in the food chain, wolves feed on deer, and deer feed on grass, but wolves do not feed on grass.^[1] Thus, the feed on relation among life forms is intransitive, in this sense.

Another example that does not involve preference loops arises in freemasonry: in some instances lodge A recognizes lodge B, and lodge B recognizes lodge C, but lodge A does not recognize lodge C. Thus the recognition relation among Masonic lodges is intransitive.

Antitransitivity

Often the term intransitive is used to refer to the stronger property of antitransitivity.

We just saw that the feed on relation is not transitive, but it still contains some transitivity: for instance: humans feed on rabbits, rabbits feed on carrots, and humans also feed on carrots.

A relation is antitransitive if this never occurs at all, i.e.,

Many authors use the term intransitivity to mean antitransitivity.^[2][3]

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 have never played C, and therefore, A has not defeated C.

By transposition, each of the following formulas is equivalent to antitransitivity of R:

An antitransitive relation is always irreflexive.

An irreflexive and left- (or right-)unique relation is always anti-transitive.

Cycles

The term intransitivity is often 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 is preferred to B
• B is preferred to C
• C is preferred to A

Rock, paper, scissors; Nontransitive dice; Intransitive machines^[4]; and Penney's game are examples. Real combative relations of competing species^[5], strategies of individual animals^[6], and fights of remote-controlled vehicles in BattleBots shows ("robot Darwinism")^[7] can be cyclic as well.

Assuming no option is preferred to itself i.e. the relation is irreflexive, 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.

Notice that a cycle is neither necessary nor sufficient for a binary relation to be not transitive. For example, an equivalence relation possesses cycles but is transitive. Now, consider the relation "is an enemy of" and suppose that the relation is symmetric and satisfies the condition that for any country, any enemy of an enemy of the country is not itself an enemy of the country. This is an example of an antitransitive relation that does not have any cycles. In particular, by virtue of being antitransitive the relation is not transitive.

The game of rock, paper, scissors is an example. The relation over rock, paper, and scissors is "defeats", and the standard rules of the game are such that rock defeats scissors, scissors defeats paper, and paper defeats rock. Furthermore, it is also true that scissors does not defeat rock, paper does not defeat scissors, and rock does not defeat paper. Finally, it is also true that no option defeats itself. This information can be depicted in a table:

In such cases intransitivity reduces to a broader equation of numbers of people and the weights of their units of measure in assessing candidates.

Such as:

• 30% favor 60/40 weighting between social consciousness and fiscal conservatism
• 50% favor 50/50 weighting between social consciousness and fiscal conservatism
• 20% favor a 40/60 weighting between social consciousness and fiscal conservatism

While each voter may not assess the units of measure identically, the trend then becomes a single vector on which the consensus agrees is a preferred balance of candidate criteria.

References

• {{cite book|last1=Anand|first1=P|year=1993|title=Foundations of Rational Choice Under Risk|location=Oxford|publisher=Oxford University Press}}.

• [https://www.researchgate.net/publication/225909889_How_vicious_are_cycles_of_intransitive_choice Bar-Hillel, M., & Margalit, A. (1988). How vicious are cycles of intransitive choice? Theory and Decision, 24(2), 119-145.]
• [https://doi.org/10.1007/s11518-014-5245-x Klimenko, A. Y. (2014). Complexity and intransitivity in technological development. Journal of Systems Science and Systems Engineering, 23(2), 128-152.]
• [https://doi.org/10.3390/e17064364 Klimenko, A. Y. (2015). Intransitivity in theory and in the real world. Entropy, 17(6), 4364-4412.]
• [https://ssrn.com/abstract=2657942 Poddiakov, A., & Valsiner, J. (2013). Intransitivity cycles and their transformations: How dynamically adapting systems function. In: L. Rudolph (Ed.). Qualitative mathematics for the social sciences: Mathematical models for research on cultural dynamics. Abingdon, NY: Routledge. Pp. 343–391.]

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