Inmathematics,intransitivity (sometimes callednontransitivity) is a property ofbinary relations that are nottransitive relations. That is, we can find three values${\displaystyle a}$,${\displaystyle b}$, and${\displaystyle c}$ where the transitive condition does not hold.

Antitransitivity is astronger property which describes a relation where, for any three values, the transitivity condition never holds.

Some authors use the termintransitive to refer to antitransitivity.^[1][2]

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. A relation isintransitive if it is not transitive. Assuming the relation is named${\displaystyle R}$, it is intransitive if:

$${\displaystyle \mathrm{\neg }\left(\mathrm{\forall }a,b,c:aRb\wedge bRc⟹aRc\right).}$$

This statement is equivalent to$${\displaystyle \mathrm{\exists }a,b,c:aRb\wedge bRc\wedge \mathrm{\neg }(aRc).}$$

For example, the inequality relation,${\displaystyle \ne }$, is intransitive. This can be demonstrated by replacing${\displaystyle R}$ with${\displaystyle \ne }$ and choosing${\displaystyle a=1}$,${\displaystyle b=2}$, and${\displaystyle c=1}$. We have${\displaystyle 1\ne 2}$ and${\displaystyle 2\ne 1}$ and it is not true that${\displaystyle 1\ne 1}$.

Notice that, for a relation to be intransitive, the transitivity condition just has to be not true at some${\displaystyle a}$,${\displaystyle b}$, and${\displaystyle c}$. It can still hold for others. For example, it holds when${\displaystyle a=1}$,${\displaystyle b=2}$, and${\displaystyle c=3}$, then${\displaystyle 1\ne 2}$ and${\displaystyle 2\ne 3}$ and it is true that${\displaystyle 1\ne 3}$.

For a more complicated example of intransitivity, consider the relationR on the integers such thata R b if and only ifa is a multiple ofb or a divisor ofb. This relation is intransitive since, for example, 2R 6 (2 is a divisor of 6) and 6R 3 (6 is a multiple of 3), but 2 is neither a multiple nor a divisor of 3. This does not imply that the relation isantitransitive (see below); for example, 2R 6, 6R 12, and 2R 12 as well.

An example in biology comes from thefood chain. Wolves feed on deer, and deer feed on grass, but wolves do not feed on grass.^[3] Thus, thefeed on relation among life forms is intransitive, in this sense.

Antitransitivity for a relation says that the transitive condition does not hold for any three values.

In the example above, thefeed 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 isantitransitive if this never occurs at all. The formal definition is:$${\displaystyle \mathrm{\forall }a,b,c:aRb\wedge bRc⟹\mathrm{\neg }(aRc).}$$

For example, the relationR on the integers, such thata R b if and only ifa + b is odd, is intransitive. Ifa R b andb R c, then eithera andc are both odd andb is even, or vice-versa. In either case,a + c is even.

A second example of an antitransitive relation: thedefeated relation inknockout 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.

Bytransposition, each of the following formulas is equivalent to antitransitivity ofR:

• An antitransitive relation is alwaysirreflexive.
• An antitransitive relation on a set of ≥4 elements is neverconnex. On a 3-element set, the depicted cycle has both properties.
• An irreflexive andleft- (orright-) unique relation is always anti-transitive.^[4] An example of the former is themother relation. IfA is the mother ofB, andB the mother ofC, thenA cannot be the mother ofC.
• If a relationR is antitransitive, so is each subset ofR.

The termintransitivity 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;intransitive dice; andPenney'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 isirreflexive, 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 (orcycle) is known as anintransitivity.

Notice that a cycle is neither necessary nor sufficient for a binary relation to be not transitive. For example, anequivalence 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 ofrock, 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:

The first argument of the relation is a row and the second one is a column. Ones indicate the relation holds, zero indicates that it does not hold. Now, notice that the following statement is true for any pair of elements x and y drawn (with replacement) from the set {rock, scissors, paper}: If x defeats y, and y defeats z, then x does not defeat z. Hence the relation is antitransitive.

Thus, a cycle is neither necessary nor sufficient for a binary relation to be antitransitive.

• Intransitivity can occur undermajority rule, in probabilistic outcomes ofgame theory, and in theCondorcet voting method in which ranking several candidates can produce a loop of preference when the weights are compared (seevoting paradox).
• Intransitive dice demonstrate that the relation "dieX rolls a higher number than dieY more than half the time" need not be transitive.
• Inpsychology, intransitivity often occurs in a person'ssystem of values (orpreferences, ortastes), potentially leading to unresolvable conflicts.
• Analogously, ineconomics intransitivity can occur in a consumer'spreferences. This may lead toconsumer behaviour that does not conform to perfecteconomic rationality. Economists and philosophers have questioned whether violations of transitivity must necessarily lead to 'irrational behaviour' (see Anand (1993)).

It has been suggested thatCondorcet 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.

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 singlevector on which theconsensus agrees is a preferred balance of candidate criteria.

• Anand, P (1993).Foundations of Rational Choice Under Risk. Oxford: Oxford University Press..
• Bar-Hillel, M., & Margalit, A. (1988). How vicious are cycles of intransitive choice?Theory and Decision, 24(2), 119-145.
• Klimenko, Alexander Y. (2014)."Complexity and intransitivity in technological development"(PDF).Journal of Systems Science and Systems Engineering.23 (2):128–152.doi:10.1007/s11518-014-5245-x.S2CID 59390606.
• Klimenko, Alexander (2015)."Intransitivity in Theory and in the Real World".Entropy.17 (12):4364–4412.arXiv:1507.03169.Bibcode:2015Entrp..17.4364K.doi:10.3390/e17064364.

• Properties of binary relations

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