Intransitivity

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, that is, (if the relation in question is named $R$ ) $\lnot \left(\forall a,b,c:aRb\land bRc\implies aRc\right).$

This statement is equivalent to $\exists a,b,c:aRb\land bRc\land \lnot (aRc).$

For example, consider the relation R on the integers such that a R b if and only if a is a multiple of b or a divisor of b. This relation is intransitive since, for example, 2 R 6 (2 is a divisor of 6) and 6 R 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 is antitransitive (see below); for example, 2 R 6, 6 R 12, and 2 R 12 as well.

As another example, 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.

In the example above, 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. $\forall a,b,c:aRb\land bRc\implies \lnot (aRc).$

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

For example, the relation R on the integers, such that a R b if and only if a + b is odd, is intransitive. If a R b and b R c, then either a and c are both odd and b is even, or vice-versa. In either case, a + c is even.

A second 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: ${\begin{aligned}&\forall a,b,c:aRb\land aRc\implies \lnot (bRc)\\[3pt]&\forall a,b,c:aRc\land bRc\implies \lnot (aRb)\end{aligned}}$

Properties

An antitransitive relation is always irreflexive.
An antitransitive relation on a set of ≥4 elements is never connex. On a 3-element set, the depicted cycle has both properties.
An irreflexive and left- (or right-) unique relation is always anti-transitive.^[4] An example of the former is the mother relation. If A is the mother of B, and B the mother of C, then A cannot be the mother of C.
If a relation R is antitransitive, so is each subset of R.

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; intransitive dice; 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:

	rock	scissors	paper
rock	0	1	0
scissors	0	0	1
paper	1	0	0

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.

Occurrences in preferences

Intransitivity can occur under majority rule, in probabilistic outcomes of game theory, and in the Condorcet voting method in which ranking several candidates can produce a loop of preference when the weights are compared (see voting paradox).
Intransitive dice demonstrate that the relation "die X rolls a higher number than die Y more than half the time" need not be transitive.
In psychology, intransitivity often occurs in a person's system of values (or preferences, or tastes), potentially leading to unresolvable conflicts.
Analogously, in economics intransitivity can occur in a consumer's preferences. This may lead to consumer behaviour that does not conform to perfect economic rationality. Economists and philosophers have questioned whether violations of transitivity must necessarily lead to 'irrational behaviour' (see Anand (1993)).

Likelihood

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.

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

^ Wolves do in fact eat grass – see Engel, Cindy (2003). Wild Health: Lessons in Natural Wellness from the Animal Kingdom (paperback ed.). Houghton Mifflin. p. 141. ISBN 0-618-34068-8..
^ "Guide to Logic, Relations II". Archived from the original on 2008-09-16. Retrieved 2006-07-13.
^ "IntransitiveRelation". Archived from the original on 2016-03-03. Retrieved 2006-07-13.
^ If aRb, bRc, and aRc would hold for some a, b, c, then $a = b$ by left uniqueness, contradicting aRb by irreflexivity.
^ Kerr, Benjamin; Riley, Margaret A.; Feldman, Marcus W.; Bohannan, Brendan J. M. (2002). "Local dispersal promotes biodiversity in a real-life game of rock–paper–scissors". Nature. 418 (6894): 171–174. Bibcode:2002Natur.418..171K. doi:10.1038/nature00823. PMID 12110887. S2CID 4348391.
^ Leutwyler, K. (2000). Mating Lizards Play a Game of Rock-Paper-Scissors. Scientific American.
^ Atherton, K. D. (2013). A brief history of the demise of battle bots.

Intransitivity

Antitransitivity

Properties

Cycles

Occurrences in preferences

Likelihood

References

Further reading