Definitions

# Intransitivity

In mathematics, the term intransitivity is used for related, but different properties of binary relations:

## The property of not being transitive

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.

$negforall a, b, c: a R b wedge b R c Rightarrow a R c.$

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.

## The property of being antitransitive

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.,

$forall a, b, c: a R b wedge b R c Rightarrow neg a R c$

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.

## A cycle in a binary relation

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 is preferred to B
• B is preferred to C
• C is preferred to A

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.

## Occurrences of intransitivity 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 probabilities are not necessarily 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. In recent years, economists and philosophers have questioned whether violations of transitivity must necessarily lead to 'irrational behaviour' (see Anand (1993)).

## Likelihood of intransitivity

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 conservatisim

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

See also Anand P (1993) Foundations of Rational Choice Under Risk, Oxford, Oxford University Press

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