Quasitransitive relation

Quasitransitivity is a weakened version of transitivity that is used in social choice theory or microeconomics. Informally, a relation is quasitransitive if it is symmetric for some values and transitive elsewhere. The concept was introduced by Sen (1969) to study the consequences of Arrow's theorem.

Formal definition

A binary relation T over a set X is quasitransitive if for all a, b, and c in X the following holds:

$(a\operatorname{T}b) \wedge \neg(b\operatorname{T}a) \wedge (b\operatorname{T}c) \wedge \neg(c\operatorname{T}b) \Rightarrow (a\operatorname{T}c) \wedge \neg(c\operatorname{T}a).$

If the relation is also antisymmetric, T is transitive.

Alternately, for a relation T, define the asymmetric or "strict" part P:

$(a\operatorname{P}b) \Leftrightarrow (a\operatorname{T}b) \wedge \neg(b\operatorname{T}a).$

Then T is quasitransitive iff P is transitive.

Examples

Preferences are assumed to be quasitransitive (rather than transitive) in some economic contexts. The classic example is a person indifferent between 10 and 11 grams of sugar and indifferent between 11 and 12 grams of sugar, but who prefers 12 grams of sugar to 10. Similarly, the Sorites paradox can be resolved by weakening assumed transitivity of certain relations to quasitransitivity.

Properties

• Every transitive relation is quasitransitive; every quasitransitive relation is an acyclic relation. In each case the converse does not hold in general.

See also

• Intransitivity
• Reflexive relation