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.

Formal definition

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

: $(aoperatorname\{T\}b)\; wedge\; eg(boperatorname\{T\}a)\; wedge\; (boperatorname\{T\}c)\; wedge\; eg(coperatorname\{T\}b)\; Rightarrow\; (aoperatorname\{T\}c)\; wedge\; eg(coperatorname\{T\}a)$

If the relation is also antisymmetric, T is transitive.

Alternately, for a relation T, define the asymmetric part P::$(aoperatorname\{P\}b)\; Leftrightarrow\; (aoperatorname\{T\}b)\; wedge\; eg(boperatorname\{T\}a)$

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.

ee also

* Intransitivity
* Reflexive relation