Quasitransitive relation

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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.

[edit] 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 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.

[edit] 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.