Quasitransitive Relation
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Quasitransitive Relation
The quasitransitive relation x5/4y. Its symmetric and transitive part is shown in blue and green, respectively.

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:

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

If the relation is also antisymmetric, T is transitive.

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

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

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 7 and 8 grams of sugar and indifferent between 8 and 9 grams of sugar, but who prefers 9 grams of sugar to 7.[1] Similarly, the Sorites paradox can be resolved by weakening assumed transitivity of certain relations to quasitransitivity.

## Properties

• A relation R is quasi-transitive if, and only if, it is the disjoint union of a symmetric relation J and a transitive relation P.[2]J and P are not uniquely determined by a given R;[3] however, the P from the only-if part is minimal.[4]
• As a consequence, each symmetric relation is quasi-transitive, and so is each transitive relation.[5] Moreover, an anti-symmetric and quasi-transitive relation is always transitive.[6]
• The relation from the above sugar example, {(7,7), (7,8), (7,9), (8,7), (8,8), (8,9), (9,8), (9,9)}, is quasi-transitive, but not transitive.
• A quasitransitive relation needn't be acyclic: for every non-empty set A, the universal relation A×A is both cyclic and quasitransitive.

## References

1. ^ Robert Duncan Luce (Apr 1956). "Semiorders and a Theory of Utility Discrimination" (PDF). Econometrica. 24 (2): 178-191. doi:10.2307/1905751. Here: p.179; Luce's original example consists in 400 comparisons (of coffee cups with different amounts of sugar) rather than just 2.
2. ^ The naminig follows Bossert & Suzumura (2009), p.2-3. — For the only-if part, define xJy as xRy ? yRx, and define xPy as xRy ? ¬yRx. — For the if part, assume xRy ? ¬yRx ? yRz ? ¬zRy holds. Then xPy and yPz, since xJy or yJz would contradict ¬yRx or ¬zRy. Hence xPz by transitivity, ¬xJz by disjointness, ¬zJx by symmetry. Therefore, zRx would imply zPx, and, by transitivity, zPy, which contradicts ¬zRy. Altogether, this proves xRz ? ¬zRx.
3. ^ For example, if R is an equivalence relation, J may be chosen as the empty relation, or as R itself, and P as its complement.
4. ^ Given R, whenever xRy ? ¬yRx holds, the pair (x,y) can't belong to the symmetric part, but must belong to the transitive part.
5. ^ Since the empty relation is trivially both transitive and symmetric.
6. ^ The anti-symmetry of R forces J to be coreflexive; hence the union of J and the transitive P is again transitive.

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