Quasitransitive relation

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

The mathematical notion of quasitransitivity is a weakened version of transitivity that is used in social choice theory and 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:

[math]\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). }[/math]

If the relation is also antisymmetric, T is transitive.

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

[math]\displaystyle{ (a\operatorname{P}b) \Leftrightarrow (a\operatorname{T}b) \wedge \neg(b\operatorname{T}a). }[/math]

Then T is quasitransitive if and only if 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 quasitransitive 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 quasitransitive, and so is each transitive relation.[5] Moreover, an antisymmetric and quasitransitive 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 quasitransitive, 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.
  • A relation is quasitransitive if, and only if, its complement is.
  • Similarly, a relation is quasitransitive if, and only if, its converse is.

See also

References

  1. Robert Duncan Luce (Apr 1956). "Semiorders and a Theory of Utility Discrimination". Econometrica 24 (2): 178–191. doi:10.2307/1905751. https://www.imbs.uci.edu/files/personnel/luce/pre1990/1956/Luce_Econometrica_1956.pdf.  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), p.2-3. — For the only-if part, define xJy as xRyyRx, and define xPy as xRy ∧ ¬yRx. — For the if part, assume xRy ∧ ¬yRxyRz ∧ ¬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 antisymmetry of R forces J to be coreflexive; hence the union of J and the transitive P is again transitive.