Quasitransitive Relation

Get Quasitransitive Relation essential facts below. View Videos or join the Quasitransitive Relation discussion. Add Quasitransitive Relation to your PopFlock.com topic list for future reference or share this resource on social media.
## Formal definition

## Examples

## Properties

## See also

## References

This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

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.

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

If the relation is also antisymmetric, T is transitive.

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

Then T is quasitransitive iff P is transitive.

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.

- 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.

**^**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.**^**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*.**^**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.**^**Given*R*, whenever*xRy*? ¬*yRx*holds, the pair (*x*,*y*) can't belong to the symmetric part, but must belong to the transitive part.**^**Since the empty relation is trivially both transitive and symmetric.**^**The anti-symmetry of*R*forces*J*to be coreflexive; hence the union of*J*and the transitive*P*is again transitive.

- Sen, A. (1969). "Quasi-transitivity, rational choice and collective decisions".
*Rev. Econ. Stud*.**36**: 381-393. doi:10.2307/2296434. Zbl 0181.47302. - Frederic Schick (Jun 1969). "Arrow's Proof and the Logic of Preference".
*Philosophy of Science*.**36**(2): 127-144. doi:10.1086/288241. JSTOR 186166. - Amartya K. Sen (1970).
*Collective Choice and Social Welfare*. Holden-Day, Inc. - Amartya K. Sen (Jul 1971). "Choice Functions and Revealed Preference" (PDF).
*The Review of Economic Studies*.**38**(3): 307-317. doi:10.2307/2296384. - A. Mas-Colell and H. Sonnenschein (1972). "General Possibility Theorems for Group Decisions" (PDF).
*The Review of Economic Studies*.**39**: 185-192. doi:10.2307/2296870. - D.H. Blair and R.A. Pollak (1982). "Acyclic Collective Choice Rules".
*Econometrica*.**50**: 931-943. doi:10.2307/1912770. - Bossert, Walter; Suzumura, Kotaro (Apr 2005). Rational Choice on Arbitrary Domains: A Comprehensive Treatment (PDF) (Technical Report). Université de Montréal, Hitotsubashi University Tokyo.
- Bossert, Walter; Suzumura, Kotaro (Mar 2009). Quasi-transitive and Suzumura consistent relations (PDF) (Technical Report). Université de Montréal, Waseda University Tokyo.
- Bossert, Walter; Suzumura, K?tar? (2010).
*Consistency, choice and rationality*. Harvard University Press. ISBN 0674052994. - Alan D. Miller and Shiran Rachmilevitch (Feb 2014). Arrow's Theorem Without Transitivity (PDF) (Working paper). University of Haifa.

This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

Popular Products

Music Scenes

Popular Artists