Quasitransitive relation

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:

(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:

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

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.

References

^ Robert Duncan Luce (Apr 1956). "Semiorders and a Theory of Utility Discrimination" (PDF). Econometrica. 24 (2): 178–191. doi:10.2307/1905751. JSTOR 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 antisymmetry 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 (3): 381–393. doi:10.2307/2296434. JSTOR 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. S2CID 121427121.
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. JSTOR 2296384.
A. Mas-Colell and H. Sonnenschein (1972). "General Possibility Theorems for Group Decisions" (PDF). The Review of Economic Studies. 39 (2): 185–192. doi:10.2307/2296870. JSTOR 2296870. S2CID 7295776. Archived from the original (PDF) on 2018-04-12.
D.H. Blair and R.A. Pollak (1982). "Acyclic Collective Choice Rules". Econometrica. 50 (4): 931–943. doi:10.2307/1912770. JSTOR 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). Social Choice and Welfare (Technical Report). 39 (2–3). Université de Montréal, Waseda University Tokyo: 323–334. doi:10.1007/s00355-011-0600-z. S2CID 38375142. Archived from the original (PDF) on 2018-04-12.
Bossert, Walter; Suzumura, Kōtarō (2010). Consistency, choice and rationality. Harvard University Press. ISBN 978-0674052994.
Alan D. Miller and Shiran Rachmilevitch (Feb 2014). Arrow's Theorem Without Transitivity (PDF) (Working paper). University of Haifa.

[1] Robert Duncan Luce (Apr 1956). "Semiorders and a Theory of Utility Discrimination" (PDF). Econometrica. 24 (2): 178–191. doi:10.2307/1905751. JSTOR 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 antisymmetry of R forces J to be coreflexive; hence the union of J and the transitive P is again transitive.

[1]

[2]

[3]

[4]

[5]

[6]

Formal definition

Examples

Properties

See also

References