Quasitransitive relation

The quasitransitive relation . Its symmetrical part is shown in blue, its transitive part in green. ${\displaystyle x\leq {\frac {5}{4}}y}$

Quasi-transitivity is a watered- down version of transitivity used in social choice theory and microeconomics . Informally speaking, a relation is quasi-transitory if it is symmetric for some values ​​and transitive for others. The concept was introduced in 1969 by Amartya K. Sen to investigate the consequences of the Arrow theorem .

Examples

Some authors regard economic preferences as quasitransitive (and not as transitive). The classic example is a person who is undecided between 7 and 8 grams of sugar and also undecided between 8 and 9 grams of sugar, but who prefers 9 grams of sugar over 7. Similarly, the Sorites paradox can be resolved by weakening the assumed transitivity of certain relations to quasitransitivity.

Formal definition

A two-digit relation over a set is quasitransitive if the following applies to all : ${\displaystyle T}$ ${\displaystyle X}$ ${\displaystyle a,b,c\in X}$

${\displaystyle aTb\wedge \neg bTa\wedge bTc\wedge \neg cTb\Rightarrow aTc\wedge \neg cTa}$.

If the relation is also antisymmetric, T is transitive.

An alternate definition uses the asymmetrical, or “strict” part of , defined by ; thus is quasitransitive if and only if is transitive. ${\displaystyle P}$ ${\displaystyle T}$ ${\displaystyle aPb\Leftrightarrow aTb\wedge \neg bTa}$ ${\displaystyle T}$ ${\displaystyle P}$

properties

• A relation is quasi-transitive if and only if it is the disjoint union of a symmetric relation and a transitive relation . and are not clearly determined by; however, that is minimal from the " " evidence. ${\displaystyle R}$ ${\displaystyle J}$ ${\displaystyle P}$ ${\displaystyle J}$ ${\displaystyle P}$ ${\displaystyle R}$ ${\displaystyle P}$ ${\displaystyle \Rightarrow }$

• As a result, every symmetric relation is quasi-transitive, just like every transitive relation. In addition, an antisymmetric and quasi-transitive relation is always also transitive.

• The relation {(7.7), (7.8), (7.9), (8.7), (8.8), (8.9), (9.8), (9.9) } from the sugar example above is quasi-transitive, but not transitive.

• A quasitransitive relation does not have to correspond to an acyclic graph : for every non-empty set the universal relation is the Cartesian product both cyclic and quasitransitive. ${\displaystyle A}$ ${\displaystyle A\times A}$

credentials

1. ↑ Amartya K. Sen: Quasi-Transitivity, Rational Choice and Collective Decisions . In: The Review of Economic Studies . tape 36 , 1969, p. 381-393 .
2. ↑ Robert Duncan Luce: Semi Orders and a Theory of Utility Discrimination . In: Econometrica . tape 24 , no. 2 , April 1956, p. 178–191 ( online [PDF]). Here: p. 179; Luce's original example consists of 400 comparisons (of coffee cups with different amounts of sugar) instead of just 2.
3. ↑ The naming follows (Bossert, Suzumura 2009), pp. 2–3. - " ": Define and , then is symmetrical by construction and the transitivity of is identical to the definition of the quasitransitivity of . - " ": Let be the disjoint union of symmetrical and transitive and hold , then follows and , since or would contradict the assumptions or . Hence it follows because of transitivity, because of disjointness, because of symmetry. would therefore imply and, because of transitivity, also what contradicts. Overall this proves . ${\displaystyle \Rightarrow }$ ${\displaystyle xJy:\Leftrightarrow xRy\land yRx}$ ${\displaystyle xPy:\Leftrightarrow xRy\land \lnot yRx}$ ${\displaystyle J}$ ${\displaystyle P}$ ${\displaystyle R}$ ${\displaystyle \Leftarrow }$ ${\displaystyle R}$ ${\displaystyle J}$ ${\displaystyle P}$ ${\displaystyle xRy\land \lnot yRx\land yRz\land \lnot zRy}$ ${\displaystyle xPy}$ ${\displaystyle yPz}$ ${\displaystyle xJy}$ ${\displaystyle yJz}$ ${\displaystyle \lnot yRx}$ ${\displaystyle \lnot zRy}$ ${\displaystyle xPz}$ ${\displaystyle \lnot xJz}$ ${\displaystyle \lnot zJx}$ ${\displaystyle zRx}$ ${\displaystyle zPx}$ ${\displaystyle zPy}$ ${\displaystyle \lnot zRy}$ ${\displaystyle xRz\land \lnot zRx}$
4. ↑ If R z. B. is an equivalence relation , J can be chosen as the empty relation or as R itself and P each as complement R \ J.
5. ↑ If xRy ∧ ¬ yRx holds for a given R , the pair ( x , y ) cannot belong to the symmetric part, it must therefore belong to the transitive part in any case.
6. ↑ Since the empty relation is trivially both transitive and symmetric.
7. ↑ The antisymmetry of R makes the coreflexivity ^{( s )} of J necessary, so the union of J and the transitive part P is again transitive.

• Frederic Schick: Arrow's Proof and the Logic of Preference . In: Philosophy of Science . tape 36 , no. 2 , June 1969, p. 127-144 , JSTOR : 186166 (English).
• Walter Bossert, Kotaro Suzumura: Quasi-transitive and Suzumura consistent relations . March 2009 (English, technical report [PDF] Université de Montréal, Waseda University Tokyo).