Quasitransitive relation

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The quasitransitive relation x<=
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5/4y. Its symmetric and transitive part is shown in blue and green, respectively. Quasitransitive 25 percent margin.gif
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.

Contents

Formal definition

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 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

See also

Related Research Articles

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Arrow's impossibility theorem, the general possibility theorem or Arrow's paradox is an impossibility theorem in social choice theory that states that when voters have three or more distinct alternatives (options), no ranked voting electoral system can convert the ranked preferences of individuals into a community-wide ranking while also meeting the specified set of criteria: unrestricted domain, non-dictatorship, Pareto efficiency, and independence of irrelevant alternatives. The theorem is often cited in discussions of voting theory as it is further interpreted by the Gibbard–Satterthwaite theorem. The theorem is named after economist and Nobel laureate Kenneth Arrow, who demonstrated the theorem in his doctoral thesis and popularized it in his 1951 book Social Choice and Individual Values. The original paper was titled "A Difficulty in the Concept of Social Welfare".

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<span class="mw-page-title-main">Sorites paradox</span> Logical paradox from vague predicates

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<span class="mw-page-title-main">Weak ordering</span> Mathematical ranking of a set

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<span class="mw-page-title-main">Relation (mathematics)</span> Relationship between two sets, defined by a set of ordered pairs

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In order theory, the Szpilrajn extension theorem, proved by Edward Szpilrajn in 1930, states that every partial order is contained in a total order. Intuitively, the theorem says that any method of comparing elements that leaves some pairs incomparable can be extended in such a way that every pair becomes comparable. The theorem is one of many examples of the use of the axiom of choice in the form of Zorn's lemma to find a maximal set with certain properties.

<span class="mw-page-title-main">Semiorder</span> Numerical ordering with a margin of error

In order theory, a branch of mathematics, a semiorder is a type of ordering for items with numerical scores, where items with widely differing scores are compared by their scores and where scores within a given margin of error are deemed incomparable. Semiorders were introduced and applied in mathematical psychology by Duncan Luce (1956) as a model of human preference. They generalize strict weak orderings, in which items with equal scores may be tied but there is no margin of error. They are a special case of partial orders and of interval orders, and can be characterized among the partial orders by additional axioms, or by two forbidden four-item suborders.

Prasanta Kumar Pattanaik, is an Indian-American emeritus professor at the Department of Economics at the University of California. He is a Fellow of the Econometric Society.

References

  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 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.