The mathematical notion of quasitransitivity is a weakened version of transitivity that is used in

social choice theory Social choice theory or social choice is a theoretical framework for analysis of combining individual opinions, preferences, interests, or welfares to reach a ''collective decision'' or ''social welfare'' in some sense.Amartya Sen (2008). "Soci ...

and

microeconomics Microeconomics is a branch of mainstream economics that studies the behavior of individuals and firms in making decisions regarding the allocation of scarce resources and the interactions among these individuals and firms. Microeconomics fo ...

. Informally, a relation is quasitransitive if it is

symmetric Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...

for some values and transitive elsewhere. The concept was introduced by to study the consequences of

Arrow's theorem 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 ...

Formal definition

binary relation In mathematics, a binary relation associates elements of one set, called the ''domain'', with elements of another set, called the ''codomain''. A binary relation over Set (mathematics), sets and is a new set of ordered pairs consisting of ele ...

T over a

set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...

''X'' is quasitransitive if for all ''a'', ''b'', and ''c'' in ''X'' the following holds: :

(a\operatornameb) \wedge \neg(b\operatornamea) \wedge (b\operatornamec) \wedge \neg(c\operatornameb) \Rightarrow (a\operatornamec) \wedge \neg(c\operatornamea).

If the relation is also antisymmetric, T is transitive. Alternately, for a relation T, define the asymmetric or "strict" part P: :

(a\operatornameb) \Leftrightarrow (a\operatornameb) \wedge \neg(b\operatornamea).

Then T is quasitransitive if and only if P is transitive.

Examples

Preference In psychology, economics and philosophy, preference is a technical term usually used in relation to choosing between alternatives. For example, someone prefers A over B if they would rather choose A than B. Preferences are central to decision theo ...

s 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. Similarly, the

Sorites paradox The sorites paradox (; sometimes known as the paradox of the heap) is a paradox that results from vague predicates. A typical formulation involves a heap of sand, from which grains are removed individually. With the assumption that removing a sing ...

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 In mathematics, a disjoint union (or discriminated union) of a family of sets (A_i : i\in I) is a set A, often denoted by \bigsqcup_ A_i, with an injection of each A_i into A, such that the images of these injections form a partition of A (th ...

of a symmetric relation ''J'' and a transitive relation ''P''. ''J'' and ''P'' are not uniquely determined by a given ''R''; however, the ''P'' from the ''only-if'' part is minimal. * As a consequence, each symmetric relation is quasitransitive, and so is each transitive relation. Moreover, an antisymmetric and quasitransitive relation is always transitive.The antisymmetry of ''R'' forces ''J'' to be coreflexive; hence the union of ''J'' and the transitive ''P'' is again transitive. * The relation from the above sugar example, , is quasitransitive, but not transitive. * A quasitransitive relation needn't be acyclic: for every non-empty set ''A'', the

universal relation In mathematics, a homogeneous relation (also called endorelation) over a set ''X'' is a binary relation over ''X'' and itself, i.e. it is a subset of the Cartesian product . This is commonly phrased as "a relation on ''X''" or "a (binary) relation ...

''A'' ×''A'' is both cyclic and quasitransitive. * A relation is quasitransitive if, and only if, its

complement A complement is something that completes something else. Complement may refer specifically to: The arts * Complement (music), an interval that, when added to another, spans an octave ** Aggregate complementation, the separation of pitch-class ...

is. * Similarly, a relation is quasitransitive if, and only if, its

converse Converse may refer to: Mathematics and logic * Converse (logic), the result of reversing the two parts of a definite or implicational statement ** Converse implication, the converse of a material implication ** Converse nonimplication, a logical c ...

is.

References

* * * * * * * * * * {{cite report , url=http://econ.haifa.ac.il/~admiller/ArrowWithoutTransitivity.pdf , author=Alan D. Miller and Shiran Rachmilevitch , title=Arrow's Theorem Without Transitivity , institution=University of Haifa , type=Working paper , date=Feb 2014 Binary relations Social choice theory

Formal definition

Examples

Properties

See also

References