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

social choice theory Social choice theory is a branch of welfare economics that extends the Decision theory, theory of rational choice to collective decision-making. Social choice studies the behavior of different mathematical procedures (social welfare function, soc ...

and

microeconomics Microeconomics is a branch of economics that studies the behavior of individuals and Theory of the firm, firms in making decisions regarding the allocation of scarcity, scarce resources and the interactions among these individuals and firms. M ...

. Informally, a relation is quasitransitive if it is

symmetric Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is invariant under some transformations ...

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

Arrow's theorem Arrow's impossibility theorem is a key result in social choice theory showing that no Ordinal utility, ranked-choice procedure for group decision-making can satisfy the requirements of rational choice. Specifically, Kenneth Arrow, Arrow showed no ...

Formal definition

binary relation In mathematics, a binary relation associates some elements of one Set (mathematics), set called the ''domain'' with some elements of another set called the ''codomain''. Precisely, a binary relation over sets X and Y is a set of ordered pairs ...

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 the ...

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 s ...

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, the disjoint union (or discriminated union) A \sqcup B of the sets and is the set formed from the elements of and labelled (indexed) with the name of the set from which they come. So, an element belonging to both and appe ...

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) on a set ''X'' is a binary relation between ''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 is. * Similarly, a relation is quasitransitive if, and only if, its converse 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 Properties of binary relations Social choice theory

Formal definition

Examples

Properties

See also

References