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). "So ...

and microeconomics. 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 definit ...

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

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 ''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 Asymmetric may refer to: *Asymmetry in geometry, chemistry, and physics Computing * Asymmetric cryptography, in public-key cryptography *Asymmetric digital subscriber line, Internet connectivity * Asymmetric multiprocessing, in computer architect ...

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

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. 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 ...

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 ''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-clas ...

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 Binary relations Social choice theory

Formal definition

Examples

Properties

See also

References