thumb|The quasitransitive relation ''x''≤{{sfrac|5|4}}''y''. 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 {{harvtxt|Sen|1969}} to study the consequences of Arrow's theorem.

==Formal definition== A binary relation T over a set ''X'' is '''quasitransitive''' if for all ''a'', ''b'', and ''c'' in ''X'' the following holds:

: <math>(a\operatorname{T}b) \wedge \neg(b\operatorname{T}a) \wedge (b\operatorname{T}c) \wedge \neg(c\operatorname{T}b) \Rightarrow (a\operatorname{T}c) \wedge \neg(c\operatorname{T}a).</math>

If the relation is also antisymmetric, T is transitive.

Alternately, for a relation T, define the asymmetric or "strict" part P: :<math>(a\operatorname{P}b) \Leftrightarrow (a\operatorname{T}b) \wedge \neg(b\operatorname{T}a).</math>

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.<ref>{{cite journal | url=https://scholar.archive.org/work/63pkkfwm4nechh6c7gxxmuusda | author=Robert Duncan Luce | author-link=Robert Duncan Luce | title=Semiorders and a Theory of Utility Discrimination | journal=Econometrica | volume=24 | number=2 | pages=178–191 | date=Apr 1956 | 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.</ref> Similarly, the Sorites paradox 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 of a symmetric relation ''J'' and a transitive relation ''P''.<ref>The naming follows {{harvtxt|Bossert|Suzumura|2009}}, p.2-3.<!---'J' is used instead of 'I' to increase readability with sans-serif fonts---> &mdash; For the ''only-if'' part, define ''xJy'' as ''xRy'' ∧ ''yRx'', and define ''xPy'' as ''xRy'' ∧ ¬''yRx''. &mdash; For the ''if'' part, assume ''xRy'' ∧ ¬''yRx'' ∧ ''yRz'' ∧ ¬''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''.</ref> ''J'' and ''P'' are not uniquely determined by a given ''R'';<ref>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.</ref> however, the ''P'' from the ''only-if'' part is minimal.<ref>Given ''R'', whenever ''xRy'' ∧ ¬''yRx'' holds, the pair (''x'',''y'') can't belong to the symmetric part, but must belong to the transitive part.</ref> * As a consequence, each symmetric relation is quasitransitive, and so is each transitive relation.<ref>Since the empty relation is trivially both transitive and symmetric.</ref> Moreover, an antisymmetric and quasitransitive relation is always transitive.<ref>The antisymmetry of ''R'' forces ''J'' to be coreflexive; hence the union of ''J'' and the transitive ''P'' is again transitive.</ref> * The relation from the above sugar example, {(7,7), (7,8), (7,9), (8,7), (8,8), (8,9), (9,8), (9,9)}, 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 is. * Similarly, a relation is quasitransitive if, and only if, its converse is.

==See also== * Intransitivity * Reflexive relation

==References== {{reflist}} * {{cite journal | last=Sen | first=A. | author-link=Amartya Sen | title=Quasi-transitivity, rational choice and collective decisions | zbl=0181.47302 | journal=Rev. Econ. Stud. | volume=36 | pages=381–393 | year=1969 | issue=3 | doi=10.2307/2296434 | jstor=2296434 }} * {{cite journal | jstor=186166 | author=Frederic Schick | title=Arrow's Proof and the Logic of Preference | journal=Philosophy of Science | volume=36 | number=2 | pages=127–144 | date=Jun 1969 | doi=10.1086/288241| s2cid=121427121 }} * {{cite book | author=Amartya K. Sen | title=Collective Choice and Social Welfare | publisher=Holden-Day, Inc. | year=1970 }} * {{cite journal | url=https://www.ihs.ac.at/publications/eco/visit_profs/blume/sen.pdf | author=Amartya K. Sen | title=Choice Functions and Revealed Preference | journal=The Review of Economic Studies | volume=38 | number=3 | pages=307–317 | date=Jul 1971 | doi=10.2307/2296384 | jstor=2296384 | archive-date=2016-09-10 | access-date=2018-04-11 | archive-url=https://web.archive.org/web/20160910121449/http://www.ihs.ac.at/publications/eco/visit_profs/blume/sen.pdf | url-status=dead }} * {{cite journal | url=https://pdfs.semanticscholar.org/f66c/6beda52f00373ca04509fd9ad27e6763055f.pdf | archive-url=https://web.archive.org/web/20180412082300/https://pdfs.semanticscholar.org/f66c/6beda52f00373ca04509fd9ad27e6763055f.pdf | url-status=dead | archive-date=2018-04-12 | author=A. Mas-Colell and H. Sonnenschein | title=General Possibility Theorems for Group Decisions | journal=The Review of Economic Studies | volume=39 | pages=185–192 | year=1972 | issue=2 | doi=10.2307/2296870| jstor=2296870 | s2cid=7295776 }} * {{cite journal | author=D.H. Blair and R.A. Pollak | title=Acyclic Collective Choice Rules | journal=Econometrica | volume=50 | pages=931–943 | year=1982 | issue=4 | doi=10.2307/1912770| jstor=1912770 }} * {{cite report | url=http://faculty.arts.ubc.ca/pnorman/CETC/Papers/bossert.pdf | first1=Walter | last1=Bossert | first2=Kotaro | last2=Suzumura | title=Rational Choice on Arbitrary Domains: A Comprehensive Treatment | institution=Université de Montréal, Hitotsubashi University Tokyo | type=Technical Report | date=Apr 2005 | archive-date=2018-04-12 | access-date=2018-04-11 | archive-url=https://web.archive.org/web/20180412082240/http://faculty.arts.ubc.ca/pnorman/CETC/Papers/bossert.pdf | url-status=dead }} * {{cite journal | url=https://pdfs.semanticscholar.org/240e/97a4f812ff51317a68c7b72d0f1e84eb8266.pdf | archive-url=https://web.archive.org/web/20180412082302/https://pdfs.semanticscholar.org/240e/97a4f812ff51317a68c7b72d0f1e84eb8266.pdf | url-status=dead | archive-date=2018-04-12 | first1=Walter | last1=Bossert | first2=Kotaro | last2=Suzumura | title=Quasi-transitive and Suzumura consistent relations | journal=Social Choice and Welfare | institution=Université de Montréal, Waseda University Tokyo | type=Technical Report | date=Mar 2009 | volume=39 | issue=2–3 | pages=323–334 | doi=10.1007/s00355-011-0600-z | s2cid=38375142 }} * {{cite book | title=Consistency, choice and rationality | first1=Walter | last1=Bossert | first2=Kōtarō | last2=Suzumura | publisher=Harvard University Press | year=2010 | isbn=978-0674052994 }} * {{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 }}

Category:Properties of binary relations Category:Social choice theory