Quasitransitive relation

Key takeaways

Quasitransitive relation belongs to science; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Quasitransitive relation to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Quasitransitive relation from memory before moving on to harder problems.

Reference excerpt

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.

Formal definition A binary relation T over a set X is quasitransitive if for all a, b, and c in X the following holds:

( a T ⁡ b ) ∧ ¬ ( b T ⁡ a ) ∧ ( b T ⁡ c ) ∧ ¬ ( c T ⁡ b ) ⇒ ( a T ⁡ c ) ∧ ¬ ( c T ⁡ a ) . {\displaystyle (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).}

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

( a P ⁡ b ) ⇔ ( a T ⁡ b ) ∧ ¬ ( b T ⁡ a ) . {\displaystyle (a\operatorname {P} b)\Leftrightarrow (a\operatorname {T} b)\wedge \neg (b\operatorname {T} a).}

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 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. 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 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

Sen, A. (1969). "Quasi-transitivity, rational choice and collective decisions". Rev. Econ. Stud. 36 (3): 381–393. doi:10.2307/2296434. JSTOR 2296434. Zbl 0181.47302. Frederic Schick (Jun 1969). "Arrow's Proof and the Logic of Preference". Philosophy of Science. 36 (2): 127–144. doi:10.1086/288241. JSTOR 186166. S2CID 121427121. Amartya K. Sen (1970). Collective Choice and Social Welfare. Holden-Day, Inc. Amartya K. Sen (Jul 1971). "Choice Functions and Revealed Preference" (PDF). The Review of Economic Studies. 38 (3): 307–317. doi:10.2307/2296384. JSTOR 2296384. Archived from the original (PDF) on 2016-09-10. Retrieved 2018-04-11. A. Mas-Colell and H. Sonnenschein (1972). "General Possibility Theorems for Group Decisions" (PDF). The Review of Economic Studies. 39 (2): 185–192. doi:10.2307/2296870. JSTOR 2296870. S2CID 7295776. Archived from the original (PDF) on 2018-04-12. D.H. Blair and R.A. Pollak (1982). "Acyclic Collective Choice Rules". Econometrica. 50 (4): 931–943. doi:10.2307/1912770. JSTOR 1912770. Bossert, Walter; Suzumura, Kotaro (Apr 2005). Rational Choice on Arbitrary Domains: A Comprehensive Treatment (PDF) (Technical Report). Université de Montréal, Hitotsubashi University Tokyo. Archived from the original (PDF) on 2018-04-12. Retrieved 2018-04-11. Bossert, Walter; Suzumura, Kotaro (Mar 2009). "Quasi-transitive and Suzumura consistent relations" (PDF). Social Choice and Welfare (Technical Report). 39 (2–3). Université de Montréal, Waseda University Tokyo: 323–334. doi:10.1007/s00355-011-0600-z. S2CID 38375142. Archived from the original (PDF) on 2018-04-12. Bossert, Walter; Suzumura, Kōtarō (2010). Consistency, choice and rationality. Harvard University Press. ISBN 978-0674052994. Alan D. Miller and Shiran Rachmilevitch (Feb 2014). Arrow's Theorem Without Transitivity (PDF) (Working paper). University of Haifa.

Illustrations

Worked examples

Example 1 — a first encounter with Quasitransitive relation

Start with the simplest possible case. Write down what Quasitransitive relation claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In science, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Quasitransitive relation before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Quasitransitive relation ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Quasitransitive relation

In research: Quasitransitive relation appears in science research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Quasitransitive relation in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Quasitransitive relation is common in secondary-school and first-year university syllabi. It links to neighbouring topics Properties of binary relations, Social choice theory, so understanding it makes those chapters shorter.
In everyday life: Look for Quasitransitive relation outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Quasitransitive relation in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Quasitransitive relation means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Quasitransitive relation out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Quasitransitive relation in simple terms?

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.

Why does Quasitransitive relation matter?

Because it connects several science ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Quasitransitive relation?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Quasitransitive relation.