ECON30019 Chap.2 Preferences and the Theory of Rational Choice
Preferences and the Theory of Rational Choice
Everything formal in the subject is built on one object: a binary relation over a set of alternatives. A preference is a relation ordering alternatives by how much they are wanted, and treating it as a relation rather than as a feeling is what makes it testable, because relations have properties and properties can fail.
The chapter fixes the three symbols, noting that only weak preference is primitive while strict preference and indifference are defined from it, then states the two rationality axioms, completeness and transitivity, and works through the six properties the derived relations inherit.
It closes with the representation theorem on a finite set, which says a ranking is rational exactly when a utility function representing it exists, and with the consequence that the representation is ordinal, so differences and ratios of utility carry no meaning.
What this chapter covers
- 01
Binary relations, the universe, and what makes a preference a special case
- 02
Weak preference as the primitive, with strict preference and indifference defined from it
- 03
Completeness, and why indifference satisfies it rather than violating it
- 04
Transitivity, and why a cycle makes a decision maker exploitable
- 05
The two-item and three-item statements of rationality found in the subject
- 06
Three properties the strict relation inherits, and three the indifference relation inherits
- 07
Indifference classes, and the picture a utility function will encode
- 08
Two proof shapes: direct proof and proof by contradiction
- 09
Utility representation and the finite-set theorem, in both directions
- 10
Ordinality, and the two claims it forbids you from making
Prove that indifference is transitive
- 1Unpack the first premise by the definition of indifference: x is weakly preferred to y, and y is weakly preferred to x.
- 1Unpack the second premise the same way: y is weakly preferred to z, and z is weakly preferred to y.
- 1Chain the forward pair by transitivity of the weak relation to obtain that x is weakly preferred to z, then chain the backward pair to obtain that z is weakly preferred to x.
- 1Recombine by the definition of indifference: both weak preferences hold between x and z, so x is indifferent to z. Note that transitivity was used twice and completeness was not needed.
Key terms
- Binary relation
- A statement of how one thing stands to another, defined on a stated universe. Preference is one example among many, and the formalism is what makes it testable.
- Weak preference
- The primitive relation, read as at least as good as. Strict preference and indifference are both defined from it rather than assumed alongside it.
- Completeness
- The condition that any two alternatives can be compared, so at least one weak preference holds between them. Indifference is the case where both hold, which satisfies it twice over.
- Transitivity
- The condition that rankings chain: if the first is at least as good as the second and the second at least as good as the third, then the first is at least as good as the third.
- Utility representation
- An assignment of numbers to alternatives whose ordering reproduces the ranking exactly, in both directions. On a finite set one exists precisely when the ranking is rational.
- Ordinality
- The property that any increasing relabelling of a utility function represents the same preferences, so differences and ratios of utility, and comparisons across people, carry no meaning.
Preferences and the Theory of Rational Choice FAQ
Does rationality mean acting in your own interest?
No. In this subject rationality is a property of a relation, not of a motive. It asserts only that any two alternatives can be compared and that the rankings chain. Someone with unusual, self-destructive or purely altruistic tastes can hold a perfectly rational preference relation, and nothing in the definition rules that out.
The subject seems to give two different definitions of rationality. Which is right?
Both appear in the subject's own materials: one lecture states completeness and transitivity, and the following recap adds reflexivity. In practice they describe the same relations, since reflexivity follows from completeness applied with the same alternative on both sides. If a question asks you to select the rationality properties from a list, say in one clause which definition you are using.
Why can I not say one option gives twice as much utility as another?
Because the representation is ordinal. Utilities of one, two and three and utilities of one, forty and forty-one represent exactly the same preferences, so any statement about differences or ratios is an artefact of the numbers you happened to choose. The same argument rules out adding or comparing utilities across two people.
Exam move
This is the chapter to over-learn, because everything later is defined against it and the marks are definitional. Recite the three symbol definitions and the two axioms until they are exact, and practise the two proof shapes on properties the subject does not work for you, such as transitivity of strict preference.
When you write a proof, label every line with the definition or axiom that justifies it, and say explicitly when an axiom was not needed, since that demonstrates you know which assumption is doing the work rather than reciting both.
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