06/06/2026, 16:19 ECON 2291 — Preference Theory & EUT Revision Guide
E C O N 2 2 9 1 — E C O N O M I C T H E O RY | D U R H A M U N I V E R S I T Y
Preference Theory & Expected Utility
Theory
A complete first-class revision guide — covering all lectures, textbook proofs, past paper
answers, and practice questions
C ON T E N T S
1. Part I — Preference Theory: Axioms, Utility, and Indifference Curves
2. Part II — Special Utility Functions and Monotonic Transformations
3. Part III — Revealed Preference and WARP
4. Part IV — Expected Utility Theory: Lotteries and the VNM Framework
5. Part V — Risk Attitudes and the Shape of the Bernoulli Function
6. Part VI — Violations of EUT: The Allais Paradox
7. Part VII — Prospect Theory
8. Part VIII — Past Paper Questions: Full Model Answers
9. Part IX — Practice Questions with Answers
PA R T I
Preference Theory: Axioms, Utility, and
Indifference Curves
What are Preferences?
Preferences describe an individual's mental ranking of alternatives, without reference to
the intensity of feeling. In formal terms, a binary relation ≽ on a set X encodes the
answer to "how do you compare x and y?" for every pair of alternatives. From the weak
preference relation ≽ ("at least as good as"), we derive two further relations:
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,06/06/2026, 16:19 ECON 2291 — Preference Theory & EUT Revision Guide
— Strict preference: x≻y if x≽y but not y≽x
— Indifference: x~y if x≽y and y≽x
The Axioms of Preference
For ≽ to constitute a proper preference relation, it must satisfy at minimum
completeness and transitivity. Additional properties are frequently assumed.
COMPLETENESS TRANSITIVITY
For any x, y ∈ X , either x ≽ y or y ≽ x For any x, y, z ∈ X , if x ≽ y and y ≽ z ,
(or both). The consumer can always rank then x ≽ z . Preferences are consistent —
any two bundles. Note: completeness no cycles. This is the most contested
implies reflexivity ( x ≽ x for all x ). axiom empirically.
REFLEXIVITY SYMMETRY
For any x ∈ X , x ≽ x . Every alternative If xRy then yRx . Applies to the
is at least as good as itself. Often implicit indifference relation ~ . Example: "is
in completeness. married to" is symmetric; "is parent of" is
not.
CONTINUITY MONOTONICITY
If bundle A ≻ B and C is very close to Weak: if x ≥ x' and y ≥ y' , then (x,y) ≽
B , then A ≻ C . No sudden jumps in the (x',y') . Strong: additionally, if at least one
preference ordering. Formally: the sets {x inequality is strict, then (x,y) ≻ (x',y') .
: x ≽ y} and {x : y ≽ x} are closed. Captures "more is better."
CONVEXITY LOCAL NON-SATIATION
Averages are preferred to extremes. If x~ For any bundle, there is always a nearby
y , then for any α ∈ (0,1) , the mixture αx bundle that is strictly preferred. Weaker
+ (1-α)y ≽ y . Implies diminishing than monotonicity — it rules out thick
marginal utility and bowed-in indifference curves without requiring that
indifference curves. "more is always better."
DE F I N I T I O N — P R E F E R E NC E R E L AT I O N ( O S B O R N E & RU B I N ST E I N )
A preference relation on X is a complete and transitive binary relation on X . These
two conditions are the minimum required. Completeness ensures the agent can
file:///C:/Users/tyala/Downloads/ECON2291_Revision_Guide (2).html#part9 2/28
, 06/06/2026, 16:19 ECON 2291 — Preference Theory & EUT Revision Guide
always decide; transitivity ensures consistency.
Equivalence Relations and Indifference Sets
The indifference relation ~ derived from any preference relation is an equivalence
relation: it is reflexive, symmetric, and transitive. This matters because equivalence
relations partition X into disjoint indifference sets (indifference curves in two-good
space). All bundles within one indifference set are equally preferred; bundles from
different sets are not.
C O M M O N M I STA K E
Transitivity violations are more common in humans than we expect. Osborne &
Rubinstein (2023) report that in a study of ~1,300 participants, only 15% exhibited no
transitivity violation across a 36-question preference questionnaire. The median
participant violated transitivity across 6 triples. The point: the axiom is a
simplification, not a description of reality.
From Preferences to Utility Functions
A utility function U:X→ℝ represents a preference relation ≽ if
x ≽ y if and only if U(x) ≥ U(y)
Critically, the numbers assigned by a utility function have no cardinal meaning — only
their ranking matters. If U(a) = 10 and U(b) = 5 , we know a ≻ b but cannot say "a is
twice as good."
P RO P O S I T I O N 1 . 1 ( R E P R E S E N TAT I O N T H E O R E M ) — O S B O R N E & RU B I N ST E I N
Every preference relation on a finite set can be represented by a utility function. The
proof constructs the function by repeatedly identifying and removing the minimal
elements of the remaining set, assigning increasing utility values at each stage.
P RO P O S I T I O N 1 . 2 — I N F I N I T E S E T S A N D FA I LU R E O F R E P R E S E N TAT I O N
Not all preference relations on infinite sets can be represented by a utility function.
The canonical counterexample is lexicographic preferences on the unit square {(x₁,
x₂) : x₁, x₂ ∈ [0,1]} . The proof uses Cantor's diagonal argument: representation would
file:///C:/Users/tyala/Downloads/ECON2291_Revision_Guide (2).html#part9 3/28
E C O N 2 2 9 1 — E C O N O M I C T H E O RY | D U R H A M U N I V E R S I T Y
Preference Theory & Expected Utility
Theory
A complete first-class revision guide — covering all lectures, textbook proofs, past paper
answers, and practice questions
C ON T E N T S
1. Part I — Preference Theory: Axioms, Utility, and Indifference Curves
2. Part II — Special Utility Functions and Monotonic Transformations
3. Part III — Revealed Preference and WARP
4. Part IV — Expected Utility Theory: Lotteries and the VNM Framework
5. Part V — Risk Attitudes and the Shape of the Bernoulli Function
6. Part VI — Violations of EUT: The Allais Paradox
7. Part VII — Prospect Theory
8. Part VIII — Past Paper Questions: Full Model Answers
9. Part IX — Practice Questions with Answers
PA R T I
Preference Theory: Axioms, Utility, and
Indifference Curves
What are Preferences?
Preferences describe an individual's mental ranking of alternatives, without reference to
the intensity of feeling. In formal terms, a binary relation ≽ on a set X encodes the
answer to "how do you compare x and y?" for every pair of alternatives. From the weak
preference relation ≽ ("at least as good as"), we derive two further relations:
file:///C:/Users/tyala/Downloads/ECON2291_Revision_Guide (2).html#part9 1/28
,06/06/2026, 16:19 ECON 2291 — Preference Theory & EUT Revision Guide
— Strict preference: x≻y if x≽y but not y≽x
— Indifference: x~y if x≽y and y≽x
The Axioms of Preference
For ≽ to constitute a proper preference relation, it must satisfy at minimum
completeness and transitivity. Additional properties are frequently assumed.
COMPLETENESS TRANSITIVITY
For any x, y ∈ X , either x ≽ y or y ≽ x For any x, y, z ∈ X , if x ≽ y and y ≽ z ,
(or both). The consumer can always rank then x ≽ z . Preferences are consistent —
any two bundles. Note: completeness no cycles. This is the most contested
implies reflexivity ( x ≽ x for all x ). axiom empirically.
REFLEXIVITY SYMMETRY
For any x ∈ X , x ≽ x . Every alternative If xRy then yRx . Applies to the
is at least as good as itself. Often implicit indifference relation ~ . Example: "is
in completeness. married to" is symmetric; "is parent of" is
not.
CONTINUITY MONOTONICITY
If bundle A ≻ B and C is very close to Weak: if x ≥ x' and y ≥ y' , then (x,y) ≽
B , then A ≻ C . No sudden jumps in the (x',y') . Strong: additionally, if at least one
preference ordering. Formally: the sets {x inequality is strict, then (x,y) ≻ (x',y') .
: x ≽ y} and {x : y ≽ x} are closed. Captures "more is better."
CONVEXITY LOCAL NON-SATIATION
Averages are preferred to extremes. If x~ For any bundle, there is always a nearby
y , then for any α ∈ (0,1) , the mixture αx bundle that is strictly preferred. Weaker
+ (1-α)y ≽ y . Implies diminishing than monotonicity — it rules out thick
marginal utility and bowed-in indifference curves without requiring that
indifference curves. "more is always better."
DE F I N I T I O N — P R E F E R E NC E R E L AT I O N ( O S B O R N E & RU B I N ST E I N )
A preference relation on X is a complete and transitive binary relation on X . These
two conditions are the minimum required. Completeness ensures the agent can
file:///C:/Users/tyala/Downloads/ECON2291_Revision_Guide (2).html#part9 2/28
, 06/06/2026, 16:19 ECON 2291 — Preference Theory & EUT Revision Guide
always decide; transitivity ensures consistency.
Equivalence Relations and Indifference Sets
The indifference relation ~ derived from any preference relation is an equivalence
relation: it is reflexive, symmetric, and transitive. This matters because equivalence
relations partition X into disjoint indifference sets (indifference curves in two-good
space). All bundles within one indifference set are equally preferred; bundles from
different sets are not.
C O M M O N M I STA K E
Transitivity violations are more common in humans than we expect. Osborne &
Rubinstein (2023) report that in a study of ~1,300 participants, only 15% exhibited no
transitivity violation across a 36-question preference questionnaire. The median
participant violated transitivity across 6 triples. The point: the axiom is a
simplification, not a description of reality.
From Preferences to Utility Functions
A utility function U:X→ℝ represents a preference relation ≽ if
x ≽ y if and only if U(x) ≥ U(y)
Critically, the numbers assigned by a utility function have no cardinal meaning — only
their ranking matters. If U(a) = 10 and U(b) = 5 , we know a ≻ b but cannot say "a is
twice as good."
P RO P O S I T I O N 1 . 1 ( R E P R E S E N TAT I O N T H E O R E M ) — O S B O R N E & RU B I N ST E I N
Every preference relation on a finite set can be represented by a utility function. The
proof constructs the function by repeatedly identifying and removing the minimal
elements of the remaining set, assigning increasing utility values at each stage.
P RO P O S I T I O N 1 . 2 — I N F I N I T E S E T S A N D FA I LU R E O F R E P R E S E N TAT I O N
Not all preference relations on infinite sets can be represented by a utility function.
The canonical counterexample is lexicographic preferences on the unit square {(x₁,
x₂) : x₁, x₂ ∈ [0,1]} . The proof uses Cantor's diagonal argument: representation would
file:///C:/Users/tyala/Downloads/ECON2291_Revision_Guide (2).html#part9 3/28
