A Theory of Temptation without Uncertainty Madhav Chandrasekher - - PowerPoint PPT Presentation

Preliminaries Representation Theorems Discussion

A Theory of Temptation without Uncertainty

Madhav Chandrasekher

Arizona State University

NASM 2009

Preliminaries Representation Theorems Discussion

A Menu Choice Problem

1 Period 1 (ex ante): DM is offered an allowance (i.e. a

collection of consumption/savings budget plans).

2 Period 2 (ex post): DM chooses a consumption/savings plan

(i.e. she selects a consumption path). GP (2005) perspective: Interpret each menu as an (implicit) dynamic optimization problem.

Preliminaries Representation Theorems Discussion

Why Study Menu Choice?

Two interesting issues arise in evaluating menus: DM may have (revealed) ex ante uncertainty about ex post preferences. DM may view some ex post preferences as harmful (ex ante) (i.e. an ex post taste for high consumption). Say that the DM has a ‘self-control’ problem in latter case.

Preliminaries Representation Theorems Discussion

DLR (2001)

The EU Model: U(A) =

s∈S ps·maxx∈A us(x)

Ingredients:

1 S = state space. 2 ps = signed measure. 3 us(·) = state-dependent consumption utilities.

Preliminaries Representation Theorems Discussion

The Question

1 Two distinct conceptual issues:

Does the agent have a self-control problem? Does the agent have ex ante uncertainty about ex post preferences?

Preliminaries Representation Theorems Discussion

The Question

1 Two distinct conceptual issues:

Does the agent have a self-control problem? Does the agent have ex ante uncertainty about ex post preferences?

2 Conflation of uncertainty and existence of self-control

problems in the EU model.

Preliminaries Representation Theorems Discussion

The Question

1 Two distinct conceptual issues:

Does the agent have a self-control problem? Does the agent have ex ante uncertainty about ex post preferences?

2 Conflation of uncertainty and existence of self-control

problems in the EU model.

3 Goal: Provide foundations for temptation preferences without

uncertainty.

Preliminaries Representation Theorems Discussion

Choice Environment

1 X = {x1, . . ., xn} = prize space. 2 2X = space of menus. 3 P(X) = space of preference relations on 2X.

Behavioral Primitive: ∈ P(X).

Preliminaries Representation Theorems Discussion

GP (2005)

The Strotz Model v : X → R (put Av = arg maxx∈A v(x)) u : X → R UST(A) := maxx∈Av u(x)

Preliminaries Representation Theorems Discussion

Limitations of Strotz, after DLR (2008)

Example 1: Multi-Dimensional Temptation Put X = {x, y, z}, x = broccoli, y = vegan cookie, z = cookie. Assume that

1

{x} ≻ {y} ≻ {z}.

2

{x, y} ∼ {x}, {x, z} ∼ {z}, {y, z} ∼ {y}.

3

{x, y, z} ∼ {y}.

Example 2: Temptation Aggregation Put X = {x, y, z} (as above). Assume that

1

{x} ≻ {y} ≻ {z}.

2

{x, y} ∼ {x}, {x, z} ∼ {x}.

3

{x, y, z} ∼ {y}.

Preliminaries Representation Theorems Discussion

The Category Model

Ingredients

1 A collection of sets Ci ⊆ X such that:

(Completeness) ∪i Ci = X (Non-Redundance) Ci ⊆ ∪j=i Cj

2 A (strict) normative ranking u(·) : X → R

Using these ingredients we define: UC(A) = maxCi minx∈A∩Ci u(x)

Preliminaries Representation Theorems Discussion

The Category Model, cont’d

Example 1 - Multi-Dimensional Temptation

1 Put C1 = {x, z}, C2 = {y}, C := {C1, C2}. 2 u(x) > u(y) > u(z). 3 UC({x, z}) = u(z), UC({x, y}) = u(x), UC({y, z}) = u(y).

Example 2 - Temptation Aggregation

1 Put C1 = {x, y}, C2 = {x, z} 2 u(x) > u(y) > u(z). 3 UC({x, y, z}) = u(y), UC({x, y}) = u(x) = UC({x, z}).

Preliminaries Representation Theorems Discussion

Variations, part I

The Partition Model

1 A collection of sets B ≡ {Bi} such that

Bi ∩ Bj = ∅, ∀(i, j). ∪i Bi = X.

2 UB(A) := maxBi∈B minx∈A∩Bi u(x)

The Rigid Category Model

1 An index set A and a collection {Cx}x∈A such that

∪x∈A Cx = X. x ∈ sup(Cx). If x ∈ A, then x / ∈ Cy, ∀y = x.

2 UCA(A) = maxCx:x∈A minz∈A∩Cx u(z)

Preliminaries Representation Theorems Discussion

Variations, part II

The Local Partition Model

1 A family of partitions B∗ ≡ {BA}A∈2X such that

(Completeness) ∪B∈BA B = A. (Coherence) A ⊆ A′ ⇒ ∀D ∈ BA, ∃D′ ∈ BA′, D ⊆ D′.

2 UB∗(A) := maxB∈BA minx∈B u(x)

The Local Category Model

1 A menu indexed collection of categories C∗ ≡ {CA}A∈2X such

that

(Completeness) ∪D∈CA D = A. (Non-Redundance) inf(D) ⊆ ∪D′∈CA:D=D′ D′ (Coherence) A ⊆ A′ ⇒ ∀D ∈ CA, ∃D′ ∈ CA′, D ⊆ D′.

2 UC∗(A) := maxD∈CA minx∈D u(x).

Preliminaries Representation Theorems Discussion

Representing the Strotz Model

DSB Axiom: A ∪ B ∼ A or A ∪ B ∼ B, ∀A, B. Definition: Call an x ∈ A a Strong Equivalent if

1 x ∼ A 2 B ∼ A, ∀B ⊆ A s.t. x ∈ B.

A1′: If A = ∅, then Σs(A) = ∅. (Σs(A) = strong equivalents in A)

Preliminaries Representation Theorems Discussion

Representing the Strotz Model

DSB Axiom: A ∪ B ∼ A or A ∪ B ∼ B, ∀A, B. Definition: Call an x ∈ A a Strong Equivalent if

1 x ∼ A 2 B ∼ A, ∀B ⊆ A s.t. x ∈ B.

A1′: If A = ∅, then Σs(A) = ∅. (Σs(A) = strong equivalents in A) Lemma: A1′ is equivalent to DSB.

Preliminaries Representation Theorems Discussion

Representing the Strotz Model, cont’d.

Theorem (GP (2005)) A preference ∈ P(X) satisfies A1′ if and only if there is a pair (u, v) such that UST(A) := maxx∈Av u(x) represents .

Preliminaries Representation Theorems Discussion

Representing the Local Category Model

Definition: An element x ∈ A is called a Weak Equivalent if the following properties hold:

1 x ∼ A 2 B A whenever B ⊆ A and x ∈ B.

A1: If A = ∅, then Σ(A) = ∅. (Σ(A) = weak equivalents)

Preliminaries Representation Theorems Discussion

Representing the Local Category Model, cont’d.

Theorem A (strict) preference ∈ P(X) satisfies A1 if and only if it admits a representation by the local category model.

Preliminaries Representation Theorems Discussion

Connections with GP (2005)

Proposition Assume that ∈ P(X) admits a UST(·) representation, then there is a local category {CA} such that UC∗(·) ≡ UST(·). Note: The content of the Proposition is that given a pair (u, v) we can explicitly construct a local category {CA} such that the equality of the Proposition holds.

Preliminaries Representation Theorems Discussion

Axioms for the Partition Model

Say that x →t y if x ≻ y and {x, y} ∼ y. Call a menu A temptation-free if x →t y, ∀x, y ∈ A.

Preliminaries Representation Theorems Discussion

Axioms for the Partition Model

Say that x →t y if x ≻ y and {x, y} ∼ y. Call a menu A temptation-free if x →t y, ∀x, y ∈ A. A1∗: (No Aggregation - NAG) If A ∪ B is temptation free, then either A ∪ B ∼ A or A ∪ B ∼ B.

Preliminaries Representation Theorems Discussion

Axioms for the Partition Model

Say that x →t y if x ≻ y and {x, y} ∼ y. Call a menu A temptation-free if x →t y, ∀x, y ∈ A. A1∗: (No Aggregation - NAG) If A ∪ B is temptation free, then either A ∪ B ∼ A or A ∪ B ∼ B. A2∗:(Reduction) If x, y ∈ A and x →t y, then A ∼ A\x.

Preliminaries Representation Theorems Discussion

Axioms for the Partition Model

Say that x →t y if x ≻ y and {x, y} ∼ y. Call a menu A temptation-free if x →t y, ∀x, y ∈ A. A1∗: (No Aggregation - NAG) If A ∪ B is temptation free, then either A ∪ B ∼ A or A ∪ B ∼ B. A2∗:(Reduction) If x, y ∈ A and x →t y, then A ∼ A\x. A3∗: If x ≻ y ≻ z and x →t z, y →t z, then x →t y. A4∗: If x ≻ y ≻ z and x →t y, x →t z, then y →t z.

Preliminaries Representation Theorems Discussion

Representing the Partition Model

Theorem A (strict) preference ∈ P(X) satisfies A1∗ − A4∗ if and only if it admits a representation by the Partition Model. Corollary Let (B1, u1(·)), (B2, u2(·)) be two partition representations of a given ∈ P(X). Then, B1 ≡ B2 and u1(·) and u2(·) are ordinally equiv- alent.

Preliminaries Representation Theorems Discussion

Axioms for the Rigid Category Model

Definition of Max-Min Relation Fix any cardinal U(·) that represents and put umin(x, A) := min{U(B) : B ⊆ A, x ∈ B} This yields a well-defined relation A on elements of A.

Preliminaries Representation Theorems Discussion

Axioms for the Rigid Category Model

Definition of Max-Min Relation Fix any cardinal U(·) that represents and put umin(x, A) := min{U(B) : B ⊆ A, x ∈ B} This yields a well-defined relation A on elements of A. Let A∗ := {inf(I 1

A), . . ., inf(I k A)} denote the -minimal

elements in the A indifference classes. Call a menu A simple if A = A∗.

Preliminaries Representation Theorems Discussion

Axioms for the Rigid Category Model, cont’d.

A2: (Strong Reduction) If A∗ ⊆ B ⊆ A, then B ∼ A.

Preliminaries Representation Theorems Discussion

Axioms for the Rigid Category Model, cont’d.

A2: (Strong Reduction) If A∗ ⊆ B ⊆ A, then B ∼ A. A3: (Consistency) There is a maximal simple menu A such that

1 For any simple menu B, if x ∈ B ∩ A and x ∼B∪z z, then

x ∼B∪z y, ∀y ∈ B.

Preliminaries Representation Theorems Discussion

Axioms for the Rigid Category Model, cont’d.

A2: (Strong Reduction) If A∗ ⊆ B ⊆ A, then B ∼ A. A3: (Consistency) There is a maximal simple menu A such that

1 For any simple menu B, if x ∈ B ∩ A and x ∼B∪z z, then

x ∼B∪z y, ∀y ∈ B.

2 If x ∼ B, then ∃B′ ⊇ B s.t. x ∼ B′ and IB′(x) ∩ A = ∅.

IA(x) denotes the A indifference class of x.

Preliminaries Representation Theorems Discussion

Axioms, cont’d.

A3′: (Strong Consistency) Let B be any simple set. If x ∼B∪z z and y ∼B∪z z, then x ∼B∪z y.

Preliminaries Representation Theorems Discussion

Representing the Rigid Category Model

Theorem A (strict) preference ∈ P(X) satisfies A1 − A3 if and only if it can be represented by the Rigid Category Model. Corollary Assume that (A1, CA1, u1(·)) and (A2, CA2, u2(·)) are two rigid cat- egory representations of ∈ P(X). Then, A1 = A2, CA1 ≡ CA2, and u1(·) is ordinally equivalent to u2(·).

Preliminaries Representation Theorems Discussion

Connections with GP (2005), cont’d.

Proposition Assume ∈ P(X) admits a UST(·) representation (w/ pair (u, v)). Then, there is a (non-canonical) rigid category Cu,v such that UCu,v (·) ≡ UST(·).

Preliminaries Representation Theorems Discussion

Connections with GP (2005), cont’d.

Proposition Assume ∈ P(X) admits a UST(·) representation (w/ pair (u, v)). Then, there is a (non-canonical) rigid category Cu,v such that UCu,v (·) ≡ UST(·). Put Bt(x) = {z : x →t z} and let Cx := {x} ∪ Bt(x), C ≡ {Cx}x∈X Then, the (possibly redundant) rigid category UC(·) represents and, moreover, UC(·) = UST(·).

Preliminaries Representation Theorems Discussion

Problems with Categories

Category representations satisfy “transitivity of temptation” x →t y →t z ⇒ x →t z

Preliminaries Representation Theorems Discussion

Problems with Categories

Category representations satisfy “transitivity of temptation” x →t y →t z ⇒ x →t z Implies counterfactual reasoning about self-control problems.

Preliminaries Representation Theorems Discussion

Problems with Categories

Category representations satisfy “transitivity of temptation” x →t y →t z ⇒ x →t z Implies counterfactual reasoning about self-control problems. Wanted: A generalization of the category model where the DM is not fully aware of her self-control problem (e.g. cannot reason counterfactually).