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 p s · max x ∈ A u s ( x ) Ingredients : 1 S = state space. 2 p s = signed measure. 3 u s ( · ) = 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 = { x 1 , . . ., x n } = prize space. 2 2 X = space of menus. 3 P ( X ) = space of preference relations on 2 X . Behavioral Primitive : � ∈ P ( X ).

Preliminaries Representation Theorems Discussion GP (2005) The Strotz Model v : X → R (put A v = arg max x ∈ A v ( x )) u : X → R U ST ( A ) := max x ∈ A v 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 { x } ≻ { y } ≻ { z } . 1 { x , y } ∼ { x } , { x , z } ∼ { z } , { y , z } ∼ { y } . 2 { x , y , z } ∼ { y } . 3 Example 2: Temptation Aggregation Put X = { x , y , z } (as above). Assume that { x } ≻ { y } ≻ { z } . 1 { x , y } ∼ { x } , { x , z } ∼ { x } . 2 { x , y , z } ∼ { y } . 3

Preliminaries Representation Theorems Discussion The Category Model Ingredients 1 A collection of sets C i ⊆ X such that: (Completeness) ∪ i C i = X (Non-Redundance) C i �⊆ ∪ j � = i C j 2 A (strict) normative ranking u ( · ) : X → R Using these ingredients we define: U C ( A ) = max C i min x ∈ A ∩C i u ( x )

Preliminaries Representation Theorems Discussion The Category Model, cont’d Example 1 - Multi-Dimensional Temptation 1 Put C 1 = { x , z } , C 2 = { y } , C := {C 1 , C 2 } . 2 u ( x ) > u ( y ) > u ( z ). 3 U C ( { x , z } ) = u ( z ) , U C ( { x , y } ) = u ( x ) , U C ( { y , z } ) = u ( y ). Example 2 - Temptation Aggregation 1 Put C 1 = { x , y } , C 2 = { x , z } 2 u ( x ) > u ( y ) > u ( z ). 3 U C ( { x , y , z } ) = u ( y ) , U C ( { x , y } ) = u ( x ) = U C ( { x , z } ).

Preliminaries Representation Theorems Discussion Variations, part I The Partition Model 1 A collection of sets B ≡ { B i } such that B i ∩ B j = ∅ , ∀ ( i , j ). ∪ i B i = X . 2 U B ( A ) := max B i ∈B min x ∈ A ∩B i u ( x ) The Rigid Category Model 1 An index set A and a collection {C x } x ∈A such that ∪ x ∈A C x = X . x ∈ sup( C x ). If x ∈ A , then x / ∈ C y , ∀ y � = x . 2 U C A ( A ) = max C x : x ∈A min z ∈ A ∩C x u ( z )

Preliminaries Representation Theorems Discussion Variations, part II The Local Partition Model 1 A family of partitions B ∗ ≡ {B A } A ∈ 2 X such that (Completeness) ∪ B ∈B A B = A . (Coherence) A ⊆ A ′ ⇒ ∀ D ∈ B A , ∃ D ′ ∈ B A ′ , D ⊆ D ′ . 2 U B ∗ ( A ) := max B ∈B A min x ∈ B u ( x ) The Local Category Model 1 A menu indexed collection of categories C ∗ ≡ {C A } A ∈ 2 X such that (Completeness) ∪ D ∈C A D = A . (Non-Redundance) inf( D ) �⊆ ∪ D ′ ∈C A : D � = D ′ D ′ (Coherence) A ⊆ A ′ ⇒ ∀ D ∈ C A , ∃ D ′ ∈ C A ′ , D ⊆ D ′ . 2 U C ∗ ( A ) := max D ∈C A min x ∈ 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 Σ s ( A ) � = ∅ . (Σ s ( 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 Σ s ( A ) � = ∅ . (Σ s ( A ) = strong equivalents in A ) Lemma : A 1 ′ is equivalent to DSB.

Preliminaries Representation Theorems Discussion Representing the Strotz Model, cont’d. Theorem (GP (2005)) A preference � ∈ P ( X ) satisfies A 1 ′ if and only if there is a pair ( u , v ) such that U ST ( A ) := max x ∈ A v 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 A 1 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 U ST ( · ) representation, then there is a local category {C A } such that U C ∗ ( · ) ≡ U ST ( · ). Note: The content of the Proposition is that given a pair ( u , v ) we can explicitly construct a local category {C A } 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 A 1 ∗ − A 4 ∗ if and only if it admits a representation by the Partition Model. Corollary Let ( B 1 , u 1 ( · )) , ( B 2 , u 2 ( · )) be two partition representations of a given � ∈ P ( X ). Then, B 1 ≡ B 2 and u 1 ( · ) and u 2 ( · ) 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 u min ( 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 u min ( 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 ∗ .