Choosing with the Worst in Mind: A Reference-Dependent Model Gerelt - PowerPoint PPT Presentation

Choosing with the Worst in Mind: A Reference-Dependent Model Gerelt Tserenjigmid Caltech Bounded Rationality in Choice Conference 2015 Gerelt Tserenjigmid A Reference-Dependent Model

Motivation A new axiomatic model of reference-dependent preferences in which reference points are menu-dependent. Preference Reversal: a over b in E1, but b over a in E2. Purpose: Explicitly model reference points, As restrictive as possible, but consistent with two well-known preference reversals (compromise and attraction effects). Gerelt Tserenjigmid A Reference-Dependent Model

Motivation Two Preference Reversals ( x , p ) is chosen over ( y , q ) from binary menu { ( x , p ) , ( y , q ) } , but ( y , q ) is chosen over ( x , p ) when ( z , · ) is added. A2 A2 ( y , q ) ( y , q ) q q ( x , p ) ( x , p ) p p z y x z y x A1 A1 Compromise Effect Attraction Effect well documented in experimental and marketing studies: consumer choice, risky choice, choice over policy issues

Motivation Two Preference Reversals ( x , p ) is chosen over ( y , q ) from binary menu { ( x , p ) , ( y , q ) } , but ( y , q ) is chosen over ( x , p ) when ( z , · ) is added. A2 A2 ( z , r ) r ( y , q ) ( y , q ) q q ( z , t ) t ( x , p ) ( x , p ) p p z y x z y x A1 A1 Compromise Effect Attraction Effect well documented in experimental and marketing studies: consumer choice, risky choice, choice over policy issues Gerelt Tserenjigmid A Reference-Dependent Model

Today ... Basic Setup and Model. Diminishing Sensitivity. Predictions. Application. Representation Theorem. Related Literature. Gerelt Tserenjigmid A Reference-Dependent Model

Basic Setup and Model Gerelt Tserenjigmid A Reference-Dependent Model

Basic Setup X = R 2 + the set of all alternatives with two attributes. α ⊆ 2 X \ { (0 , 0) } a collection of compact subsets (menus) of X . The primitive: A choice correspondence C : α ⇒ X where C ( A ) ⊆ A and C ( A ) � = ∅ for each A ∈ α . Gerelt Tserenjigmid A Reference-Dependent Model

Models of Reference-Dependent Preferences Exogenous Reference Points: Tversky and Kahneman (1991), the utility of ( x , p ) for given reference point r =( r 1 , r 2 ) is � � � � � � V r x , p = f u ( x ) − u ( r 1 ) + g w ( p ) − w ( r 2 ) . (1) Gerelt Tserenjigmid A Reference-Dependent Model

Models of Reference-Dependent Preferences Exogenous Reference Points: Tversky and Kahneman (1991), the utility of ( x , p ) for given reference point r =( r 1 , r 2 ) is � � � � � � V r x , p = f u ( x ) − u ( r 1 ) + g w ( p ) − w ( r 2 ) . (1) Menu-Dependent Reference Point: for each A ∈ α , ( x A , p A ) ≡ � � ( x , p ) ∈ A x , min ( x , p ) ∈ A p min . Our Model: C is a MinMin reference dependent choice if ∃ strictly increasing functions u , w , and f such that for any A ∈ α , � �� u ( x ) − u ( x A ) w ( p ) − w ( p A ) � � � C ( A ) = arg max f + f . (2) ( x , p ) ∈ A transitivity Gerelt Tserenjigmid A Reference-Dependent Model

Why Minimums ? Our Model: For any A ∈ α , � �� u ( x ) − u ( x A ) w ( p ) − w ( p A ) � � � C ( A ) = arg max f + f . ( x , p ) ∈ A In both the Compromise and Attraction Effects, ( z , · ) is added to { ( x , p ) , ( y , q ) } . As restrictive as possible: f and ( x A , p A ), compared to standard Additive Utility Model, u ( x ) + w ( p ). Gerelt Tserenjigmid A Reference-Dependent Model

Why Minimums ? Our Model: For any A ∈ α , � �� u ( x ) − u ( x A ) w ( p ) − w ( p A ) � � � C ( A ) = arg max f + f . ( x , p ) ∈ A In both the Compromise and Attraction Effects, ( z , · ) is added to { ( x , p ) , ( y , q ) } . As restrictive as possible: f and ( x A , p A ), compared to standard Additive Utility Model, u ( x ) + w ( p ). General Menu-Dependence ? no predictive power. Gerelt Tserenjigmid A Reference-Dependent Model

Why Minimums ? Our Model: For any A ∈ α , � �� u ( x ) − u ( x A ) w ( p ) − w ( p A ) � � � C ( A ) = arg max f + f . ( x , p ) ∈ A In both the Compromise and Attraction Effects, ( z , · ) is added to { ( x , p ) , ( y , q ) } . As restrictive as possible: f and ( x A , p A ), compared to standard Additive Utility Model, u ( x ) + w ( p ). General Menu-Dependence ? no predictive power. Maximums ? cannot have the Attraction Effect. Average ? i) increasing in minimum; ii) more parameters. Gerelt Tserenjigmid A Reference-Dependent Model

Diminishing Sensitivity – Tversky and Kahneman (1991) MRS x , p is increasing in the reference for the first dimension – strict concavity of f . I (1 , 1) A2 √ x − 1 + √ p − 1 = 2 . 7 7 5 3 1 1 3 5 7 A1

Diminishing Sensitivity – Tversky and Kahneman (1991) MRS x , p is increasing in the reference for the first dimension – strict concavity of f . I (1 , 1) A2 √ x − 1 + √ p − 1 = 2 . 7 √ x − 3 + √ p − 1 = 2 7 I (3 , 1) 5 3 1 1 3 5 7 A1 Gerelt Tserenjigmid A Reference-Dependent Model

Diminishing Sensitivity ⇒ Compromise and Attraction Effects Gerelt Tserenjigmid A Reference-Dependent Model

A2 A2 r ( y , q ) ( y , q ) q q t ( x , p ) ( x , p ) p p ( y , p ) ( y , p ) I ( y , p ) I ( y , p ) z y x z y x A1 A1 Compromise Effect Attraction Effect detail sufficiency and necessity of diminishing sensitivity

A2 A2 ( z , r ) r ( y , q ) ( y , q ) q q ( z , t ) t ( x , p ) ( x , p ) p p ( z , p ) ( y , p ) ( z , p ) ( y , p ) I ( y , p ) I ( y , p ) z y x z y x A1 A1 Compromise Effect Attraction Effect detail sufficiency and necessity of diminishing sensitivity

I ( z , p ) I ( z , p ) A2 A2 ( z , r ) r ( y , q ) ( y , q ) q q ( z , t ) t ( x , p ) ( x , p ) p p ( z , p ) ( y , p ) ( z , p ) ( y , p ) I ( y , p ) I ( y , p ) z y x z y x A1 A1 Compromise Effect Attraction Effect detail sufficiency and necessity of diminishing sensitivity Gerelt Tserenjigmid A Reference-Dependent Model

Bounds on Preference Reversals Gerelt Tserenjigmid A Reference-Dependent Model

Two Decoy Effect and Symmetric Dominance I ( z , p ) A2 ( y , q ) q ( z , t ) t ( x , p ) p s ( k , s ) I ( y , p ) z y x k A1 Two Decoy Effect Teppan and Felfernig (2009)

Two Decoy Effect and Symmetric Dominance I ( z , p ) A2 ( y , q ) q ( z , t ) t ( x , p ) p ( z , p ) s ( z , s ) ( k , s ) I ( y , p ) z y x k A1 Two Decoy Effect Teppan and Felfernig (2009)

Two Decoy Effect and Symmetric Dominance I ( z , p ) A2 I ( z , s ) ( y , q ) q ( z , t ) t ( x , p ) p ( z , p ) s ( z , s ) ( k , s ) I ( y , p ) z y x k A1 Two Decoy Effect Teppan and Felfernig (2009)

Two Decoy Effect and Symmetric Dominance I ( z , p ) I ( z , p ) A2 A2 I ( z , s ) I ( z , s ) ( y , q ) ( y , q ) q q ( z , t ) t ( x , p ) ( x , p ) p p ( z , p ) ( y , p ) s ( z , s ) ( k , s ) s ( z , s ) I ( y , p ) I ( y , p ) z y x z y x k A1 A1 Two Decoy Effect Symmetric Dominance Teppan and Felfernig (2009) Masatlioglu and Uler (2013) Gerelt Tserenjigmid A Reference-Dependent Model

A numerical example A2 √ r ∗ = 47 x − x A + � p − p A V A ( x , p ) = r ∗ = 22 . 1 CE ( y , q ) q =20 AE ( x , p ) p =11 SD s ∗ = 5 . 61 y =9 x =20 z ∗ = 80 A1 9 Gerelt Tserenjigmid A Reference-Dependent Model

Application to Intertemporal Consumption Gerelt Tserenjigmid A Reference-Dependent Model

The Effect of Non Binding Borrowing Constraint Today y 2 y 1 + 1+ r = M y 1 y 2 Tomorrow graph

The Effect of Non Binding Borrowing Constraint Today y 1 + ¯ b y 2 y 1 + 1+ r = M y 1 y 2 Tomorrow graph

The Effect of Non Binding Borrowing Constraint Today y 1 + ¯ b y 2 y 1 + 1+ r = M y 1 y 2 Tomorrow graph

The Effect of Non Binding Borrowing Constraint 1 = α ( y 1 + ¯ ⋆ c ∗ b ) < α M Today ⋆ Excess Sensitivity of Consumption to Current Income y 1 + ¯ b y 2 y 1 + 1+ r = M y 1 y 2 Tomorrow graph Gerelt Tserenjigmid A Reference-Dependent Model

Representation Theorem Gerelt Tserenjigmid A Reference-Dependent Model

Representation Theorem Notation Notation: For any a , b , c ∈ X , � � � � i) a � b if a ∈ C { a , b } and a ≻ b if a = C { a , b } , � � � � ii) a � c b if a ∈ C { a , b , c } and a ≻ c b if a ∈ C { a , b , c } and � � ∈ C { a , b , c } b / . Gerelt Tserenjigmid A Reference-Dependent Model

Representation Theorem 3 Standard Axioms Axiom (Regularity) i) Monotonicity, ii) Continuity, and iii) Solvability. Solvability: For any ( y , q ) and p , ∃ x ∈ R + s.t ( x , p ) ≻ ( y , q ). Axiom (Transitivity) � and � c are transitive for any c . Gerelt Tserenjigmid A Reference-Dependent Model

Representation Theorem 3 Standard Axioms R satisfies Cancellation if for any ( x 1 , p 1 ) , ( x 2 , p 2 ) , ( x 3 , p 3 ), if ( x 1 , p 1 ) R ( x 2 , p 3 ) and ( x 2 , p 2 ) R ( x 3 , p 1 ) , then ( x 1 , p 2 ) R ( x 3 , p 3 ) . Axiom (Cancellation) � and � c satisfy Cancellation for any c . Gerelt Tserenjigmid A Reference-Dependent Model

Representation Theorem 2 New Axioms WARP: ∀ A ∈ α , if a � = C ( A ), then C ( A \{ a } )= C ( A ) \{ a } . Axiom (Independence of Non Extreme Alternatives) > ( x A , p A ), then C ( A \{ a } )= C ( A ) \{ a } . ∀ A ∈ α , if a � = C ( A ) and a > Gerelt Tserenjigmid A Reference-Dependent Model