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 A1 x y z p q (x, p) (y, q) Compromise Effect A2 A1 x y z p q (x, p) (y, q) 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 A1 x y z p q (x, p) (y, q) Compromise Effect A2 A1 x y z p q (x, p) (y, q) Attraction Effect (z, r) r (z, t) t

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 = R2

+ the set of all alternatives with two attributes.

α ⊆ 2X \ {(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=(r1, r2) is Vr

• x, p
• = f
• u(x) − u(r1)
• + g
• w(p) − w(r2)
• .

(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=(r1, r2) is Vr

• x, p
• = f
• u(x) − u(r1)
• + g
• w(p) − w(r2)
• .

(1) Menu-Dependent Reference Point: for each A ∈ α, (xA, pA) ≡

• min

(x, p)∈A x,

min

(x, p)∈A p

• .

Our Model: C is a MinMin reference dependent choice if ∃ strictly increasing functions u, w, and f such that for any A ∈ α, C(A) = arg max

(x, p)∈A

• f
• u(x) − u(xA)
• + f
• w(p) − w(pA)
• . (2)

transitivity Gerelt Tserenjigmid A Reference-Dependent Model

Why Minimums ?

Our Model: For any A ∈ α, C(A) = arg max

(x, p)∈A

• f
• u(x) − u(xA)
• + f
• w(p) − w(pA)
• .

In both the Compromise and Attraction Effects, (z, · ) is added to {(x, p), (y, q)}. As restrictive as possible: f and (xA, pA), compared to standard Additive Utility Model, u(x) + w(p).

Gerelt Tserenjigmid A Reference-Dependent Model

Why Minimums ?

Our Model: For any A ∈ α, C(A) = arg max

(x, p)∈A

• f
• u(x) − u(xA)
• + f
• w(p) − w(pA)
• .

In both the Compromise and Attraction Effects, (z, · ) is added to {(x, p), (y, q)}. As restrictive as possible: f and (xA, pA), 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, C(A) = arg max

(x, p)∈A

• f
• u(x) − u(xA)
• + f
• w(p) − w(pA)
• .

In both the Compromise and Attraction Effects, (z, · ) is added to {(x, p), (y, q)}. As restrictive as possible: f and (xA, pA), 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)

MRSx, p is increasing in the reference for the first dimension – strict concavity of f . A2 A1 1 3 5 7 1 3 5 7 I(1, 1) √x − 1 + √p − 1 = 2.7

Diminishing Sensitivity – Tversky and Kahneman (1991)

MRSx, p is increasing in the reference for the first dimension – strict concavity of f . A2 A1 1 3 5 7 1 3 5 7 I(1, 1) √x − 1 + √p − 1 = 2.7 √x − 3 + √p − 1 = 2 I(3, 1)

Gerelt Tserenjigmid A Reference-Dependent Model

Diminishing Sensitivity ⇒ Compromise and Attraction Effects

Gerelt Tserenjigmid A Reference-Dependent Model

A2 A1 I(y, p) x y z p q r (x, p) (y, q) (y, p) Compromise Effect A2 A1 I(y, p) x y z p q t (x, p) (y, q) (y, p) Attraction Effect

sufficiency and necessity of diminishing sensitivity detail

A2 A1 I(y, p) x y z p q r (x, p) (y, q) (y, p) Compromise Effect A2 A1 I(y, p) x y z p q t (x, p) (y, q) (y, p) Attraction Effect (z, r) (z, p) (z, t) (z, p)

sufficiency and necessity of diminishing sensitivity detail

A2 A1 I(y, p) x y z p q r (x, p) (y, q) (y, p) Compromise Effect A2 A1 I(y, p) x y z p q t (x, p) (y, q) (y, p) Attraction Effect (z, r) (z, p) (z, t) (z, p) I(z, p) I(z, p)

sufficiency and necessity of diminishing sensitivity detail Gerelt Tserenjigmid A Reference-Dependent Model

Bounds on Preference Reversals

Gerelt Tserenjigmid A Reference-Dependent Model

Two Decoy Effect and Symmetric Dominance

A2 A1 I(y, p) I(z, p) x y z k p q t s (x, p) (y, q) (z, t) (k, s) Two Decoy Effect

Teppan and Felfernig (2009)

Two Decoy Effect and Symmetric Dominance

A2 A1 I(y, p) I(z, p) x y z k p q t s (x, p) (y, q) (z, t) (k, s) Two Decoy Effect

Teppan and Felfernig (2009)

(z, p) (z, s)

Two Decoy Effect and Symmetric Dominance

A2 A1 I(y, p) I(z, p) x y z k p q t s (x, p) (y, q) (z, t) (k, s) Two Decoy Effect

Teppan and Felfernig (2009)

(z, p) (z, s) I(z, s)

Two Decoy Effect and Symmetric Dominance

A2 A1 I(y, p) I(z, p) x y z k p q t s (x, p) (y, q) (z, t) (k, s) Two Decoy Effect

Teppan and Felfernig (2009)

(z, p) (z, s) I(z, s) A2 A1 I(y, p) I(z, p) x y z p q s (x, p) (y, q) (y, p) (z, s) Symmetric Dominance

Masatlioglu and Uler (2013)

I(z, s)

Gerelt Tserenjigmid A Reference-Dependent Model

A numerical example

A2 A1 (y, q) (x, p) CE AE SD VA(x, p) = √ x − xA +

• p − pA

z∗=80

9

y =9 x =20 s∗ = 5.61 p =11 q =20 r∗ = 22.1 r∗ = 47

Gerelt Tserenjigmid A Reference-Dependent Model

Application to Intertemporal Consumption

Gerelt Tserenjigmid A Reference-Dependent Model

The Effect of Non Binding Borrowing Constraint

Today Tomorrow y1 +

y2 1+r = M

y2 y1

graph

The Effect of Non Binding Borrowing Constraint

Today Tomorrow y1 +

y2 1+r = M

y2 y1 y1 + ¯ b

graph

The Effect of Non Binding Borrowing Constraint

Today Tomorrow y1 +

y2 1+r = M

y2 y1 y1 + ¯ b

graph

The Effect of Non Binding Borrowing Constraint

Today Tomorrow y1 +

y2 1+r = M

y2 y1 y1 + ¯ b ⋆ c∗

1 = α (y1 + ¯

b) < α M ⋆ Excess Sensitivity of Consumption to Current Income

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) ac b if a∈C

• {a, b, c}
• and a≻c b if a ∈ C
• {a, b, c}
• and

b / ∈ C

• {a, b, c}
• .

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 (x1, p1), (x2, p2), (x3, p3), if (x1, p1) R (x2, p3) and (x2, p2) R (x3, p1), then (x1, p2) R (x3, p3). 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) ∀A ∈ α, if a=C(A) and a> >(xA, pA), then C(A\{a})=C(A)\{a}.

Gerelt Tserenjigmid A Reference-Dependent Model

Representation Theorem

2 New Axioms – Translation Invariance

A2 I(z, p) p q x y z (x, p) (y, q) (z, p)

Representation Theorem

2 New Axioms – Translation Invariance

A2 I(z, p) p q x y z (x, p) (y, q) (z, p) A1 x′ y′ z′

Representation Theorem

2 New Axioms – Translation Invariance

A2 I(z, p) p q x y z (x, p) (y, q) (z, p) A1 x′ y′ z′ I(z′, p) (x′, p) (y′, q) (z′, p)

Gerelt Tserenjigmid A Reference-Dependent Model

Representation Theorem

2 New Axioms – Translation Invariance

A2 A1 I(z, p) I(z′, p) p q x y z (x, p) (y, q) (z, p) x′ y′ z′ (x′, p) (y′, q) (z′, p) Axiom (Translation Invariance) For any x, y, z, x′, y′, z′, if [x, y]D[x′, y′] and [y, z]D[y′, z′], then (x, p) ∼(z, p) (y, q) if and only if (x′, p) ∼(z′, p) (y′, q). [x, y]D[x′, y′] if for any p′, q′, (x, p′)∼(y, q′)⇔(x′, p′)∼(y′, q′).

Gerelt Tserenjigmid A Reference-Dependent Model

Representation Theorem

2 New Axioms – Translation Invariance

A2 A1 I(z, p) I(z′, p) p q x y z (x, p) (y, q) (z, p) x′ y′ z′ (x′, p) (y′, q) (z′, p) Axiom (Translation Invariance) For any x, y, z, x′, y′, z′, if [x, y]D[x′, y′] and [y, z]D[y′, z′], then (x, p) ∼✟✟

✟

(z, p) (y, q) if and only if (x′, p) ∼✘✘

✘

(z′, p) (y′, q).

[x, y]D[x′, y′] if for any p′, q′, (x, p′)∼(y, q′)⇔(x′, p′)∼(y′, q′).

Gerelt Tserenjigmid A Reference-Dependent Model

Representation Theorem

Theorem C satisfies Regularity, Transitivity, Cancellation, Independence

• f Non Extreme Alternatives, and Translation Invariance iff

∃ continuous and strictly increasing functions f , u, w s.t f (0) = u(0) = w(0) = 0 and for any A, C(A) = arg max

(x, p)∈A

• f
• u(x) − u(xA)
• + f
• w(p) − w(pA)
• .

(3)

Gerelt Tserenjigmid A Reference-Dependent Model

Representation Theorem

Theorem C satisfies Regularity, Transitivity, Cancellation, Independence

• f Non Extreme Alternatives, and Translation Invariance iff

∃ continuous and strictly increasing functions f , u, w s.t f (0) = u(0) = w(0) = 0 and for any A, C(A) = arg max

(x, p)∈A

• f
• u(x) − u(xA)
• + f
• w(p) − w(pA)
• .

(3) Proposition (Uniqueness) For any two vectors (f , u, w) and (f ′, u′, w′) that satisfy (3), if f (1)=f ′(1) and u(1)=u′(1), then (f , u, w) = (f ′, u′, w′).

Gerelt Tserenjigmid A Reference-Dependent Model

Related Literature

Reference-Dependent Models:

Status Quo, Initial Endowment: Masatlioglu & Ok (2005, 2013), Sagi (2006), and Apesteguia & Ballester (2009). Expectations-based: Koszegi & Rabin (2006, 2007). Revealed (P)Reference: Ok, Ortoleva, & Riella (2015).

Explaining Compromise and Attraction Effects with Menu-Dependent or Context-Dependent Preferences:

Simonson & Tversky (1992), Wernerfelt (1995), Kamenica (2008), and Bordalo, Gennaioli, & Shleifer (2013).

Choice Theory and Weakening WARP:

Manzini & Mariotti (2007, 2012 (lexicographic)), Ehlers & Sprumont (2008), Eliaz, Richter, & Rubinstein (2011), and Apesteguia & Ballester (2013). Manzini & Mariotti (2012 (categorize)) and Cherepanov, Feddersen, & Sandroni (2013). De Clippel & Eliaz (2012), Masatlioglu, Nakajima, & Ozbay (2012), and Manzini, Mariotti, & Tyson (2013).

Gerelt Tserenjigmid A Reference-Dependent Model

Contributions

Explicitly Model Reference Points. Diminishing Sensitivity is necessary and sufficient for the Attraction Effect (Compromise Effect). Bounds on Preference Reversals:

Two Decoy Effect, Symmetric Dominance.

Representation Theorem.

Gerelt Tserenjigmid A Reference-Dependent Model

Thank You !!!

Gerelt Tserenjigmid A Reference-Dependent Model

Representation Theorem – Role of INEA and TI

• 0. Regularity: Exist menu-dependent utility functions VA(x, p).
• 1. Trans & Cancel on : V{(x,p), (y,q)}(x, p) = u(x) + w(p);

(x, p) (y, q) ⇔ u(x) + w(p) ≥ u(y) + w(q).

• 2. –/

/– on (z, s): V{(x, p), (y, q), (z, s)}(x, p)=u(z, s)(x) + w(z, s)(p); (x, p)(z, s) (y, q) ⇔ u(z, s)(x) + w(z, s)(p)≥u(z, s)(y) + w(z, s)(q). INEA and ED Connect utilities for different menus,

• 3. INEA: VA(x, p) = V{(x, p), (xA, p′), (x′, pA)}(x, p).
• 4. TI: u(z, s)(x)=f
• u(x) − u(z)
• and w(z, s)(p)=f
• w(p) − w(s)
• .

Gerelt Tserenjigmid A Reference-Dependent Model

Transitivity

(x, p) is chosen over (y, q) iff f (u(x)−u(y))+✭✭✭✭✭✭✭

✭

f (w(p) − w(p)) > ✭✭✭✭✭✭✭

✭

f (u(y) − u(y))+f (w(q)−w(p)) iff u(x) + w(p) > u(y) + w(q).

back Gerelt Tserenjigmid A Reference-Dependent Model

Diminishing Sensitivity and Attraction Effect

Axiom (Advantage of Extreme Alternatives (AE)) For any x, y, z, l and p, q such that x > y > z > l and p < q, if C({(x, p), (y, q), (z, p)}) = {(x, p), (y, q)}, then C({(x, p), (y, q)})=(x, p) and C({(x, p), (y, q), (l, p)})=(y, q). Proposition (Characterizing Diminishing Sensitivity) Suppose C = C(f , u, w) for strictly increasing, continuous functions f , u, w. Then C satisfies AE if and only if f is strictly concave.

back Gerelt Tserenjigmid A Reference-Dependent Model

Violations of Transitivity and Diminishing Sensitivity

A Generalization: VA(x, p) = f

• u(x) − u(xA)
• + g
• w(p) − w(pA)
• .

Violations of Transitivity and Diminishing Sensitivity: if f is more diminishing sensitive than g, then we observe (x, p) ≻ (y, q) ≻ (z, r) ≻ (x, p), is called Similarity Cycle. if g is more diminishing sensitive than f , then we observe (x, p) ≺ (y, q) ≺ (z, r) ≺ (x, p), is called Regret Cycle.

Gerelt Tserenjigmid A Reference-Dependent Model

Attanasio and Weber (2010)

back Gerelt Tserenjigmid A Reference-Dependent Model

More Applications

Application to Risky Choice – binary lottery (x, p) Observing Different Degrees of Risk-Aversion Estimating Risk Preferences from Binary vs Non Binary Menus

Gerelt Tserenjigmid A Reference-Dependent Model