Random Utility without Regularity Michel Regenwetter Department of - - PowerPoint PPT Presentation

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Random Utility without Regularity Michel Regenwetter Department of - - PowerPoint PPT Presentation

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Random Utility without Regularity Michel Regenwetter Department of Psychology, University of Illinois at Urbana-Champaign Winer Memorial Lectures 2018 Work w. J. Dana, C. Davis-Stober, J. Mller-Trede, M.Robinson Thanks: NSF-DRMS SES-10-62045

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SLIDE 1

Random Utility without Regularity

Michel Regenwetter

Department of Psychology, University of Illinois at Urbana-Champaign

Winer Memorial Lectures 2018

Work w. J. Dana, C. Davis-Stober, J. Müller-Trede, M.Robinson

Thanks: NSF-DRMS SES-10-62045 & SES-14-59866.

Regenwetter RUM without Regularity Winer Memorial Lectures 2018 1 / 40

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SLIDE 2

Outline

1

Context ()

2

Random Utility & Random Preference

3

Context-Dependent Random Utility & Random Preference

4

Random Utility without Regularity

5

Conclusions

Regenwetter RUM without Regularity Winer Memorial Lectures 2018 2 / 40

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SLIDE 3

Context ()

Outline

1

Context ()

2

Random Utility & Random Preference

3

Context-Dependent Random Utility & Random Preference

4

Random Utility without Regularity

5

Conclusions

Regenwetter RUM without Regularity Winer Memorial Lectures 2018 3 / 40

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SLIDE 4

Context ()

Context ()

Rationality of decision making.

Regenwetter RUM without Regularity Winer Memorial Lectures 2018 4 / 40

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SLIDE 5

Context ()

Context ()

Rationality of decision making. Huge amounts of heterogeneity within and across decision makers.

Regenwetter RUM without Regularity Winer Memorial Lectures 2018 4 / 40

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SLIDE 6

Context ()

Context ()

Rationality of decision making. Huge amounts of heterogeneity within and across decision makers. Psychological constructs (e.g., preferences) are moving targets!

Regenwetter RUM without Regularity Winer Memorial Lectures 2018 4 / 40

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SLIDE 7

Context ()

Context ()

Rationality of decision making. Huge amounts of heterogeneity within and across decision makers. Psychological constructs (e.g., preferences) are moving targets! They are subject to genuine qualitative variation.

Regenwetter RUM without Regularity Winer Memorial Lectures 2018 4 / 40

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SLIDE 8

Context ()

Context ()

Rationality of decision making. Huge amounts of heterogeneity within and across decision makers. Psychological constructs (e.g., preferences) are moving targets! They are subject to genuine qualitative variation. I only use classical probability theory.

Regenwetter RUM without Regularity Winer Memorial Lectures 2018 4 / 40

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SLIDE 9

Random Utility & Random Preference

Outline

1

Context ()

2

Random Utility & Random Preference

3

Context-Dependent Random Utility & Random Preference

4

Random Utility without Regularity

5

Conclusions

Regenwetter RUM without Regularity Winer Memorial Lectures 2018 5 / 40

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SLIDE 10

Random Utility & Random Preference

Random Utility Model

Finite set A Noncoincident RVs: ∀a, b ∈ A, a = b, Pr(Ua = Ub) = 0

Regenwetter RUM without Regularity Winer Memorial Lectures 2018 6 / 40

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SLIDE 11

Random Utility & Random Preference

Random Utility Model

Finite set A Noncoincident RVs: ∀a, b ∈ A, a = b, Pr(Ua = Ub) = 0 Random Utility Model for Best-Choice PX(x) = Pr(Ux = max

y∈X Uy),

(x ∈ X ⊆ A).

Regenwetter RUM without Regularity Winer Memorial Lectures 2018 6 / 40

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SLIDE 12

Random Utility & Random Preference

Random Utility Model

Finite set A Noncoincident RVs: ∀a, b ∈ A, a = b, Pr(Ua = Ub) = 0 Random Utility Model for Best-Choice PX(x) = Pr(Ux = max

y∈X Uy),

(x ∈ X ⊆ A). Random Utility Model for Best-Worst-Choice PX(x, y) = Pr(Ux = max

v∈X Uv, Uy = min w∈X Uw),

(x = y ∈ X ⊆ A),

Regenwetter RUM without Regularity Winer Memorial Lectures 2018 6 / 40

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SLIDE 13

Random Utility & Random Preference

Random Utility ↔ Random Preference

Every joint realization of noncoincident RVs (Ux)x∈A generates a linear order ≻ on A. Linear Order: Transitive, Asymmetric, Complete.

Regenwetter RUM without Regularity Winer Memorial Lectures 2018 7 / 40

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SLIDE 14

Random Utility & Random Preference

Random Utility ↔ Random Preference

Every joint realization of noncoincident RVs (Ux)x∈A generates a linear order ≻ on A. Linear Order: Transitive, Asymmetric, Complete. Every probability distribution on linear orders on A can be represented with noncoincident RVs (Ux)x∈A.

Regenwetter RUM without Regularity Winer Memorial Lectures 2018 7 / 40

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SLIDE 15

Random Utility & Random Preference

Random Preference Model

L: the collection of all linear orders on A

Regenwetter RUM without Regularity Winer Memorial Lectures 2018 8 / 40

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SLIDE 16

Random Utility & Random Preference

Random Preference Model

L: the collection of all linear orders on A Random Preference Model for Best-Choice PX(x) =

  • ≻∈L

BX (≻)=x

P(≻), (∀x ∈ X ⊆ A).

Regenwetter RUM without Regularity Winer Memorial Lectures 2018 8 / 40

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SLIDE 17

Random Utility & Random Preference

Random Preference Model

L: the collection of all linear orders on A Random Preference Model for Best-Choice PX(x) =

  • ≻∈L

BX (≻)=x

P(≻), (∀x ∈ X ⊆ A). Random Preference Model for Best-Worst-Choice PX(x, y) =

  • ≻∈L

BWX (≻)=(x,y)

P(≻), (∀x = y ∈ X ⊆ A).

Regenwetter RUM without Regularity Winer Memorial Lectures 2018 8 / 40

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SLIDE 18

Random Utility & Random Preference

Random Preference Model for Binary Choice

L: the collection of all linear orders on A Random Preference Model for Best-Choice P{x,y}(x) =

  • ≻∈L

x≻y

P(≻), (∀x = y ∈ A). Random Preference Model for Best-Worst-Choice P{x,y}(x, y) =

  • ≻∈L

x≻y

P(≻), (∀x = y ∈ A).

Regenwetter RUM without Regularity Winer Memorial Lectures 2018 9 / 40

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SLIDE 19

Random Utility & Random Preference

Binary Choice & Linear Ordering Polytope

Regenwetter RUM without Regularity Winer Memorial Lectures 2018 10 / 40

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SLIDE 20

Random Utility & Random Preference

Binary Choice & Linear Ordering Polytope

Triangle Inequalities (Block & Marschak, book, 1960) P{x,y}(x) + P{y,z}(y) − P{x,z}(x) ≤ 1 (∀x, y, z)

Regenwetter RUM without Regularity Winer Memorial Lectures 2018 11 / 40

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SLIDE 21

Random Utility & Random Preference

Binary Choice & Linear Ordering Polytope

Triangle Inequalities (Block & Marschak, book, 1960) P{x,y}(x) + P{y,z}(y) − P{x,z}(x) ≤ 1 (∀x, y, z) |A|: 3 4 5 6 7 8 9 # FDI’s: 2 10 20 910 87,472 > 4.8 × 108 unknown

Regenwetter RUM without Regularity Winer Memorial Lectures 2018 11 / 40

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SLIDE 22

Random Utility & Random Preference

Binary Choice & Linear Ordering Polytope

Test of Rationality (Transitivity) of Preference: Regenwetter, Dana, Davis-Stober (Psychological Review, 2011).

Regenwetter RUM without Regularity Winer Memorial Lectures 2018 12 / 40

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SLIDE 23

Context-Dependent Random Utility & Random Preference

Outline

1

Context ()

2

Random Utility & Random Preference

3

Context-Dependent Random Utility & Random Preference

4

Random Utility without Regularity

5

Conclusions

Regenwetter RUM without Regularity Winer Memorial Lectures 2018 13 / 40

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SLIDE 24

Context-Dependent Random Utility & Random Preference

Description-Experience Gap

Binary choice among lotteries H: Win $4 with probability .8, otherwise $0. L: Win $3 for sure.

Regenwetter RUM without Regularity Winer Memorial Lectures 2018 14 / 40

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SLIDE 25

Context-Dependent Random Utility & Random Preference

Description-Experience Gap

Binary choice among lotteries H: Win $4 with probability .8, otherwise $0. L: Win $3 for sure. Description-Experience Gap (Hertwig et al., Psych. Science, 2004) Decision makers “overweight” small probabilities in description. Decision makers “underweight” small probabilities in experience.

Regenwetter RUM without Regularity Winer Memorial Lectures 2018 14 / 40

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SLIDE 26

Context-Dependent Random Utility & Random Preference

Description-Experience Gap

Binary choice among lotteries H: Win $4 with probability .8, otherwise $0. L: Win $3 for sure. Description-Experience Gap (Hertwig et al., Psych. Science, 2004) Decision makers “overweight” small probabilities in description. Decision makers “underweight” small probabilities in experience. How about “context:” Description vs. Experience

Regenwetter RUM without Regularity Winer Memorial Lectures 2018 14 / 40

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SLIDE 27

Context-Dependent Random Utility & Random Preference

Context-Dependent Random Preference for DE

Let R{D,E} denote a finite collection of pairs of binary preference relations of the form (≻D, ≻E), where x ≻D y denotes that x is preferred to y in description x ≻E y denotes that x is preferred to y in experience according to context-dependent preference pattern (≻D, ≻E) ∈ R{D,E}.

Regenwetter RUM without Regularity Winer Memorial Lectures 2018 15 / 40

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SLIDE 28

Context-Dependent Random Utility & Random Preference

Context-Dependent Random Preference for DE

CONTEXT-DEPENDENT RANDOM-PREFERENCE MODEL There is a probability distribution over R{D,E} such that PD

xy

=

  • (≻D,≻E )∈R{D,E}s.t.

x≻Dy

P(≻D,≻E), and PE

xy

=

  • (≻D,≻E )∈R{D,E}s.t.

x≻E y

P(≻D,≻E).

Regenwetter RUM without Regularity Winer Memorial Lectures 2018 16 / 40

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SLIDE 29

Context-Dependent Random Utility & Random Preference

Random Preference Model

Regenwetter RUM without Regularity Winer Memorial Lectures 2018 17 / 40

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SLIDE 30

Context-Dependent Random Utility & Random Preference

Random Preference Model

Derived possible preferences from Cumulative Prospect Theory (Tversky & Kahneman, J. of Risk & Uncertainty, 1992)

Regenwetter RUM without Regularity Winer Memorial Lectures 2018 17 / 40

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SLIDE 31

Context-Dependent Random Utility & Random Preference

Context-Independent RP of CPT with γ, δ < 1

CPT with overweighting and 0 ≤ γD = γE, δD = δE < 1.

Regenwetter RUM without Regularity Winer Memorial Lectures 2018 18 / 40

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SLIDE 32

Context-Dependent Random Utility & Random Preference

Context-Independent RP of CPT with γ, δ < 1

CPT with overweighting and 0 ≤ γD = γE, δD = δE < 1.

Description Experience 1 2 3 4 5 6 1 2 3 4 5 6 OV1 OV2 1 1 OV3 1 1 OV4 1 1 1 1 OV5 1 1 OV6 1 1 1 1 OV7 1 1 1 1 OV8 1 1 1 1 1 1 OV9 1 1 . . . OV32 1 1 1 1 1 1 1 1 1 1 1 1

Regenwetter RUM without Regularity Winer Memorial Lectures 2018 18 / 40

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SLIDE 33

Context-Dependent Random Utility & Random Preference

Context-Independent RP of CPT with γ, δ < 1

CPT with overweighting and 0 ≤ γD = γE, δD = δE < 1.

Description Experience 1 2 3 4 5 6 1 2 3 4 5 6 OV1 OV2 1 1 OV3 1 1 OV4 1 1 1 1 OV5 1 1 OV6 1 1 1 1 OV7 1 1 1 1 OV8 1 1 1 1 1 1 OV9 1 1 . . . OV32 1 1 1 1 1 1 1 1 1 1 1 1

PHL(D6) = PHL(E6) ≥ PHL(D5) = PHL(E5)

Regenwetter RUM without Regularity Winer Memorial Lectures 2018 18 / 40

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SLIDE 34

Context-Dependent Random Utility & Random Preference

FDIs Context-Independent RP of CPT with γ, δ < 1

Necessary and sufficient conditions for context-independent random preference model of CPT with overweighting. PHL(D6) = PHL(E6) ≥ PHL(D5) = PHL(E5), PHL(D2) = PHL(E2) ≥ PHL(D1) = PHL(E1), PHL(D2) ≥ PHL(D5), PHL(D3) = PHL(E3), PHL(D4) = PHL(E4).

Regenwetter RUM without Regularity Winer Memorial Lectures 2018 19 / 40

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SLIDE 35

Context-Dependent Random Utility & Random Preference

Context-Dependent RP of CPT with γD, γE, δD, δE < 1

Description Experience 1 2 3 4 5 6 1 2 3 4 5 6 OO1 1 1 1 1 OO2 1 1 1 1 OO3 1 1 1 OO4 1 1 1 OO5 1 1 1 OO6 1 1 OO7 1 1 OO8 1 1 OO9 1 1 OO10 1 1 1 . . . OO659 1 1 1 1 1 1 1 1 1 1 OO660 1 1 1 1 1 1 1 1 1 1 1 1

Regenwetter RUM without Regularity Winer Memorial Lectures 2018 20 / 40

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SLIDE 36

Context-Dependent Random Utility & Random Preference

Context-Dependent RP of CPT with γD, γE, δD, δE < 1

Description Experience 1 2 3 4 5 6 1 2 3 4 5 6 OO1 1 1 1 1 OO2 1 1 1 1 OO3 1 1 1 OO4 1 1 1 OO5 1 1 1 OO6 1 1 OO7 1 1 OO8 1 1 OO9 1 1 OO10 1 1 1 . . . OO659 1 1 1 1 1 1 1 1 1 1 OO660 1 1 1 1 1 1 1 1 1 1 1 1

max

  • PHL(E1), PHL(D1), PHL(D5)
  • ≤ PHL(D2)

Regenwetter RUM without Regularity Winer Memorial Lectures 2018 20 / 40

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SLIDE 37

Context-Dependent Random Utility & Random Preference

FDIs Context-Dep. RP of CPT with γD, γE, δD, δE < 1

Necessary and sufficient conditions for context-dependent random preference model of CPT with overweighting.

max

  • PHL(E1), PHL(D1), PHL(D5)
  • ≤

PHL(D2), PHL(E1) + PHL(E5) ≤ PHL(E2) + PHL(D1) + PHL(D6), PHL(D6) + PHL(E5) ≤ 1 + PHL(D2), PHL(E1) + PHL(E6) ≤ 1 + PHL(D1), +PHL(D6), PHL(D6) + PHL(E2) + PHL(E5) ≤ 1 + PHL(D2) + PHL(E6), PHL(E1) + PHL(E6) + PHL(D2) ≤ 1 + PHL(E2) + PHL(D1) + PHL(D6), PHL(D3) + PHL(D4) ≤ 1 + PHL(E3) + PHL(E4), and first 7 Conditions holdwiththe labels E and D swapped.

Regenwetter RUM without Regularity Winer Memorial Lectures 2018 21 / 40

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SLIDE 38

Context-Dependent Random Utility & Random Preference

Context-Dependent RP γD, δD < 1 < γE, δE

Description Experience 1 2 3 4 5 6 1 2 3 4 5 6 OU1 1 1 1 OU2 1 1 1 1 OU3 1 1 1 1 OU4 1 1 1 1 1 OU5 1 1 1 1 1 1 OU6 1 1 1 1 OU7 1 1 1 1 1 OU8 1 1 1 1 1 OU9 1 1 1 1 1 1 OU10 1 1 1 1 1 1 . . . OU249 1 1 1 1 1 1 1 OU250 1 1 1 1 1 1 1 1

Regenwetter RUM without Regularity Winer Memorial Lectures 2018 22 / 40

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SLIDE 39

Context-Dependent Random Utility & Random Preference

Context-Dependent RP γD, δD < 1 < γE, δE

Description Experience 1 2 3 4 5 6 1 2 3 4 5 6 OU1 1 1 1 OU2 1 1 1 1 OU3 1 1 1 1 OU4 1 1 1 1 1 OU5 1 1 1 1 1 1 OU6 1 1 1 1 OU7 1 1 1 1 1 OU8 1 1 1 1 1 OU9 1 1 1 1 1 1 OU10 1 1 1 1 1 1 . . . OU249 1 1 1 1 1 1 1 OU250 1 1 1 1 1 1 1 1

max

  • PHL(E1), PHL(D1), PHL(D5)
  • ≤ PHL(D2)

Regenwetter RUM without Regularity Winer Memorial Lectures 2018 22 / 40

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SLIDE 40

Context-Dependent Random Utility & Random Preference

FDIs Context-Dependent RP γD, δD < 1 < γE, δE

Necessary and sufficient conditions for context-dependent random preference model of CPT with overweighting.

PHL(E6) ≤ PHL(E5) ≤ PHL(E2) ≤ PHL(D2) max

  • PHL(D1), PHL(D5)
  • ≤

PHL(D2) max

  • PHL(D5), PHL(E5)
  • ≤

PHL(D6) PHL(E2) ≤ PHL(E1) PHL(E3) ≤ PHL(E4) PHL(D1) + PHL(E5) ≤ PHL(D2) + PHL(D5) PHL(D1) + PHL(D5) ≤ PHL(D2) + PHL(E1) PHL(D6) + PHL(E1) ≤ 1 + PHL(D2) PHL(D4) + PHL(E3) ≤ 1 + PHL(D3) PHL(D3) + PHL(D4) ≤ 1 + PHL(E4) PHL(D1) + PHL(D6) ≤ 1 + PHL(E1) PHL(D1) + PHL(D6) + PHL(E5) ≤ 1 + PHL(D5) + PHL(E1)

Regenwetter RUM without Regularity Winer Memorial Lectures 2018 23 / 40

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SLIDE 41

Context-Dependent Random Utility & Random Preference

Statistical Analysis

Bayes Factors on Hertwig et al. (2004) data. Context-independent overweighting: ∼ 10−8

Regenwetter RUM without Regularity Winer Memorial Lectures 2018 24 / 40

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SLIDE 42

Context-Dependent Random Utility & Random Preference

Statistical Analysis

Bayes Factors on Hertwig et al. (2004) data. Context-independent overweighting: ∼ 10−8 Context-dependent overweighting: 0.002

Regenwetter RUM without Regularity Winer Memorial Lectures 2018 24 / 40

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SLIDE 43

Context-Dependent Random Utility & Random Preference

Statistical Analysis

Bayes Factors on Hertwig et al. (2004) data. Context-independent overweighting: ∼ 10−8 Context-dependent overweighting: 0.002 Overweighting in description, underweighting in experience: 300 Regenwetter & Robinson (Psychological Review, 2017).

Regenwetter RUM without Regularity Winer Memorial Lectures 2018 24 / 40

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SLIDE 44

Random Utility without Regularity

Outline

1

Context ()

2

Random Utility & Random Preference

3

Context-Dependent Random Utility & Random Preference

4

Random Utility without Regularity

5

Conclusions

Regenwetter RUM without Regularity Winer Memorial Lectures 2018 25 / 40

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SLIDE 45

Random Utility without Regularity

Asymmetric Dominance & Regularity

Asymmetric Dominance Jim shops for a new TV.

Regenwetter RUM without Regularity Winer Memorial Lectures 2018 26 / 40

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SLIDE 46

Random Utility without Regularity

Asymmetric Dominance & Regularity

Asymmetric Dominance Jim shops for a new TV. Faced with two options, he is unsure which one to buy.

Regenwetter RUM without Regularity Winer Memorial Lectures 2018 26 / 40

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SLIDE 47

Random Utility without Regularity

Asymmetric Dominance & Regularity

Asymmetric Dominance Jim shops for a new TV. Faced with two options, he is unsure which one to buy. Option t (the “target”) has better picture quality.

Regenwetter RUM without Regularity Winer Memorial Lectures 2018 26 / 40

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SLIDE 48

Random Utility without Regularity

Asymmetric Dominance & Regularity

Asymmetric Dominance Jim shops for a new TV. Faced with two options, he is unsure which one to buy. Option t (the “target”) has better picture quality. Option c (the “competitor”) has better reliability.

Regenwetter RUM without Regularity Winer Memorial Lectures 2018 26 / 40

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SLIDE 49

Random Utility without Regularity

Asymmetric Dominance & Regularity

Asymmetric Dominance Jim shops for a new TV. Faced with two options, he is unsure which one to buy. Option t (the “target”) has better picture quality. Option c (the “competitor”) has better reliability. Only when Jim is shown a “decoy” option d that resembles t but is slightly worse, he feels inclined to choose t over both c and d.

Regenwetter RUM without Regularity Winer Memorial Lectures 2018 26 / 40

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SLIDE 50

Random Utility without Regularity

Asymmetric Dominance & Regularity

Asymmetric Dominance Jim shops for a new TV. Faced with two options, he is unsure which one to buy. Option t (the “target”) has better picture quality. Option c (the “competitor”) has better reliability. Only when Jim is shown a “decoy” option d that resembles t but is slightly worse, he feels inclined to choose t over both c and d. Regularity: X ⊆ Y ⇒ PX(x) ≥ PY(x) (∀x ∈ X).

Regenwetter RUM without Regularity Winer Memorial Lectures 2018 26 / 40

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SLIDE 51

Random Utility without Regularity

Asymmetric Dominance & Regularity

Asymmetric Dominance Jim shops for a new TV. Faced with two options, he is unsure which one to buy. Option t (the “target”) has better picture quality. Option c (the “competitor”) has better reliability. Only when Jim is shown a “decoy” option d that resembles t but is slightly worse, he feels inclined to choose t over both c and d. Regularity: X ⊆ Y ⇒ PX(x) ≥ PY(x) (∀x ∈ X). Violations of regularity are broadly viewed as violations of random utility models and random preference models in general.

Regenwetter RUM without Regularity Winer Memorial Lectures 2018 26 / 40

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SLIDE 52

Random Utility without Regularity

Best-Choice

Random Utility Model for Best-Choice PX(x) = Pr(Ux = max

y∈X Uy),

(x ∈ X ⊆ A). Random Preference Model for Best-Choice PX(x) =

  • ≻∈L

BX (≻)=x

P(≻), (∀x ∈ X ⊆ A).

Regenwetter RUM without Regularity Winer Memorial Lectures 2018 27 / 40

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SLIDE 53

Random Utility without Regularity

Best-Choice

Falmagne (JMP,1978); Barberá & Pattanaik (Econometrica, 1986): Necessary and sufficient conditions regardless of |A|.

Regenwetter RUM without Regularity Winer Memorial Lectures 2018 28 / 40

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SLIDE 54

Random Utility without Regularity

Best-Choice

Falmagne (JMP,1978); Barberá & Pattanaik (Econometrica, 1986): Necessary and sufficient conditions regardless of |A|. Equality constraints such as

x∈X PX(x) = 1.

Inequality constraints

  • Y : X⊆Y⊆A

(−1)|Y\X| PY(x) ≥ 0, (for all possible x ∈ X ⊆ A). Block-Marschak Polynomials

Regenwetter RUM without Regularity Winer Memorial Lectures 2018 28 / 40

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SLIDE 55

Random Utility without Regularity

Best-Choice

Block-Marschak Polynomials for A = {a, b, c} PA(x) ≥ 0, (∀x ∈ A), PX(x) ≥ PA(x), (∀X ⊂ A, |X| = 2), 1 − P{x,y}(x) − P{x,z}(x) + PA(x) ≥ 0, (∀{x, y, z} = {a, b, c}), (using P{x}(x) = 1)

Regenwetter RUM without Regularity Winer Memorial Lectures 2018 29 / 40

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SLIDE 56

Random Utility without Regularity

Best-Choice

Block-Marschak Polynomials for A = {a, b, c} PX(x) ≥ PA(x), (∀X ⊂ A, |X| = 2),

Regenwetter RUM without Regularity Winer Memorial Lectures 2018 30 / 40

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SLIDE 57

Random Utility without Regularity

Linear Ordering Polytope & Regularity

Fiorini (JMP, 2004) gave FDI’s of Linear Ordering Polytope. V1 : dct; V3 : cdt, ctd; V4 : dtc; V7 : tdc; V8 : tcd

Regenwetter RUM without Regularity Winer Memorial Lectures 2018 31 / 40

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SLIDE 58

Random Utility without Regularity

Context-Dependence with Dominance

t ≻ d, t ⊲ d

Regenwetter RUM without Regularity Winer Memorial Lectures 2018 32 / 40

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SLIDE 59

Random Utility without Regularity

Context-Dependence with Dominance

t ≻ d, t ⊲ d Binary Best ch. joint random utility i choice from A in binary and best choice from A 1 t ≻ d ≻ c t ⊲ d ⊲ c p1 [Ut > Ud > Uc] ∩ [Vt > Vd > Vc] 2 t ≻ d ≻ c t ⊲ c ⊲ d p2 [Ut > Ud > Uc] ∩ [Vt > Vc > Vd] 3 t ≻ d ≻ c c ⊲ t ⊲ d p3 [Ut > Ud > Uc] ∩ [Vc > Vt > Vd] 4 t ≻ c ≻ d t ⊲ d ⊲ c p4 [Ut > Uc > Ud] ∩ [Vt > Vd > Vc] 5 t ≻ c ≻ d t ⊲ c ⊲ d p5 [Ut > Uc > Ud] ∩ [Vt > Vc > Vd] 6 t ≻ c ≻ d c ⊲ t ⊲ d p6 [Ut > Uc > Ud] ∩ [Vc > Vt > Vd] 7 c ≻ t ≻ d t ⊲ d ⊲ c p7 [Uc > Ut > Ud] ∩ [Vt > Vd > Vc] 8 c ≻ t ≻ d t ⊲ c ⊲ d p8 [Uc > Ut > Ud] ∩ [Vt > Vc > Vd] 9 c ≻ t ≻ d c ⊲ t ⊲ d p9 [Uc > Ut > Ud] ∩ [Vc > Vt > Vd]

Regenwetter RUM without Regularity Winer Memorial Lectures 2018 32 / 40

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SLIDE 60

Random Utility without Regularity

Context-Dependence with Dominance

t ≻ d, t ⊲ d P{d,t} = PA(d) = 0; P{c,t}(t) ≥ P{c,d}(d). V4 : t ≻ d ≻ c, c ⊲ t ⊲ d; V5 : c ≻ t ≻ d, t ⊲ d ∧ t ⊲ c

Regenwetter RUM without Regularity Winer Memorial Lectures 2018 33 / 40

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SLIDE 61

Random Utility without Regularity

Context-Dependence with Asymmetric Dominance

t ≻ d, t ⊲ d.

Regenwetter RUM without Regularity Winer Memorial Lectures 2018 34 / 40

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SLIDE 62

Random Utility without Regularity

Context-Dependence with Asymmetric Dominance

t ≻ d, t ⊲ d. Require that t ≻ c ⇒ t ⊲ c.

Regenwetter RUM without Regularity Winer Memorial Lectures 2018 34 / 40

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SLIDE 63

Random Utility without Regularity

Context-Dependence with Asymmetric Dominance

t ≻ d, t ⊲ d. Require that t ≻ c ⇒ t ⊲ c. Binary Best ch. joint random utility i choice from A in binary and best choice from A 1 t ≻ d ≻ c t ⊲ d ⊲ c p1 [Ut > Ud > Uc] ∩ [Vt > Vd > Vc] 2 t ≻ d ≻ c t ⊲ c ⊲ d p2 [Ut > Ud > Uc] ∩ [Vt > Vc > Vd] 4 t ≻ c ≻ d t ⊲ d ⊲ c p4 [Ut > Uc > Ud] ∩ [Vt > Vd > Vc] 5 t ≻ c ≻ d t ⊲ c ⊲ d p5 [Ut > Uc > Ud] ∩ [Vt > Vc > Vd] 7 c ≻ t ≻ d t ⊲ d ⊲ c p7 [Uc > Ut > Ud] ∩ [Vt > Vd > Vc] 8 c ≻ t ≻ d t ⊲ c ⊲ d p8 [Uc > Ut > Ud] ∩ [Vt > Vc > Vd] 9 c ≻ t ≻ d c ⊲ t ⊲ d p9 [Uc > Ut > Ud] ∩ [Vc > Vt > Vd]

Regenwetter RUM without Regularity Winer Memorial Lectures 2018 34 / 40

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SLIDE 64

Random Utility without Regularity

Context-Dependence with Asymmetric Dominance

t ≻ d, t ⊲ d. Require that t ≻ c ⇒ t ⊲ c.

Regenwetter RUM without Regularity Winer Memorial Lectures 2018 35 / 40

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SLIDE 65

Random Utility without Regularity

Context-Dependence with Asymmetric Dominance

t ≻ d, t ⊲ d. Require that t ≻ c ⇒ t ⊲ c. P{t,c}(t) ≥ P{d,c}(d); P{t,c}(t) ≤ PA(t).

Regenwetter RUM without Regularity Winer Memorial Lectures 2018 35 / 40

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SLIDE 66

Random Utility without Regularity

Context defined by absence or presence of d.

Context defined by absence or presence of d. P{c,d}(d) ≥ PA(t)

Regenwetter RUM without Regularity Winer Memorial Lectures 2018 36 / 40

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SLIDE 67

Random Utility without Regularity

Absence or presence of d; Asymm. dom.

Context defined by absence or presence of d. Require t ≻ c ⇒ t ⊲ c. P{c,d}(d) ≥ PA(t) P{t,c}(t) ≤ PA(t).

Regenwetter RUM without Regularity Winer Memorial Lectures 2018 37 / 40

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SLIDE 68

Random Utility without Regularity

Statistical Analysis

Bayes Factors Context-independent RUM (regularity): .004 Model 1A: 2.00 Model 1B (reverse regularity): 6.01 Model 2A: 1.99 Model 2B (reverse regularity): 3.00 Work with Johannes Müller-Trede.

Regenwetter RUM without Regularity Winer Memorial Lectures 2018 38 / 40

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SLIDE 69

Conclusions

Outline

1

Context ()

2

Random Utility & Random Preference

3

Context-Dependent Random Utility & Random Preference

4

Random Utility without Regularity

5

Conclusions

Regenwetter RUM without Regularity Winer Memorial Lectures 2018 39 / 40

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SLIDE 70

Conclusions

Context-Dependent Random Utility Model

Context-dependent Random Utility Model for Best-Choice PΓ

X(x) = Pr(UΓ x = max y∈X UΓ y),

(for all possible x ∈ X ⊆ A),

Regenwetter RUM without Regularity Winer Memorial Lectures 2018 40 / 40

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SLIDE 71

Conclusions

Context-Dependent Random Utility Model

Context-dependent Random Utility Model for Best-Choice PΓ

X(x) = Pr(UΓ x = max y∈X UΓ y),

(for all possible x ∈ X ⊆ A), Context-dependent Random Utility Model for Best-Worst-Choice PΓ

X(x, y) = Pr(UΓ x = max v∈X UΓ v, UΓ y = min w∈X UΓ w),

(x = y ∈ X ⊆ A),

Regenwetter RUM without Regularity Winer Memorial Lectures 2018 40 / 40

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SLIDE 72

Conclusions

Conclusions

Building a context-dependent RUM or RP model www.regenwetterlab.org regenwet@illinois.edu

Regenwetter RUM without Regularity Winer Memorial Lectures 2018 41 / 40

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SLIDE 73

Conclusions

Conclusions

Building a context-dependent RUM or RP model List every permissible best (best-worst) choice for every context. www.regenwetterlab.org regenwet@illinois.edu

Regenwetter RUM without Regularity Winer Memorial Lectures 2018 41 / 40

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SLIDE 74

Conclusions

Conclusions

Building a context-dependent RUM or RP model List every permissible best (best-worst) choice for every context. These patterns define the vertices of a convex polytope. www.regenwetterlab.org regenwet@illinois.edu

Regenwetter RUM without Regularity Winer Memorial Lectures 2018 41 / 40

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SLIDE 75

Conclusions

Conclusions

Building a context-dependent RUM or RP model List every permissible best (best-worst) choice for every context. These patterns define the vertices of a convex polytope. Use math or software to characterize facet-structure. www.regenwetterlab.org regenwet@illinois.edu

Regenwetter RUM without Regularity Winer Memorial Lectures 2018 41 / 40

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SLIDE 76

Conclusions

Conclusions

Building a context-dependent RUM or RP model List every permissible best (best-worst) choice for every context. These patterns define the vertices of a convex polytope. Use math or software to characterize facet-structure. Use order-constrained freq. or Bayesian inference (e.g., QTEST) www.regenwetterlab.org regenwet@illinois.edu

Regenwetter RUM without Regularity Winer Memorial Lectures 2018 41 / 40