1

Regions of rationality: Maps for bounded agents

(Decision Analysis, in press)

Robin M. HogarthICREA & Universitat Pompeu Fabra, Barcelona

&Natalia Karelaia

H.E.C., Université de Lausanne

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“Regions of rationality”The starting point:

– “heuristics and biases” (Kahneman, Slovic, & Tversky, 1982)

– simple decision rules can rival the predictive ability of complex algorithms (e.g., regression) (e.g., TTB: Gigerenzer, Todd, & the ABC Research Group, 1999;

EW: Dawes & Corrigan, 1974).

Idea: – Attention as a scarce resource (Simon, 1978) ->

how much information to seek & how to combine the pieces to make decisions in different “regions”: identify decision rules that are appropriate to each region

• multiple-cue prediction (multi-attribute choice)• cues are probabilistically related to the criterion

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A theoretical approach…

1. Effectiveness of several heuristics: the probability that the best of m alternatives (with k cues) is identified;

the environmental conditions favoring various heuristics, e.g.:

• differential weighting of cues

• inter-correlations of cues

• continuous/binary cues (c/b)

• noise in the environment

• interactions of these factors

2. Illustration: 20 “artificial” and 4 empirical environments

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Models

• Single Variable (SV) models1. Lexicographic – SVc2. Lexicographic – SVb3. DEBA (binary cues)

• Equal weight (EW) models4. EWc5. EWb

• Hybrid models6. EW/DEBA7. EW/SVb

• Domran (DR) models (lower benchmark) 8. DRc9. DRb

• Multiple regression (MR) (upper benchmark) 10. MRc11. MRb

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Method Single Variable, continuous cues - SVc

• Choosing between A & B• Y = criterion and X = cue

• Assume: Y and X are N(0,1), >0

= error, , N(0, ),

• Question:

0),( jicor 0),( Xcor

iiyxi XY 21 yx

yx

? bbaaba xXxXYYP

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Prob {SVc chooses the best b/w A & B}

bbyxb

aayxa

XY

XY

,

ba YY

if 0 babayx XX

or bayxab xx

Note: ab is N 212,0 yx .

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Therefore,

1,0: Nz

Prob {SVc chooses the best b/w A & B}

pdf = probability density function

dzl

pdf

xxzP

xxP

xXxXYYP

ab

z

l

yx

bayx

bayxab

bbaaba

ab

212

1

1

21

21

zV

)0,0(z

- z1 and z2 are bivariate N

Prob {SVc chooses the best from A, B, & C}

,

,

bbyxb

aayxa

XY

XY

and ccyxc XY

ccaacabbaaba xXxXYYxXxXYYP

acab lzlzP 21

ab acl l

z dzpdf

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SVc: generalizing to the case of m alternatives (m>3)

1121

21

121

1...

.........

...1

),0,...,0(

),,...,,(

mm

z

z

m

V

zzzz

where

1,1

,12 2

*

mi

dd

yx

iyxi

(m-1) between-alternative

comparisons

*1

*1

...

... 12121

d d

z

m

dzpdf

YYYYYYP

m

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Overall probability of correct choice by SVc • Random sampling of m=3 from the underlying population of

alternatives. • Either A, B, or C is chosen -> overall probability is:

3 P{((Xa>Xb) & (Xa>Xc))&((Ya>Yb)&(Ya>Yc))}

integrated across : D1 = Xa - Xb > 0, and D2 = Xa - Xc > 0

where , .

dddzpdfpdfd d

zd

*1

*2

00

3

1

1

21

21

zV),,(

),,(

21

21

ddd

zzz

21

12dV

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Overall probability of correct choice by SVc: generalizing to m>3

where

.

2...1

.........

1...2

,

1...2/1

.........

2/1...1

),,...,,(),,...,,(

,........

121121

00

*1

*1

dz

mm

d d

zd

VV

ddddzzzz

dddzpdfpdfmm

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Other models: EWc & MRc

iix

xyi vXY

iii uYY ˆ

)1,0(: 2xyNv )1,0(: 2

adjRNu

Model:

Error:

Vd

xx

xx

2...

.........

...2

2

2

adjadj

adjadj

RR

RR

2...

.........

...2

2

2

di* 212 xyx

ixy d

)1(2 2adj

i

R

d

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Models with binary cues - SVb

where

Therefore,

WaYw

ywSVb

)1,0(:

0

,5.0

1,0

2yw

yw

w

N

W

WaY ywSVb 2

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Models with binary cues - SVb choosing 1 of 2

where

22 12

2

12

2

yw

ywab

yw

baywab

h

z

bbaaba

hww

h

dzpdf

wWwWYYP

ab

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Models with binary cues - DEBA & Hybrids

• Prob {a given alternative is chosen correctly}= the joint probability that the sequence of decisions (or eliminations) made at each stage is correct.

• Three key notions: 1. Appropriate model for each stage

2. Partial correlations:

and partial st. deviations:

3. Probability theory to calculate sequence of correct eliminations

wywwwywwywyw ... 1213121,,,

wwwwwww .1.. ,,,213121

Illustration: 20 “artificial” environments

- Choosing the best from 2, 3, and 4 alternatives - n=40

)min()max(

ii yxyx

1yx

ji xx

xy2R

kn

n

1

k

subcases 1 ... 5 1 ... 5 1 ... 5 1 ... 5

0,3 ... 0,7 0,3 ... 0,7 0,1 ... 0,4 0,0 ... 0,5

0,4 ... 0,8 0,4 ... 0,8 0,3 ... 0,6 0,3 ... 0,8

0,5 ... 0,7 0,3 ... 0,5 0,5 ... 0,8 0,4 ... 0,5

0,4 ... 0,8 0,3 ... 0,8 0,4 ... 0,9 0,3 ... 0,8

Case A Case B Case C Case D

1,3

0,0 0,1 0,60,5

1,1 1,1 1,3

3 3 5 5

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1 2 3 4 540

45

50

55

60

65

70

75

perc

enta

ge c

orre

ct

Case A

subcases

MRc

SVc

EWc

DEBA

DRb

1 2 3 4 540

45

50

55

60

65

70

75

perc

enta

ge c

orre

ct

Case C

subcases

EWc

MRc

SVc

DEBA

DRb

1 2 3 4 540

45

50

55

60

65

70

75

perc

enta

ge c

orre

ct

Case B

subcases

MRc

SVc EWc

DEBA

DRb

1 2 3 4 540

45

50

55

60

65

70

75

perc

enta

ge c

orre

ct

Case D

subcases

SVc

MRc

DEBA

EWc

DRb

Low inter-cue corr High inter-cue corr

3 cues 3 cues

5 cues5 cues

High inter-cue corr Low inter-cue corr

Choosing the best from 3

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(1) Similarity of models’ performance – agreement between models (average between all

pairs, A-D)=63% (vs. 33.(3)% of random agreement), lower when lower inter-cue corr.

(2) Model with continuous cues outperform their binary counterparts (except DR).

– DRb > DRc. Choosing at random: DRb = in 51%, DRc = in 81%.

(3) Larger inter-cue correlation reduces performance of all models (except SV).

Some results

Regression of model performance

Models: SVc SVb DEBA EWc EWb

Regression coefficients* for:

Constant 42 47 44 42 44

Dummy1 (1 of 3) -12 -14 -13 -13 -14

Dummy2 (1 of 4) -7 -9 -8 -7 -8

1

50 29 11

32 50 37

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Regression statistics:

R2 0,99 0,99 0,99 0,98 0,99

Estimated standard error 1,30 0,72 1,13 1,43 1,04

* All regression coefficients are statistically significant (p < .001).

xy1yx

jixx

k

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Illustration: 4 empirical datasets 1)  Golf all-around ranking, N=60

1. Birdie average (*-1)2. Scoring average3. Putting average

  

 2) Golf earnings, N=60 1. Top 10 finishes

2. All-around ranking (*-1) 3. Consecutive cuts

  

3) PhD economics programs: ratings-1993, http://www.phds.org, N=107

  1. # of PhDs for the academic year 87-88 to 91-922.  Total # of program citations 88-92/ number program faculty3. % Faculty with research support

  

4) Consumer reports:test score for digital cameras, http://sub.which.net,N=49 

1. Image quality 2. Picture download time3. Focusing

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Illustration: empirical datasets

)min()max(ii yxyx

1yx

ji xx

xy

2R

kn

n

1

Golf Golf Economics Consumer reports:rankings earnings PhD programs Digital cameras

0.23 0.29 0.07 0.38

0.78 0.86 0.81 0.79

0.46 0.46 0.60 0.20

0.78 0.84 0.89 0.80

0.68 0.81 0.81 0.73

1.07 1.07 1.04 1.09

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one of two one of three one of four

50

55

60

65

70

75

80

85pe

rcen

tage

cor

rect

Golf rankings

SVc

MRc

EWc

DEBA

DRb

one of two one of three one of four

50

55

60

65

70

75

80

85

perc

enta

ge c

orre

ct

Golf earnings

SVc

MRc

EWc

DEBA

DRb

one of two one of three one of four

50

55

60

65

70

75

80

85

perc

enta

ge c

orre

ct

Economics PhD programs

EWc, MRc

DEBA

DRb

SVc

one of two one of three one of four

50

55

60

65

70

75

80

85

perc

enta

ge c

orre

ct

Consumer reports: Digital cameras

MRc EWc

DRb

SVc

DEBA

Golf ranking Golf earnings

Economics PhD programs Consumer reports

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(1) Our contributions

– Analytical analysis

– Regions of rationality: a multidimensional terrain

(2) Further research & implications

– Non-random sampling of alternatives

– Hybrids with categorical & continuous variables

– Different loss functions

– Predicting consumer preferences

– Bounded rationality and expertise: how do people build maps of their decision making terrain?

Discussion

? cYYP ba