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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
2
“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
3
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
4
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
5
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
6
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 .
7
Therefore,
1,0: Nz
Prob {SVc chooses the best b/w A & B}
pdf = probability density function
dzl
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
9
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
10
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
11
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
12
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
13
Models with binary cues - SVb
where
Therefore,
WaYw
ywSVb
)1,0(:
0
,5.0
1,0
2yw
yw
w
N
W
WaY ywSVb 2
14
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
15
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
17
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
18
(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
4
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
20
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
21
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
22
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
23
(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