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University of Warwick, April 2012. Sequential Expected Utility Theory: Sequential Sampling in Economic Decision Making under Risk. Andrea Isoni (Warwick) Graham Loomes ( Warwick ) Daniel Navarro-Martinez (LSE). Introduction. - PowerPoint PPT Presentation
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Sequential Expected Utility Theory: Sequential Sampling in Economic
Decision Making under Risk
Andrea Isoni (Warwick)
Graham Loomes (Warwick)
Daniel Navarro-Martinez (LSE)
University of Warwick, April 2012
Introduction• Modern economics is largely silent about decision making
processes (e.g., EUT, PT)• Psychologists have dedicated substantial efforts to study decision
processes• Psychological process models: Decision Field Theory, Decision by
Sampling, Query Theory, Elimination by Aspects, Priority Heuristic• Some of the models/evidence suggest the idea of a deliberation
process• Sequential sampling models (e.g., Decision Field Theory, Decision
by Sampling)• Explain decision times (e.g., decision time decreases significantly
as choice probability approaches certainty)• Virtually all economic decision models are silent about
deliberation processes and decision times
Introduction• In this paper: We take EUT and introduce in it the idea of
sequential sampling (deliberation). We show what such a model can do. We investigate experimentally some aspects of it.
• Similarity to Decision Field Theory
• Presentation: Explain the Sequential EUT model Illustrate its implications (simulation) Show some experimental evidence
-0.2 0.0 0.2 0.4 0.6 0.8
0.0
0.5
1.0
1.5
r
Den
sity
= 0.3 = 1.0
The Model: Sequential EUT• Binary choice
• Based on a random preference EUT specification:
,)()(1
n
iii xupLEU r
ii xxu 1)(
),(aSpecialBet~ r
2)3,3(Beta~ r
)021Pr()21Pr( EUEULL
• People sample repeatedly from the choice options to accumulate evidence, until it is judged to be enough to make a choice
• Use certainty equivalent (CE) differences:
• After each sample, individuals conduct a sort of internal test of the null hypothesis that D(L1, L2) is zero
• If the hypothesis is not rejected, sampling goes on; if it’s rejected, sampling stops and the individual chooses the favoured option
Introducing sequential sampling
21)2,1(),(1 CECELLDEUUCE
)21Pr()021Pr()021Pr( LLEUEUCECE
• After each sample k, an evidence statistic Ek is computed:
• The null hypothesis of zero difference is rejected if:
• Essentially a sequential two-tailed t-test of the null hypothesis that the difference in value between the options is zero
• Sampling is psychologically costly, so CONF decreases with sampling:
• We assume C = 1
• Only one additional free parameter (d)
Introducing sequential sampling
kLLDE
LLDk /
)2,1(
)2,1(
2/)1()]([ CONFEabsF kk
)1()( kdCkfCCONF
• The model can address 4 main behavioural constructs:
1) Choice probabilities: , probability that the null hypothesis is rejected with Ek > 0 instead of Ek < 0
2) Response times (RTs): Increasing function of the samples taken to reach the threshold (n) and the number of outcomes:
3) Confidence (CONF): The desired level of confidence in the last test
4) Strength of preference (SoP): Absolute value of the average CE difference sampled:
The model constructs
)21Pr( LL
outcomesofnumbernRT
nLLD
SoP)2,1(
abs
• Simulation (50,000 runs per choice)
• Three main aspects:
Comparing a risky lottery to an increasing sequence of sure amounts
Effects of changes in the three free parameters (α, β, d)Behaviour in specific lottery structures (dominance,
deviations from EUT)
Illustration of the model’s implications
• Choosing between a fixed lottery and a series of monotonically increasing amounts of money
Increasing sure amount
Lot. 1 Lot. 2 α β d Pr(1, 2) CONF SoP RT Pr Core
(20, 1) (40, 0.8; 0, 0.2) 0.35 1.00 0.10 0.00 0.85 7.91 7.60 0.04
(22, 1) (40, 0.8; 0, 0.2) 0.35 1.00 0.10 0.00 0.82 6.00 8.31 0.08
(24, 1) (40, 0.8; 0, 0.2) 0.35 1.00 0.10 0.02 0.78 4.11 9.59 0.15
(26, 1) (40, 0.8; 0, 0.2) 0.35 1.00 0.10 0.12 0.72 2.43 11.50 0.26
(28, 1) (40, 0.8; 0, 0.2) 0.35 1.00 0.10 0.52 0.66 1.69 13.14 0.45
(30, 1) (40, 0.8; 0, 0.2) 0.35 1.00 0.10 0.94 0.72 2.72 11.50 0.73
(32, 1) (40, 0.8; 0, 0.2) 0.35 1.00 0.10 1.00 0.81 4.46 8.76 0.97
(34, 1) (40, 0.8; 0, 0.2) 0.35 1.00 0.10 1.00 0.84 6.37 7.67 1.00
Lot. 1 Lot. 1 α β d Pr(1, 2) CONF SoP RT Pr Core
(30, 1) (40, 0.8; 0, 0.2) 0.05 1.00 0.10 0.04 0.75 1.62 10.48 0.20
(30, 1) (40, 0.8; 0, 0.2) 0.10 1.00 0.10 0.12 0.71 1.31 11.67 0.28
(30, 1) (40, 0.8; 0, 0.2) 0.15 1.00 0.10 0.28 0.68 1.11 12.70 0.36
(30, 1) (40, 0.8; 0, 0.2) 0.20 1.00 0.10 0.51 0.66 1.13 13.13 0.45
(30, 1) (40, 0.8; 0, 0.2) 0.25 1.00 0.10 0.71 0.67 1.43 12.99 0.55
(30, 1) (40, 0.8; 0, 0.2) 0.30 1.00 0.10 0.86 0.69 1.97 12.39 0.64
(30, 1) (40, 0.8; 0, 0.2) 0.35 1.00 0.10 0.94 0.72 2.73 11.53 0.73
(30, 1) (40, 0.8; 0, 0.2) 0.40 1.00 0.10 0.98 0.74 3.66 10.70 0.80
• Changing the location of the distribution of risk aversion coefficients (α)
Changing the free parameters (1)
Lot. 1 Lot. 2 α β d Pr(1, 2) CONF SoP RT Pr Core
(30, 1) (40, 0.8; 0, 0.2) 0.30 0.25 0.10 1.00 0.81 0.99 8.63 0.94
(30, 1) (40, 0.8; 0, 0.2) 0.30 0.40 0.10 0.98 0.76 1.11 10.24 0.82
(30, 1) (40, 0.8; 0, 0.2) 0.30 0.55 0.10 0.95 0.72 1.27 11.27 0.75
(30, 1) (40, 0.8; 0, 0.2) 0.30 0.70 0.10 0.91 0.71 1.46 11.84 0.70
(30, 1) (40, 0.8; 0, 0.2) 0.30 0.85 0.10 0.88 0.69 1.70 12.17 0.66
(30, 1) (40, 0.8; 0, 0.2) 0.30 1.00 0.10 0.86 0.69 1.97 12.38 0.64
(30, 1) (40, 0.8; 0, 0.2) 0.30 1.15 0.10 0.84 0.68 2.32 12.51 0.62
(30, 1) (40, 0.8; 0, 0.2) 0.30 1.30 0.10 0.83 0.68 2.69 12.54 0.61
• Changing the range of the distribution of risk aversion coefficients (β)
Changing the free parameters (2)
Lot. 1 Lot. 2 α β d Pr(1, 2) CONF SoP RT Pr Core
(30, 1) (40, 0.8; 0, 0.2) 0.30 1.00 0.05 0.93 0.71 1.91 20.41 0.64
(30, 1) (40, 0.8; 0, 0.2) 0.30 1.00 0.10 0.86 0.69 1.98 12.33 0.64
(30, 1) (40, 0.8; 0, 0.2) 0.30 1.00 0.15 0.82 0.66 2.01 9.83 0.64
(30, 1) (40, 0.8; 0, 0.2) 0.30 1.00 0.20 0.80 0.63 2.01 8.57 0.64
(30, 1) (40, 0.8; 0, 0.2) 0.30 1.00 0.25 0.79 0.60 2.00 7.75 0.64
(30, 1) (40, 0.8; 0, 0.2) 0.30 1.00 0.30 0.78 0.56 2.04 7.41 0.64
(30, 1) (40, 0.8; 0, 0.2) 0.30 1.00 0.35 0.77 0.53 2.08 7.01 0.64
(30, 1) (40, 0.8; 0, 0.2) 0.30 1.00 0.40 0.77 0.52 2.07 6.61 0.64
• Changing the confidence level decrease rate (d)
Changing the free parameters (3)
• Dominance (α = 0.24, β = 1.00, d = 0.05)
Specific lottery structures
Lot. 1 Lot. 2 Pr(1, 2) CONF SoP RT Pr Core
(50, 0.5; 0, 0.5) (60, 0.5; 0, 0.5) 0.00 0.92 3.94 10.12 0.00
(50, 0.5; 0, 0.5) (51, 0.5; 0, 0.5) 0.00 0.92 0.39 10.14 0.00
• Common ratio (α = 0.24, β = 1.00, d = 0.05)
Specific lottery structures
Lot. 1 Lot. 2 Pr(1, 2) CONF SoP RT Pr Core
(30, 1) (40, 0.8; 0, 0.2) 0.75 0.67 1.20 23.02 0.53
(30, 0.25; 0, 0.75) (40, 0.2; 0, 0.8) 0.37 0.68 0.15 29.95 0.53
• Deviations from EUT (common ratio and common consequence effects)
• Kahneman and Tversky (1979)
• Common consequence (α = 0.26, β = 1.00, d = 0.05)
Lot. 1 Lot. 2 Pr(1, 2) CONF SoP RT Pr Core
(24, 1) (25, 0.33; 24, 0.66; 0, 0.01) 0.66 0.66 0.05 31.10 0.48
(24, 0.34; 0, 0.66) (25, 0.33; 0, 0.67) 0.31 0.67 0.03 30.40 0.48
CR choice 1
CE difference
Freq
uenc
y
-10 -5 0 5 10
040
0080
00
CR choice 2
CE difference
Freq
uenc
y
-1.0 -0.5 0.0 0.5 1.0
040
0010
000
CC choice 1
CE difference
Freq
uenc
y
-0.6 -0.2 0.2 0.6
040
0010
000
CC choice 2
CE difference
Freq
uenc
y
-0.15 -0.05 0.05 0.15
020
0050
00
Distribution of CE differences
Experimental evidence
• Focus on one experiment: 44 students, University of Warwick
• Focus on subset of choice structures: Common ratio Dominance
• 4 different tasks: Binary choice (with response times) Confidence Strength of preference Monetary strength of preference
The choices
• Common ratio:
• Dominance:
Choices Lottery A Lottery B
1 (30, 1) (40, .80; 0, .20)
2 (30, .95; 0, .05) (40, .76; 0, .24)
3 (30, .25; 0, .75) (40, .20; 0, .80)
4 (30, .05; 0, .95) (40, .04; 0, .96)
Choices Lottery A Lottery B
1 (35, .35; 0, .65) (36, .35; 0, .65)
2 (35, .35; 0, .65) (45, .35; 0, .65)
3 (35, .35; 0, .65) (35, .36; 0, .64)
4 (35, .35; 0, .65) (35, .45; 0, .55)
The tasks (1)
The tasks (2)
The tasks (3)
The tasks (4)
Results
• Common ratio:
• Dominance:
Choices Lottery A Lottery B Prop. A.
1 (30, 1) (40, .80; 0, .20) 0.842 (30, .95; 0, .05) (40, .76; 0, .24) 0.843 (30, .25; 0, .75) (40, .20; 0, .80) 0.434 (30, .05; 0, .95) (40, .04; 0, .96) 0.16
Choices Lottery A Lottery B Prop. A
1 (35, .35; 0, .65) (36, .35; 0, .65) 0.002 (35, .35; 0, .65) (45, .35; 0, .65) 0.003 (35, .35; 0, .65) (35, .36; 0, .64) 0.004 (35, .35; 0, .65) (35, .45; 0, .55) 0.00
CR Dom CR Dom
CR vs. Dominance
Choice Problems
Mea
n V
alue
s
02
46
810
1214
conf
rtime
1 2 1 2 1 2 1 2
Dominance
Choice Problem
Mea
n V
alue
s
02
46
810
1214
conf
sop
msop
1 2 1 2 1 2 1 2
Dominance
Choice Problem
Mea
n V
alue
s
02
46
810
1214
conf
sop
msop
rtime
1 2 1 2 1 2 1 2
Dominance
Choice Problem
Mea
n V
alue
s
02
46
810
1214
conf
1 2 1 2 1 2 1 2
Dominance
Choice Problem
Mea
n V
alue
s
02
46
810
1214
conf
sop
05
1015
Common Ratio Sequence
Choice Problem
Mea
n V
alue
s
1(scaled up) 2 3 4(scaled down)
confsopmsoprtime
05
1015
Common Ratio Sequence
Choice Problem
Mea
n V
alue
s
1(scaled up) 2 3 4(scaled down)
conf
05
1015
Common Ratio Sequence
Choice Problem
Mea
n V
alue
s
1(scaled up) 2 3 4(scaled down)
confsop
05
1015
Common Ratio Sequence
Choice Problem
Mea
n V
alue
s
1(scaled up) 2 3 4(scaled down)
confsopmsop
CR Dom CR Dom
CR vs. Dominance
Choice Problems
Mea
n V
alue
s
02
46
810
1214
conf
Parameters
α: 0.27β: 1.18d: 0.05
1 2 3 4 5 6 7 8
0.0
0.2
0.4
0.6
0.8
1.0
Choice Proportions
Choice Problem
Pro
porti
on C
hoos
ing
A
DataPredictions
05
1015
Common Ratio Sequence
Choice Problem
Mea
n V
alue
s
1(scaled up) 2 3 4(scaled down)
conf
05
1015
Common Ratio Sequence
Choice Problem
Mea
n V
alue
s
1(scaled up) 2 3 4(scaled down)
confmsop
05
1015
Common Ratio Sequence
Choice Problem
Mea
n V
alue
s
1(scaled up) 2 3 4(scaled down)
confmsoprtime
1 2 1 2 1 2
Dominance
Choice Problem
Mea
n V
alue
s
05
1015
conf
1 2 1 2 1 2
Dominance
Choice Problem
Mea
n V
alue
s
05
1015
conf
msop
1 2 1 2 1 2
Dominance
Choice Problem
Mea
n V
alue
s
05
1015
conf
msop
rtime
CR Dom CR Dom
CR vs. Dominance
Choice Problems
Mea
n V
alue
s
05
1015
conf
CR Dom CR Dom
CR vs. Dominance
Choice Problems
Mea
n V
alue
s
05
1015
conf
rtime
Conclusions• We have introduced sequential sampling (deliberation) in a
standard economic decision model (Sequential EUT)• Just one additional parameter• Can explain important deviations from EUT, by simply assuming
that people sample sequentially from EUT• Makes predictions about additional behavioural measures
related to deliberation (response times, confidence)• Experimental evidence shows that these measures follow quite
systematic patterns• Sequential EUT can explain most of the patterns obtained • Potential to be extended to other economic decision models,
and other types of tasks (e.g., CE valuation, multi-alternative choice)