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

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s

02

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1214

conf

sop

msop

1 2 1 2 1 2 1 2

Dominance

Choice Problem

Mea

n V

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s

02

46

810

1214

conf

sop

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1 2 1 2 1 2 1 2

Dominance

Choice Problem

Mea

n V

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02

46

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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

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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)