Comparing Sequential Sampling Models With Standard Random Utility Models

Comparing Sequential Sampling Models With Standard Random Utility Models

Jörg RieskampCenter for Economic PsychologyUniversity of Basel, Switzerland

4/16/2012 Warwick

Decision Making Under Risk

French mathematicians (1654)

• Rational Decision Making: Principles of Expected Value

Blaise Pascal Pierre Fermat


• St. Petersburg Paradox• Expected utility theory (1738):

Replacing the value of money by its subjective value

Nicholas Bernoulli Daniel Bernoulli

Expected Utility Theory

• Axiomatic expected utility theory von Neumann & Morgenstern, 1947

Frederick Mosteller 1916 - 2006

the authors argued that when first offering a bet with a certain probability of winning, and then increasing that probability

"there is not a sudden jump from no acceptances to all acceptances at a particular offer, just as in a hearing experiment there is not a critical loudness below which nothing is heard and above which all loudnesses are heard”

instead“the bet is taken occasionally, then more and more often, until, finally, the bet is taken nearly all the time”

Mosteller & Nogee, 1951, Journal of Political Economy, p. 374

Probabilistic Nature of Preferential Choice

– experiment conducted over 10 weeks with 3 sessions each weak

– participants repeatedly accepted or rejected gambles (N=30)

Example- the participants had to accept or reject a simple

binary gamble with a probability of 2/3 to loose 5 cents and a probability of 1/3 to win a particular amount

- the winning amount varied between 5 and 16 cents

Mosteller’s & Nogee’s Study

Results: „Subject B-I"

– Participants decided between 180 pairs of gambles– Receiving 15 Euros as a show-up fee– One gamble was selected and played at the end of the

experiment and the winning amounts were paid to the subjects

Rieskamp (2008). JEP: LMC

Experimental Study

Task

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-60

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0

20

40

60

80

100

EV(Option1)

EV

(Opt

ion2

)

Expected values of the selected gambles

Results: Expected values – Choice proportions

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0.1

0.2

0.3

0.4

0.5

0.6

0.7

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0.9

1

Expected value option 2 - Expected value option 1

Cho

ice

prop

ortio

n op

tion

2

• Consumer products

How Can We Explain the Probabilistic Character of Choice?

• Random utility theories:

identically and independently extreme value distributedi

Explaining Probabilistic Character of Choice

Logit model

BA

A

VV

V

eeeBuAupBAAp

)]()([}),{|(

iijMj ji xAu 1)(

ijM

j jA xV 1

• Random utility theories:

identically and independently normal distributed i

Probit Model

)]()([}),{|( BuAupBAAp

iijMj ji xAu 1)(

Cognitive Approach to Decision Making

• Considering the information processing steps leading to a decision

• Sequential sampling models

- Vickers, 1970; Ratcliff, 1978- Busemeyer & Townsend, 1993

- Usher & McClelland, 2004

Sequential Sampling Models

• People evaluate options by continuously sampling information about the options’ attributes

• Which attribute receives the attention of the decision maker fluctuates

• The probability that an attribute receives the attention of the decision maker is a function of the attribute‘s importance

• When the overall evaluation crosses a decision threshold a decision is made

Rieskamp, Busemeyer, & Mellers (2006) Journal of Economic Literature

(adapted from Busemeyer & Johnson, 2004)

Threshold Bound (internally controlled stopping-rule)

Dynamic Development of Preference


Dynamic Development of Preference


Time Limit (externally controlled stopping-rule)


- DFT vs. Cumulative Prospect TheoryRieskamp (2008),

JEP:LMC

- DFT vs. Proportional Difference ModelScheibehenne, Rieskamp, & Gonzalez-Vallejo, 2009,

Cognitive Science

- Hierarchical Bayesian approach examining the limitations of cumulative prospect theory

Nilsson, Rieskamp, & Wagenmakers (2011), JMP

Consumer Behavior

How good are sequential sampling models to predict consumer behavior?

- Multi-attribute decision field theory Roe, Busemeyer, & Townsend, 2001

versus

- Logit and Probit ModelStandard random utility models

Multi-attribute Decision Field Theory

Decay• The preference state decays over time

Interrelated evaluations of options• Options are compared with each other• Similar alternatives compete against each other and have

a negative influence on each other

1. Calibration Experiment – Participants (N=30) repeatedly decided between three

digital cameras (72 choices)– Each camera was described by five attributes with two

or three attribute values (e.g. mega pixel, monitor size)– Models` parameters were estimated following a

maximum likelihood approach

2. Generalization Test Experiment

Study 1

24

Task

Models’ parameters

Models Parameters

Standard Random Utilit

y

Logit

Weights given to the attributes

extreme value

distributed

Probit

Weights given to the attributes

normal distribute

dSequential

Sampling

MDFT

Attention weights

allocated to the

attributes

normal distribute

d

Determines the rate at

which similarity

declines with distance

Determines the memory

of the previous

preference state

Logit – Probit: r = .99

MDFT - Logit : r = .94

MDFT - Probit: r = .94

Attribute Weigths

Model Comparison Results: Likelihood

-5 0 5 10

MDFT

vs.

L

ogit

MDFT

vs.

P

robit

Logit

vs.

P

robit

LL Diff ModelsLikelihood Differences

Results: Bayes Factor

MDFT vs. Logit

log(BF) participant

Freq

uenc

y

-10 -5 0 5

02

46

810

MDFT vs. Probit

log(BF) participant

Freq

uenc

y

-10 -5 0 5

02

46

810

Logit vs. Probit

log(BF) participant

Freq

uenc

y

-10 -5 0 5

02

46

810

Generalization Test Experiment 2– Generating a new set of options on the basis

of the estimated parameter values of experiment 1

– Comparing models‘ predictions without fitting

Study 1 – Generalization

– Comparing the observed choice proportions with the predicted choice proportions

Distance

Results

Model DistanceLog

LikelihoodBaseline 0.19 -702Logit 0.18 -863Probit 0.11 -468MDFT 0.12 -490

Conclusion

• Calibration Design:– LL: MDFT >Logit >Probit– Bayes factor: Logit > Probit >

MDFT

• Generalization Design:– Probit ≈ MDFT > Logit

Decision Field Theory - Interrelated evaluations of options1. attention specific evaluations2. competition between similar

options

Logit / Probit - Evaluation of options are independent of each other

Study 2: Qualitative PredictionsInterrelated Evaluations of Options

Interrelated Evaluation of Options

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B

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Target Competitor Decoy


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Similarity EffectsTarget Competitor Decoy


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Similarity EffectsTarget Competitor Decoy

Tversky, 1972


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Attraction EffectsTarget Competitor Decoy

Interrelated evaluation of options

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(Huber, Payne, & Puto, 1982)


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Compromise EffectsTarget Competitor Decoy


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Compromise EffectsTarget Competitor Decoy

• Is it possible to show the interrelated evaluations of options for all three situations in a within-subject design?

• Does MDFT has a substantial advantage compared to the logit and probit model in predicting people’s decisions?

Do the choice effects really matter?

Research Question

Method: Matching Task Matching Task TARGET COMPETITOR

A B

Weight

6.5 Kg 8.0 Kg

Price

??? 3'000 CHF Break

Before the main study participants had to choose one attribute value to make both options equally attractive

Method: Matching Task Matching Task TARGET COMPETITOR

A B

Weight

6.5 Kg 8.0 Kg

Price

4'000 CHF (matched)

3'000 CHF Break

Main Study Choice Task Matching Task TARGET COMPETITOR DECOY

A B C

Weight

6.5 Kg 8.0 Kg

6.6 Kg

Price

4'000 CHF (matched)

3'000 CHF Break

4'100 CHF Break

Choice Task: To the former 2 options (target + competitor) individual specified decoys were added.

Always choices between three options.

• The decoy was added either in relationship to option A or in relationship to option B

Pecularity: Decoy position


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B

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• If the third option had no effect on the preferences for A and B the average choice proportion for the target option should be 50%

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Consumer Products: - bicycles- washing machines

- notebooks- vacuum cleaners- color printers- digital cameras

Choices: 6 products, 3 effects, 3 situations, 2 decoy positions

6 × 3 × 3 × 2 = 108 choice situations (triples)

Main Study

Results

Attraction Compromise Similiarity0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

59%

48%

38%37%33% 33%

4%

19%

28%

Overall Subjects (N = 48)

TARGETCOMPETITORDECOY

Results

A

B

Decoy

Target

Attraction Effect

Results

A

B

Decoy

A

B

Decoy

TargetTarget

Attraction Effect Compromise Effect

Results

A

B

Decoy

A

B

Decoy

A

B

Decoy

TargetTarget

Attraction Effect Compromise Effect

Target

Similarity Effect

Logit – Probit: r = .72

MDFT - Logit: r = .57

MDFT - Probit: r = .61

Attribute Weigths

Results

MDFT vs. Logit

log(BF) participant

Freq

uenc

y

0 5 10 15 20

02

46

810

MDFT vs. Probit

log(BF) participant

Freq

uenc

y

0 5 10 15 20

02

46

810

Logit vs. Probit

log(BF) participant

Freq

uenc

y

0 5 10 15 20

02

46

810

Results

MDFT vs. Logit

log(BF) participant

Freq

uenc

y

0 5 10 15 20

02

46

810

MDFT vs. Probit

log(BF) participant

Freq

uenc

y

0 5 10 15 20

02

46

810

Logit vs. Probit

log(BF) participant

Freq

uenc

y

0 5 10 15 20

02

46

810

Results

MDFT vs. Logit

log(BF) participant

Freq

uenc

y

0 5 10 15 20

02

46

810

MDFT vs. Probit

log(BF) participant

Freq

uenc

y

0 5 10 15 20

02

46

810

Logit vs. Probit

log(BF) participant

Freq

uenc

y

0 5 10 15 20

02

46

810

• Sequential sampling models provide a way to describe the probabilistic character of choices

• For random choices situations Probit and MDFT are doing equally good for predicting people’s preferences

• In situations in which the interrelated evaluations of options play a major role MDFT has a substantial advantage compared to standard random utility models

Conclusions

Thanks !

Nicolas Berkowitsch

MaximilianMatthaeus

Benjamin Scheibehenne


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(Huber, Payne, & Puto, 1982)

-100 -80 -60 -40 -20 0 20 40 60 80 100-100

-80

-60

-40

-20

0

20

40

60

80

100

EV(Option1)

EV

(Opt

ion2

)

Expected values of the selected gambles


-50 -40 -30 -20 -10 0 10 20 30 400

0.1

0.2

0.3

0.4

0.5

0.6

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0.8

0.9

1

Expected value option 2 - Expected value option 1

Cho

ice

prop

ortio

n op

tion

2

- Each models’ parameters were estimated separately for each individual.

- Goodness-of-fit: Maximum likelihood

Estimating the models’ parameter(s)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

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0.8

0.9

1

DFT predicted Prob Option 1

Obs

erve

d ch

oice

pro

porti

ons

Opt

ion

1

Results: Sequential Sampling Model

r = .83

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

CPT predicted Probability Option 1

Obs

erve

d ch

oice

pro

porti

ons

Opt

ion

1

Results: Cumulative Prospect Theory

r = .88

- For 18 participants prospect theory had a better AIC value as compared to 12 participants for whom DFT was the better model (p = .36 sign test)

- When fitting the models to the data there is a slight advantage of prospect theory in describing the data

- No strong evidence in favor of one model

Results

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

CPT predicted Probability Option 1

DFT

pre

dict

ed P

roba

bilit

y O

ptio

n 1

CPT - DFT

r = .88

• A good fit of a model does not tell us very much!

• Both cumulative prospect theory and the sequential sampling model are able to described the observed choices

Conclusions

• Goal: Conducting a study to test the models rigorously against each other

• Generalization Test: Constructing decision problems for which the two models made different predictions

Study 2: Rigorous Model Comparison Test

• Generating 10.000 pairs of gambles• for each pair of gambles an experiment was simulated

with 30 synthetic participants• for each synthetic participant DFT‘s (or CPT‘s)

parameter values were drawn with replacement from the distribution of parameter values of study 1 and the model‘s predictions were determined

• each simulated experiment was repeated 100 times• the average choice probabilities were determined for

DFT and CPT• Selecting 180 gambles with different predictions of the

two models.

• Independent Test of DFT and CPT in Study 2

Bootstrapping Method

– Thirty participants decided between 180 pairs of gambles

– One gamble was selected and played at the end of the experiment and the winning amounts were paid to the participants

Study 2: Experiment

Expected Values of Selected Gambles

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-80

-60

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-20

0

20

40

60

80

100

EV Option 1

EV O

ptio

n 2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

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0.2

0.3

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0.8

0.9

1

DFT predicted probability Option 1

CPT

pre

dict

ed p

roba

bilit

y O

ptio

n 1

Predictions: CPT - DFT

r = -.87

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0.1

0.2

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0.6

0.7

0.8

0.9

1

EV Option 2 - EV Option 1

Cho

ice

prop

ortio

n op

tion

2


r = .71

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

DFT predicted probability Option 1

Cho

ice

prop

ortio

n O

ptio

n 1

Results: Sequential Sampling Model

r = .77

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

CPT predicted probability Option 1

Cho

ice

prop

ortio

n O

ptio

n 1

Results: Cumulative Prospect Theory

r = -.67

Results Study 2

• For all 30 participants DFT reached a better goodness-of-fit than CPT

• The most likely gambles predicted by DFT were chosen in 66% of all cases, whereas the most likely gambles predicted by CPT were chosen in only 34% of all cases

Limitations

- The results depend on the estimation process for CPT‘s parameters in Study 1

- With six free parameters fitting the parameters individually might not lead to reliable estimates

Hierarchical Bayesian Approach

- Estimating the posterior distribution of prospect theories‘ parameter

Hierarchical Bayesian Approach: - The median estimates of the maximum likelihood approach did not differ for most parameters of CPT

Nilsson, Rieskamp, & Wagenmakers (in press). Journal of Mathematical Psychology



- Estimating the posterior distribution of prospect theories‘ parameter

Hierarchical Bayesian Approach: However, it is in general difficult to receive reliable estimates for the loss aversion parameter of CPT

Nilsson, Rieskamp, & Wagenmakers (in press). Journal of Mathematical Psychology

Alternative models

- Heuristic model of decision making - the priority heuristic

Rieskamp (2008), JEP:LMC

- Proportional difference model Scheibehenne, Rieskamp, & Gonzalez-Vallejo,

2009, Cognitive Science

Conclusions

- Sequential sampling models appear as valid alternatives to the the conventional expected utility and nonexpected utility approach such as CPT for explaining decision making under risk

- Sequential sampling models provide a description of the cognitive process underlying decision making

Documents

Comparing Sequential Sampling Models With Standard Random Utility Models