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Comparing Sequential Sampling Models With Standard Random Utility Models Jörg Rieskamp Center for Economic Psychology University of Basel, Switzerland 4/16/2012 Warwick
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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
Decision Making Under Risk
• 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
-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
Results: Expected values – Choice proportions
-50 -40 -30 -20 -10 0 10 20 30 400
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
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
ee
eBuAupBAAp
)]()([}),{|(
iijMj ji xAu 1)(
ij
M
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
(adapted from Busemeyer & Johnson, 2004)
Dynamic Development of Preference
(adapted from Busemeyer & Johnson, 2004)
Time Limit (externally controlled stopping-rule)
Decision Making Under Risk
- 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
d
Sequential
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
MDF
T v
s.
Log
it
MDF
T v
s.
Pro
bit
Logit
v
s.
Pro
bit
LL Diff ModelsLikelihood Differences
Results: Bayes Factor
MDFT vs. Logit
log(BF) participant
Fre
quen
cy
-10 -5 0 5
02
46
810
MDFT vs. Probit
log(BF) participant
Fre
quen
cy
-10 -5 0 5
02
46
810
Logit vs. Probit
log(BF) participant
Fre
quen
cy
-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
A
B
1500
2500
3500
4500
550046810
Pric
e in
CH
F
Weight in Kg
Target Competitor Decoy
Interrelated Evaluation of Options
A
BS
1500
2500
3500
4500
550046810
Pric
e in
CH
F
Weight in Kg
Similarity EffectsTarget Competitor Decoy
Interrelated Evaluation of Options
A
BS
1500
2500
3500
4500
550046810
Pric
e in
CH
F
Weight in Kg
Similarity EffectsTarget Competitor Decoy
Tversky, 1972
Interrelated Evaluation of Options
A
B
1500
2500
3500
4500
550046810
Pric
e in
CH
F
Weight in Kg
Target Competitor Decoy
Interrelated Evaluation of Options
A
B
1500
2500
3500
4500
550046810
Pric
e in
CH
F
Weight in Kg
Target Competitor Decoy
Interrelated Evaluation of Options
A
B
1500
2500
3500
4500
550046810
Pric
e in
CH
F
Weight in Kg
Target Competitor Decoy
A
B
D
1500
2500
3500
4500
550046810
Pric
e in
CH
F
Weight in Kg
Attraction EffectsTarget Competitor Decoy
Interrelated evaluation of options
A
B
1500
2500
3500
4500
550046810
Pric
e in
CH
F
Weight in Kg
Target Competitor Decoy
A
B
D
1500
2500
3500
4500
550046810
Pric
e in
CH
F
Weight in Kg
Attraction EffectsTarget Competitor Decoy
(Huber, Payne, & Puto, 1982)
Interrelated Evaluation of Options
A
B
1500
2500
3500
4500
550046810
Pric
e in
CH
F
Weight in Kg
Target Competitor Decoy
A
B
C
1500
2500
3500
4500
550046810
Pric
e in
CH
F
Weight in Kg
Compromise EffectsTarget Competitor Decoy
Interrelated Evaluation of Options
A
B
C
1500
2500
3500
4500
550046810
Pric
e in
CH
F
Weight in Kg
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
Interrelated Evaluation of Options
A
B
1500
2500
3500
4500
550046810
Pric
e in
CH
F
Weight in Kg
Target Competitor Decoy
Interrelated Evaluation of Options
A
B
1500
2500
3500
4500
550046810
Pric
e in
CH
F
Weight in Kg
Target Competitor Decoy
Interrelated Evaluation of Options
A
B
1500
2500
3500
4500
550046810
Pric
e in
CH
F
Weight in Kg
Target Competitor Decoy
Interrelated Evaluation of Options
• 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%
A
B
1500
2500
3500
4500
550046810
Pric
e in
CHF
Weight in Kg
Target Competitor Decoy
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
Fre
qu
en
cy
0 5 10 15 20
02
46
810
MDFT vs. Probit
log(BF) participant
Fre
qu
en
cy
0 5 10 15 20
02
46
810
Logit vs. Probit
log(BF) participant
Fre
qu
en
cy
0 5 10 15 20
02
46
810
Results
MDFT vs. Logit
log(BF) participant
Fre
qu
en
cy
0 5 10 15 20
02
46
810
MDFT vs. Probit
log(BF) participant
Fre
qu
en
cy
0 5 10 15 20
02
46
810
Logit vs. Probit
log(BF) participant
Fre
qu
en
cy
0 5 10 15 20
02
46
810
Results
MDFT vs. Logit
log(BF) participant
Fre
qu
en
cy
0 5 10 15 20
02
46
810
MDFT vs. Probit
log(BF) participant
Fre
qu
en
cy
0 5 10 15 20
02
46
810
Logit vs. Probit
log(BF) participant
Fre
qu
en
cy
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
Interrelated Evaluation of Options
A
B
1500
2500
3500
4500
550046810
Pric
e in
CH
F
Weight in Kg
Target Competitor Decoy
A
B
D
1500
2500
3500
4500
550046810
Pric
e in
CH
F
Weight in Kg
Attraction EffectsTarget Competitor Decoy
(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
Results: Expected values – Choice proportions
-50 -40 -30 -20 -10 0 10 20 30 400
0.1
0.2
0.3
0.4
0.5
0.6
0.7
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
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
DFT predicted Prob Option 1
Ob
se
rve
d c
ho
ice
pro
po
rtio
ns
Op
tio
n 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
Ob
se
rve
d c
ho
ice
pro
po
rtio
ns
Op
tio
n 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
DF
T p
red
icte
d P
rob
ab
ility
Op
tio
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
-100 -80 -60 -40 -20 0 20 40 60 80 100-100
-80
-60
-40
-20
0
20
40
60
80
100
EV Option 1
EV
Op
tio
n 2
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
DFT predicted probability Option 1
CP
T p
red
icte
d p
rob
ab
ility
Op
tio
n 1
Predictions: CPT - DFT
r = -.87
-50 -40 -30 -20 -10 0 10 20 30 40 500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
EV Option 2 - EV Option 1
Ch
oic
e p
rop
ort
ion
op
tio
n 2
Results: Expected values – Choice proportions
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
Ch
oic
e p
rop
ort
ion
Op
tio
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
Ch
oic
e p
rop
ort
ion
Op
tio
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
Hierarchical Bayesian Approach
Hierarchical Bayesian Approach
- 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
