Supplementary Material
푑
휇
푑
푝푟
푑
푝
푝
∈
1
,
...
,
푁
푠푢푏푗푒푐푡푠
푑
휇
~
푈푛푖푓
(
0
,
50
]
푑
푝푟
~
Γ
(
1
,
.
1
)
푑
푝
~
푁
푑
휇
,
푑
푝푟
∈
(
0
,
50
)
휎
휇
휎
푝푟
휎
푝
푝
∈
1
,
...
,
푁
푠푢푏푗푒푐푡푠
휎
휇
~
푈푛푖푓
(
0
,
2
]
휎
푝푟
~
Γ
(
1
,
.
1
)
휎
푝
~
N
휎
휇
,
휎
푝푟
∈
(
0
,
2
)
휔
푖
푖
trials
휔
푖
=
푑
푝
[
푔푎푧푒
퐿
푖
∗
푣
퐿
푖
−
1
−
푔푎푧푒
퐿
푖
∗
푣
푅
푖
+
휃
푝
(
1
−
푔푎푧푒
퐿
푖
∗
푣
퐿
푖
−
푔푎푧푒
퐿
푖
∗
푣
푅
푖
)
]
푦
푖
푖
trials
푦
푖
~
dwieners
(
bounds
,
NDT
i
,
b
p
,
휔
푖
,
휎
푝
)
퐵표푢푛푑푠
=
2
(i.e., at
±
1
)
푁퐷
푇
푖
(
i.e., n
on
-
decision time)
= RT
–
total ROI fixation time in
trial
푖
푏
휇
푏
휅
푏
훼
푏
훽
푏
푝
푝
∈
1
,
...
,
푁
푠푢푏푗푒푐푡푠
푏
휇
~
Beta
2
,
2
∈
(
0
,
1
)
푏
휅
~
Γ
(
1
,
.
5
)
푏
푝
~
퐵푒푡푎
푏
훼
,
푏
훽
∈
(
0
,
1
)
=
푏
휇
푏
휅
=
(
1
−
푏
휇
)
푏
휅
휃
푝
푝
∈
1
,
...
,
푁
푠푢푏푗푒푐푡푠
휃
푝
~
푁
휃
휇
,
휃
푝푟
∈
(
−
.
5
,
1
.
5
)
휃
휇
휃
푝푟
휃
휇
~
푁
.
5
,
.
3
∈
(
−
.
5
,
1
.
5
)
휃
푝푟
~
Γ
(
1
,
.
1
)
Supplementary Fig. 1.
Directed acyclic graph of hierarchical aDDM with uncorrelated priors. The
hierarchical model estimates group and individual parameters for the aDDM with uncorrelated priors. The
10 group parameters are depicted in the top row of yellow circles. The 4 individual parameters estimated
for each subject are depicted in orange in the middle row. The distribution of individual parameters as a
function of the group parameters is specified using a transformation of some of the parameters, denoted by
the dashed lines. The choice and RT outcome
y
i
of trial
i
for a subject
p
is modeled as a Drift-Diffusion-
Model with bounds at
±
1, non-decision time NDT
i
, bias
b
p
, trial specific slope
ω
i
, and noise
σ
p
. The trial
specific slope
ω
i
depends on the subject’s drift rate parameter
d
p
, attentional bias parameter
θ
p
, gaze data
for the trial
gazeL
i
, and item liking ratings for the foods used in the trial (
vL
i
,
vR
i
).
gazeL
i
denotes the
proportion of time spent fixating on the left item during the trial. The hyperpriors for the group parameters
are described at the top of the graph. “
∈
(
X,Y
)” indicates truncation to bounds.
N
subjects
= 25 in the
exploratory and confirmatory datasets, and 50 in the joint dataset.
20
푑
휇
푑
푝푟
푑
푝
푝
∈
1
,
...
,
푁
푠푢푏푗푒푐푡푠
푑
휇
~
푈푛푖푓
(
0
,
50
]
푑
푝푟
~
Γ
(
1
,
.
1
)
푑
푝
~
푁
푑
휇
,
푑
푝푟
∈
(
0
,
50
)
휔
푖
=
(
푑
푝
+
Δ
푑
푝
∗
1
퐻푖푑푑푒푛
)
[
푔푎푧푒
퐿
푖
∗
푣
퐿
푖
−
1
−
푔푎푧푒
퐿
푖
∗
푣
푅
푖
+
(
휃
푝
+
Δ
휃
푝
∗
1
퐻푖푑푑푒푛
)
)
(
1
−
푔푎푧푒
퐿
푖
∗
푣
퐿
푖
−
푔푎푧푒
퐿
푖
∗
푣
푅
푖
)
]
휔
푖
푖
trials
푦
푖
푖
trials
푦
푖
~
dwieners
(
bounds
,
NDT
i
,
(
b
p
+
Δ
푏
푝
∗
1
퐻푖푑푑푒푛
)
,
휔
푖
,
(
휎
푝
+
Δ
휎
푝
∗
1
퐻푖푑푑푒푛
)
)
퐵표푢푛푑푠
=
2
(i.e., at
±
1
)
푁퐷
푇
푖
(i.e., non
-
decision time) = RT
–
total ROI fixation time in trial
푖
푏
휇
푏
휅
푏
훼
푏
훽
푏
푝
푝
∈
1
,
...
,
푁
푠푢푏푗푒푐푡푠
푏
휇
~
Beta
2
,
2
∈
(
0
,
1
)
푏
휅
~
Γ
(
1
,
.
5
)
푏
푝
~
퐵푒푡푎
푏
훼
,
푏
훽
∈
(
0
,
1
)
=
푏
휇
푏
휅
=
(
1
−
푏
휇
)
푏
휅
휃
푝
푝
∈
1
,
...
,
푁
푠푢푏푗푒푐푡푠
휃
푝
~
푁
휃
휇
,
휃
푝푟
∈
(
−
.
5
,
1
.
5
)
휃
휇
휃
푝푟
휃
휇
~
푁
.
5
,
.
3
∈
(
−
.
5
,
1
.
5
)
휃
푝푟
~
Γ
(
1
,
.
1
)
Δ
푑
휇
Δ
푑
푝푟
Δ
푑
푝
푝
∈
1
,
...
,
푁
푠푢푏푗푒푐푡푠
Δ
푑
푚푢
~
푈푛푖푓
(
−
50
,
50
)
Δ
푑
푝푟
~
Γ
(
1
,
.
1
)
Δ
푑
푝
~
푁
Δ
푑
휇
,
Δ
푑
푝푟
Δ
휃
푝
푝
∈
1
,
...
,
푁
푠푢푏푗푒푐푡푠
Δ
휃
푝
~
푁
Δ
휃
휇
,
Δ
휃
푝푟
Δ
휃
휇
Δ
휃
푝푟
Δ
휃
휇
~
푁
0
,
.
3
Δ
휃
푝푟
~
Γ
(
1
,
.
1
)
Δ
푏
휇
Δ
푏
푝푟
Δ
푏
푝
푝
∈
1
,
...
,
푁
푠푢푏푗푒푐푡푠
Δ
푏
휇
~
푈푛푖푓
(
−
1
,
1
)
Δ
푏
푝푟
~
Γ
(
1
,
.
1
)
Δ
푏
푝
~
N
Δ
푏
휇
,
Δ
푏
푝푟
휎
휇
휎
푝푟
휎
푝
푝
∈
1
,
...
,
푁
푠푢푏푗푒푐푡푠
휎
휇
~
푈푛푖푓
(
0
,
2
]
휎
푝푟
~
Γ
(
1
,
.
1
)
휎
푝
~
N
휎
휇
,
휎
푝푟
∈
(
0
,
2
)
Δ
휎
휇
Δ
휎
푝푟
Δ
휎
푝
푝
∈
1
,
...
,
푁
푠푢푏푗푒푐푡푠
Δ
휎
휇
~
푈푛푖푓
(
−
2
,
2
)
Δ
휎
푝푟
~
Γ
(
1
,
.
1
)
Δ
휎
푝
~
N
Δ
휎
휇
,
Δ
휎
푝푟
Supplementary Fig. 2.
Directed acyclic graph of hierarchical aDDM with correlated priors. The hierar-
chical model estimates group and individual parameters for the aDDM with correlated priors. The 18 group
parameters are depicted in the top row of yellow circles. The 8 individual parameters estimated for each
subject are depicted in orange in the middle row. The distribution of individual parameters as a function of
the group parameters is specified using a transformation of some of the parameters, denoted by the dashed
lines. The choice and RT outcome
y
i
of trial
i
for a subject
p
is modeled as a Drift-Diffusion-Model with
bounds at
±
1, non-decision time NDT
i
, bias
b
p
, conditional difference in bias ∆
b
p
, trial specific slope
ω
i
,
noise
σ
p
, and conditional difference in noise ∆
σ
p
. The trial specific slope
ω
i
depends on the subject’s drift
rate parameter
d
p
, conditional difference in drift rate parameter ∆
d
p
, attentional bias parameter
θ
p
, condi-
tional difference in attentional bias parameter ∆
θ
p
, gaze data for the trial
gazeL
i
, and item liking ratings for
the foods used in the trial (
vL
i
,
vR
i
).
gazeL
i
denotes the proportion of time spent fixating on the left item
during the trial. The hyperpriors for the group parameters are described at the top of the graph. “
∈
(
X,Y
)”
indicates truncation to bounds.
N
subjects
= 25 in the exploratory and confirmatory datasets, and 50 in the
joint dataset.
21
0.00
0.25
0.50
0.75
1.00
Condition
P(First Fix. Left)
0
100
200
300
400
-4
-2
0
2
4
Net Value (Fixated – Nonfixated)
Latency to First Fix. (ms)
Visible
Hidden
Exploratory
0.00
0.25
0.50
0.75
1.00
Condition
0
100
200
300
400
-4
-2
0
2
4
Net Value (Fixated – Nonfixated)
Confirmatory
0.00
0.25
0.50
0.75
1.00
Condition
0
100
200
300
400
-4
-2
0
2
4
Net Value (Fixated – Nonfixated)
Joint
Supplementary Fig. 3.
Additional fixation properties. (Top) Probability of first fixation to the left item
in the two conditions. (Bottom) Latency to first fixation as a function of the relative rating of the fixated
item. Columns indicate which dataset generated the figures. Black error bars show standard errors of the
mean across participants. Color error bars show standard deviation across participants.
22
400
600
800
1000
1
2
3
4
5
Fixated Item Rating
Middle Fixation Duration (ms)
Visible
Hidden
400
600
800
1000
1
2
3
4
5
Nonfixated Item Rating
Middle Fixation Duration (ms)
200
300
400
500
600
1
2
3
4
5
Fixated Item Rating
First Fixation Duration (ms)
200
300
400
500
600
1
2
3
4
5
Nonfixated Item Rating
First Fixation Duration (ms)
Exploratory
400
600
800
1000
1
2
3
4
5
Fixated Item Rating
400
600
800
1000
1
2
3
4
5
Nonfixated Item Rating
200
300
400
500
600
1
2
3
4
5
Fixated Item Rating
200
300
400
500
600
1
2
3
4
5
Nonfixated Item Rating
Confirmatory
400
600
800
1000
1
2
3
4
5
Fixated Item Rating
400
600
800
1000
1
2
3
4
5
Nonfixated Item Rating
200
300
400
500
600
1
2
3
4
5
Fixated Item Rating
200
300
400
500
600
1
2
3
4
5
Nonfixated Item Rating
Joint
Supplementary Fig. 4.
Fixation durations. (Row 1) Middle fixation duration as a function of the fixated
item rating. (Row 2) Middle fixation duration as a function of the nonfixated item rating. (Row 3) First
fixation duration as a function of the fixated item rating. (Row 4) First fixation duration as a function the
nonfixated item rating. Columns indicate which dataset generated the figures. Error bars show standard
errors of the mean across participants.
23
0.00
0.25
0.50
0.75
1.00
-4 -3 -2 -1 0 1 2 3 4
Value Difference (L–R)
P(Choose Left)
Subject Pool
0
2000
4000
6000
8000
-4 -3 -2 -1 0 1 2 3 4
Value Difference (L–R)
Response Time (ms)
0.00
0.25
0.50
0.75
1.00
-4 -3 -2 -1 0 1 2 3 4
Last Seen – Other Item Rating
P(Last Fixation to Chosen)
0.00
0.25
0.50
0.75
1.00
-4 -3 -2 -1 0 1 2 3 4
Value Difference (L–R)
P(Choose Left)
0
2000
4000
6000
8000
-4 -3 -2 -1 0 1 2 3 4
Value Difference (L–R)
Response Time (ms)
Visible
Hidden
Simulated
0.00
0.25
0.50
0.75
1.00
-4 -3 -2 -1 0 1 2 3 4
Last Seen – Other Item Rating
P(Last Fixation to Chosen)
Supplementary Fig. 5.
Group level predictions in the joint dataset. We use the estimates of the hierarchi-
cal aDDM in the odd trials to make predictions out-of-sample, in the even trials, separately for each subject
and condition. For each subject, we simulate 10 observations per trial, and compare the simulated and
observed data.
Blue lines:
Behavior in the visible condition.
Red lines:
Behavior in the hidden condition.
Black dashed lines and grey areas:
Simulated behavior for the respective condition (dash = mean, grey =
SD). Error bars show standard deviations across subjects.
24
0.00
0.25
0.50
0.75
1.00
−4
−3
−2
−1
0
1
2
3
4
Value Difference (L–R)
P(Choose Left)
Subject 210
0
2000
4000
6000
−4
−3
−2
−1
0
1
2
3
4
Value Difference (L–R)
Response Time (ms)
0.00
0.25
0.50
0.75
1.00
−4
−3
−2
−1
0
1
2
3
4
Last Seen – Other Item Rating
P(Last Fixation to Chosen)
0.00
0.25
0.50
0.75
1.00
−4
−3
−2
−1
0
1
2
3
4
Value Difference (L–R)
P(Choose Left)
0
2000
4000
6000
−4
−3
−2
−1
0
1
2
3
4
Value Difference (L–R)
Response Time (ms)
Visible
Hidden
Simulated
0.00
0.25
0.50
0.75
1.00
−4
−3
−2
−1
0
1
2
3
4
Last Seen – Other Item Rating
P(Last Fixation to Chosen)
0.00
0.25
0.50
0.75
1.00
−4
−3
−2
−1
0
1
2
3
4
Value Difference (L–R)
P(Choose Left)
Subject 219
0
2000
4000
6000
−4
−3
−2
−1
0
1
2
3
4
Value Difference (L–R)
Response Time (ms)
0.00
0.25
0.50
0.75
1.00
−4
−3
−2
−1
0
1
2
3
4
Last Seen – Other Item Rating
P(Last Fixation to Chosen)
0.00
0.25
0.50
0.75
1.00
−4
−3
−2
−1
0
1
2
3
4
Value Difference (L–R)
P(Choose Left)
0
2000
4000
6000
−4
−3
−2
−1
0
1
2
3
4
Value Difference (L–R)
Response Time (ms)
0.00
0.25
0.50
0.75
1.00
−4
−3
−2
−1
0
1
2
3
4
Last Seen – Other Item Rating
P(Last Fixation to Chosen)
0.00
0.25
0.50
0.75
1.00
−4
−3
−2
−1
0
1
2
3
4
Value Difference (L–R)
P(Choose Left)
Subject 325
0
2500
5000
7500
−4
−3
−2
−1
0
1
2
3
4
Value Difference (L–R)
Response Time (ms)
0.00
0.25
0.50
0.75
1.00
−4
−3
−2
−1
0
1
2
3
4
Last Seen – Other Item Rating
P(Last Fixation to Chosen)
0.00
0.25
0.50
0.75
1.00
−4
−3
−2
−1
0
1
2
3
4
Value Difference (L–R)
P(Choose Left)
0
2500
5000
7500
−4
−3
−2
−1
0
1
2
3
4
Value Difference (L–R)
Response Time (ms)
0.00
0.25
0.50
0.75
1.00
−4
−3
−2
−1
0
1
2
3
4
Last Seen – Other Item Rating
P(Last Fixation to Chosen)
0.00
0.25
0.50
0.75
1.00
−2
−1
0
1
2
Value Difference (L–R)
P(Choose Left)
Subject 326
0
2000
4000
6000
8000
−2
−1
0
1
2
Value Difference (L–R)
Response Time (ms)
0.00
0.25
0.50
0.75
1.00
−2
−1
0
1
2
Last Seen – Other Item Rating
P(Last Fixation to Chosen)
0.00
0.25
0.50
0.75
1.00
−2
−1
0
1
2
Value Difference (L–R)
P(Choose Left)
0
2000
4000
6000
8000
−2
−1
0
1
2
Value Difference (L–R)
Response Time (ms)
0.00
0.25
0.50
0.75
1.00
−3
−2
−1
0
1
2
Last Seen – Other Item Rating
P(Last Fixation to Chosen)
Supplementary Fig. 6.
Subject-level simulations. Out-of-sample predictions versus data for four randomly
selected subjects. See Figure S5 and Supplementary Methods for details.
Blue lines:
Behavior in the visible
condition.
Red lines:
Behavior in the hidden condition.
Black dashed lines and grey areas:
Simulated
behavior for the respective condition (dash = mean, grey = SD). Error bars show standard deviations across
subjects.
25
0.00
0.25
0.50
0.75
1.00
1
2
3
4
Best Rating – Worst Rating
P(First Fixation to Best)
Visible
Hidden
0
200
400
600
800
First
Middle
Last
Fixation Type
Fixation Duration (ms)
200
400
600
800
1000
0
1
2
3
4
Best Rating – Worst Rating
Mid. Fix. Duration (ms)
300
400
500
600
0
1
2
3
4
Best Rating – Worst Rating
First Fix. Duration (ms)
-200
-100
0
100
200
300
-4
-2
0
2
4
Left Rating – Right Rating
Net Fix. Duration (L-R, ms)
Full:
q
Hidden
£
0
0.00
0.25
0.50
0.75
1.00
1
2
3
4
Best Rating – Worst Rating
P(First Fixation to Best)
0
200
400
600
800
First
Middle
Last
Fixation Type
Fixation Duration (ms)
200
400
600
800
1000
0
1
2
3
4
Best Rating – Worst Rating
Mid. Fix. Duration (ms)
300
400
500
600
0
1
2
3
4
Best Rating – Worst Rating
First Fix. Duration (ms)
-200
-100
0
100
200
300
-4
-2
0
2
4
Left Rating – Right Rating
Net Fix. Duration (L-R, ms)
Partial:
q
Hidden
>
0
Supplementary Fig. 7.
Fixation properties by attentional discounting group. (Row 1) The probability
that the first fixation is to the best item as a function of choice difficulty. (Row 2) Fixation durations by
fixation type. (Row 3) Middle fixation duration as a function of choice difficulty. (Row 4) First fixation
duration as a function of choice difficulty. (Row 5) Net fixation duration to the left item as a function of its
relative value. Columns indicate which dataset generated the figures: “
θ
Hidden
≤
0” indicates subjects with
full attentional discounting (
N
= 7), “
θ
Hidden
>
0” indicates subjects with partial attentional discounting
(
N
= 43). Data is from the joint dataset. Error bars show standard errors of the mean across participants.
26
0.00
0.25
0.50
0.75
1.00
-4
-2
0
2
4
Left Rating – Right Rating
P(Left Chosen)
-0.4
-0.2
0.0
0.2
0.4
-1.0
-0.5
0.0
0.5
1.0
Final Time Advantage Left (s)
Corr. P(Left Chosen)
-1.0
-0.5
0.0
0.5
1.0
-0.5
0.0
0.5
1.0
1.5
First Fix. Dur. – Avg. First Fix. Dur. (s)
Corr. P(First Seen Chosen)
Full:
q
Hidden
£
0
0.00
0.25
0.50
0.75
1.00
-4
-2
0
2
4
Left Rating – Right Rating
P(Left Chosen)
Last Fix. R
Last Fix. L
Visible
Hidden
-0.4
-0.2
0.0
0.2
0.4
-1.0
-0.5
0.0
0.5
1.0
Final Time Advantage Left (s)
Corr. P(Left Chosen)
-1.0
-0.5
0.0
0.5
1.0
-0.5
0.0
0.5
1.0
1.5
First Fix. Dur. – Avg. First Fix. Dur. (s)
Corr. P(First Seen Chosen)
Partial:
q
Hidden
>
0
Supplementary Fig. 8.
Choice biases by attentional discounting group. (Top) The probability of choosing
the left item as a function of its relative value, conditional on last fixation location. (Middle) The corrected
probability of choosing the left item as a function of the net fixation time to the left item. (Bottom) Corrected
probability that the first seen item is chosen as a function of the excess first fixation duration, defined as first
fixation duration minus mean first fixation duration (computed for each subject). Columns indicate which
dataset generated the figures: “
θ
Hidden
≤
0” indicates subjects with full attentional discounting (
N
= 7),
“
θ
Hidden
>
0” indicates subjects with partial attentional discounting (
N
= 43). Data is from the joint
dataset. Error bars show standard errors of the mean across participants.
27