First- and Second-Order Subjective Expectations
in Strategic Decision-Making:
Experimental Evidence
Charles Manski
Northwestern University
Claudia Neri
University of St.Gallen
First- and Second-Order Subjective Expectations
1
Motivation
How do people reason when facing a strategic task?
strategic thinking: role of subjective beliefs
1st-order beliefs: beliefs about opponent’s choice
2nd-order beliefs: beliefs about opponent’s beliefs about one’s own choice
2nd-order beliefs are likely to play a role in strategic situations involving:
asymmetric information
social preferences
deception
First- and Second-Order Subjective Expectations
2
Motivation
Questions:
1
Can subjective 2nd-order beliefs be elicited probabilistically? How?
2
What is the relationship between choice, 1st-order and 2nd-order beliefs?
3
Are decision-makers heterogeneous in their choices and beliefs? How?
First- and Second-Order Subjective Expectations
3
The game and the experimental setup
Tasks
Game in which participants complete three tasks:
make a strategic (binary) choice
express their 1st-order beliefs probabilistically
express their 2nd-order beliefs probabilistically
First- and Second-Order Subjective Expectations
4
The game and the experimental setup
Choice and beliefs
Players
player i and player j
Choice
binary: A or B
1st-order beliefs
xi = subjective probability that player i assigns to ‘player j chooses A’
xj = subjective probability that player j assigns to ‘player i chooses A’
xi , xj ∈ [0, 1]
2nd-order beliefs
pi (xj ) = subjective probability distribution that player i holds about xj
pj (xi ) = subjective probability distribution that player j holds about xi
First- and Second-Order Subjective Expectations
5
The game and the experimental setup
Decision under uncertainty
what this paper is about
decision under uncertainty
subjects hold a unique subjective probability for an unknown event
what this paper is not about
decision under ambiguity
subjects possibly hold a set of subjective probabilities
First- and Second-Order Subjective Expectations
6
The game and the experimental setup
Hide-and-Seek
The Hider has to hide a prize (a $10 banknote) in one of
two locations. The Seeker has to guess where the prize
has been hidden.
A
B
If the Seeker guesses correctly, she wins the prize.
Otherwise, the Hider keeps the prize.
The two locations are two zones in which a square field
have been divided: an inner square field and an outer
contour-shaped field. The two zones have the same area,
while they differ in shape, color and labeling.
First- and Second-Order Subjective Expectations
7
The game and the experimental setup
Timing, treatments, incentives
114 Northwestern University undergraduate students
game played in pairs: a Hider and a Seeker
game played for 4 rounds. In each round:
players randomly matched in pairs
roles randomly drawn within each pair
choice made and beliefs elicited
feedback: choice outcome and total forecast reward
treatments differ in the order in which the tasks (choice, elicitation of
1st-order beliefs, elicitation of 2nd-order beliefs) are presented
between-subjects design
monetary incentives:
show-up fee ($5)
prize to the winner ($10)
reward based on forecast accuracy (min $0, max $6)
only the winnings of a randomly-selected round are paid out
First- and Second-Order Subjective Expectations
8
This experiment in context
Beliefs elicitation
probabilistic 1st-order beliefs
Nyarko and Schotter (2002)
non-probabilistic 2nd-order beliefs
Bhatt and Camerer (2005)
Vanberg (2008)
Bellemare, Sebald and Strobel (2011)
more
hide-and-seek game
Rubinstein, Tversky and Heller (1999)
Ayton and Falk (1995)
more
First- and Second-Order Subjective Expectations
9
This experiment in context
Beliefs elicitation: probabilistic vs. non-probabilistic
Lets consider this situation: agents S (Seeker) and H (Hider) choose a binary action yS = {a, b}, yH = {A, B}.
Lets focus on agent S’s beliefs:
1st-order beliefs about agent H’s action
2nd-order beliefs about agent H’s 1st-order beliefs about agent S’s choice
agent S’s
beliefs
variable
to be forecasted
1st-order
agent H’s
binary action:
yH = {A, B}
beliefs elicitation
probabilistic
non-probabilistic
discrete prob. distr. over yH
Prob S (yH = A), Prob S (yH = B)
S
yH
= {A, B}
Bellemare et al 2011
this paper
Bhatt and Camerer 2005
agent H’s
probabilistic
1st-order beliefs:
Prob H (yS = a)
continuous prob. distr. over Prob H (yS = a)
point-forecast of Prob H (yS = a)
this paper
Bellemare et al 2011
agent H’s
non-probabilistic
1st-order beliefs:
ySH = {a, b}
discrete prob. distr. over ySH
Prob S (ySH = a), Prob S (ySH = b)
2nd-order
ySH
S
= {a, b}
Bhatt and Camerer 2005
First- and Second-Order Subjective Expectations
10
Probabilistic beliefs elicitation
1st-order beliefs
Question to the Hider
What do you think the percent chance is that the Seeker will look for the prize in A?
And in B?
Write your answers in the spaces provided below.
You can choose values between 0 and 100. The values you choose should sum to 100.
Percent chance that the Seeker will look for the prize in A: . . .
Percent chance that the Seeker will look for the prize in B: . . .
scoring rule
First- and Second-Order Subjective Expectations
11
Probabilistic beliefs elicitation
2nd-order beliefs
Two steps:
1
subjects report what they think the most likely value is for their opponent’s
1st-order beliefs
2
subjects report the probabilities with which their opponent’s 1st-order beliefs
will fall within several intervals
First- and Second-Order Subjective Expectations
12
Probabilistic beliefs elicitation
2nd-order beliefs
Question to the Hider
You are the Hider.
For sure your opponent wants to find the prize, so he or she must be trying to guess
where you will hide it.
We’ve just asked your opponent to tell us what he or she thinks.
The question we asked was: What do you think the percent chance is that the Hider
will hide the prize in A?
Your opponent has answered this question. You don’t know the answer. How do you
think your opponent has answered?
Tell us what you think the most likely value is for the answer given by your opponent.
I think the most likely value for the Seeker’s answer is: . . .
scoring rule
First- and Second-Order Subjective Expectations
13
Probabilistic beliefs elicitation
2nd-order beliefs
Question to the Hider
Now tell us something more. Please complete the following sentences.
I think that the percent chance that the Seeker’s answer is not larger than 5 is: . . .
I think that the percent chance that the Seeker’s answer is larger than 5 and not larger than 20 is: . . .
I think that the percent chance that the Seeker’s answer is larger than 20 and not larger than 50 is: . . .
I think that the percent chance that the Seeker’s answer is larger than 50 and not larger than 80 is: . . .
I think that the percent chance that the Seeker’s answer is larger than 80 and not larger than 95 is: . . .
I think that the percent chance that the Seeker’s answer is larger than 95 is: . . .
scoring rule
First- and Second-Order Subjective Expectations
14
Probabilistic beliefs elicitation
2nd-order beliefs
Why two steps?
1. point forecast (non-probabilistic)
‘most likely value for opponent’s 1st-order beliefs’
meant as an introduction for participants to better understand the second step
can be only loosely interpreted as a measure of central tendency for the
subjective distribution
not required nor used in the analysis of decision-making
2. probabilistic forecast
‘probabilities with which opponent’s 1st-order beliefs falls within several
intervals’
intervals are [0,5]%, (5,20]%, (20,50]%, (50,80]%, (80,95]% and (95,100]%
independently from the point forecast given in the first step
meant to measure several values over the subjective distribution
required and used in the analysis of decision-making
more
First- and Second-Order Subjective Expectations
15
Choice and beliefs
Decision-problem and relationship between choice and beliefs
Hider
Seeker
choice and 1st-order beliefs
consistency: choice coincides with the optimal response to 1st-order beliefs
C
H = rH (PH )
 A
B
rH (PH ) =

A or B
where
C
S = rS (PS ) where
if PS > 0.5
if PH < 0.5
 A
B
if PS < 0.5
if PH > 0.5
rS (PS ) =

A or B
if PS = 0.5
if PH = 0.5
choice and 2nd-order beliefs
consistency: choice coincides with the optimal response to 2nd-order beliefs
CH = rH (QH )
if
 A
B
if
rH (QH ) =
 A or B
if
where
 CS = rS (QS ) where
QH (0.5) > 0.5
if QS (0.5) > 0.5
 A
QH (0.5) < 0.5
B
if QS (0.5) < 0.5
rS (QS ) =

QH (0.5) = 0.5
A or B
if QS (0.5) = 0.5
1st- and 2nd-order beliefs
weak coherence: if an event is assigned, according to 1st-order beliefs, a probability higher/lower/equal to
the one assigned to another event, then the same holds according to 2nd-order beliefs
PH ≥ 0.5 and 1 − QH (0.5) ≥ 0.5
PS ≥ 0.5 and QS (0.5) ≥ 0.5
or
or
PH ≤ 0.5 and 1 − QH (0.5) ≤ 0.5
PS ≤ 0.5 and QS (0.5) ≤ 0.5
strong coherence: the probability assigned to an event according to 1st- and 2nd-order beliefs coincides
PH = 1 − QH (0.5)
PS = QS (0.5)
CH = Hider’s observed choice
rH = Hider’s optimal response to beliefs
PH = Hider’s belief that Seeker chooses A
QH (x) = Hider’s belief that Seeker’s belief PS is ≤ x
CS = Seeker’s observed choice
rS = Seeker’s optimal response to beliefs
PS = Seeker’s belief that Hider chooses A
QS (x) = Seeker’s belief that Hider’s belief PH is ≤ x
First- and Second-Order Subjective Expectations
16
Choice and beliefs
Consistency and coherence: empirical frequencies vs. theoretical probabilities
relationship between choice and 1st-order beliefs is stronger-than-random
relationship between choice and 2nd-order beliefs is not stronger-than-random
stronger-than-random relationship between choice and 1st-order beliefs
correlates with:
1st- and 2nd-order beliefs being strongly coherent, and
choice coinciding with the optimal response to 2nd-order beliefs
more
First- and Second-Order Subjective Expectations
17
Heterogeneity
Choice and beliefs
participants are not a homogeneous pool of individuals
treat each participant as a unit of observation
inspect the distribution across participants of the frequency with which choice
coincides with the optimal response to 1st- and/or 2nd-order beliefs
two ‘types’
always indifferent between A and B →
choice always coincides with optimal response to 1st- and 2nd-order beliefs
never indifferent between A and B →
choice often coincides with optimal response to 1st- and 2nd-order beliefs
more
First- and Second-Order Subjective Expectations
18
Heterogeneity
Decision rules
how can we interpret heterogeneity across participants?
post-experiment questionnaire
How did you choose your actions in the game? Please describe briefly.
How did you choose your forecasts in the game? Please describe briefly.
insight into the decision rules employed by participants
complex environment: strategic, dynamic, heterogeneous
First- and Second-Order Subjective Expectations
19
Heterogeneity
Decision rules
how to handle strategy I figured people would be most likely to look in B.., so I hid in A. (Subject 66) … as if I were playing [against] the same person. (Subject 105) how to handle dynamics does she think strategically: does she think about what the beliefs/choice of the possible opponent could be? does she consider changing her beliefs/choice a:er observing the outcome of a previous interac<on? does she think that other subjects may also think strategically? does she think that other subjects may also change their beliefs/choice dynamically? does she think that other subjects’ way of thinking strategically may differ from hers? does she think that other subjects’ way of changing their beliefs/choice dynamically may differ from hers? how to handle heterogeneity homogeneous popula<on how is the representa<ve agent: like oneself or different? heterogeneous popula<on I based them on the results of my last period, because I thought in general people choose the same way. (Subject 98) I think people end up doing a lot of reverse psychology on the decisions and will probably end up very random when aggregated. (Subject 111) how can aggrega<on be performed? First- and Second-Order Subjective Expectations
20
Conclusion
What we have proposed:
how to elicit 2nd-order beliefs probabilistically (also outside the lab),
in order to turn higher-order beliefs into an observable variable
criterium of weak and strong coherence of 1st- and 2nd-order beliefs
in order to investigate the relation between choice and probabilistic beliefs
Limitations of the current work / Extensions for future work:
binary choice set / non-binary choice set
elicitation of a univariate / multivariate continuous probability distribution
assumption of subjects’ optimizing behavior / no assumption
decision under uncertainty / ambiguity
First- and Second-Order Subjective Expectations
21
Thank You
First- and Second-Order Subjective Expectations
22
Studies eliciting non-probabilistic 2nd-order beliefs
Bhatt and Camerer (2005)
1st-order beliefs: What do you think the other player will choose?
A
B
2nd-order beliefs: What do you think the other player believes you will choose?
A
B
Vanberg (2008)
1st-order beliefs: What do you think the other player will choose?
certainly A
probably A
unsure
probably B
certainly B
2nd-order beliefs: What do you think the other player believes you will choose?
certainly A
probably A
unsure
probably B
certainly B
Bellemare, Sebald and Strobel (2011)
1st-order beliefs (of type-1-person): How many type-2-persons out of 100 will choose A and how many B?
Number of type-2-persons out of 100 that will choose A:
Number of type-2-persons out of 100 that will choose B:
2nd-order beliefs (of type-2-person): What do you think about type-1-person’s beliefs about the behavior of
type-2-persons?
Type-1-person believes that
type-2-persons out of 100 choose A.
Type-1-person believes that
type-2-persons out of 100 choose B.
back
First- and Second-Order Subjective Expectations
23
Others Hide-and-Seek games
Rubinstein, Tversky and Heller (1999)
You and another player are playing the following game. You possess a treasure which you can hide in one of four
boxes arranged in a row. The boxes are marked as follows:
A
B
A
A
Your opponent will be allowed to find the treasure by opening one (and only one) of the boxes. Where will you hide
the treasure so that it will remain in your possession?
Ayton and Falk (1995)
You have to hide a treasure in one of 25 boxes which are ordered according to the following table. Another player
will try to find the treasure. Hide the treasure in one of the boxes.
back
First- and Second-Order Subjective Expectations
24
Scoring rules to elicit 1st- and 2nd-order beliefs
back
Hider’s score for accuracy of 1st-order beliefs
SH (IA , IB , PH ) = 2 −
n
IA − P H
2
2 o
+ IB − (1 − PH )
PH = Hider’s 1st-order beliefs that Seeker chooses A
IA = 1 if Seeker chooses A, = 0 otherwise
IB = 1 if Seeker chooses B, = 0 otherwise
First- and Second-Order Subjective Expectations
25
Scoring rules to elicit 1st- and 2nd-order beliefs
back
Hider’s score for accuracy of 1st-order beliefs
SH (IA , IB , PH ) = 2 −
n
IA − P H
2
2 o
+ IB − (1 − PH )
PH = Hider’s 1st-order beliefs that Seeker chooses A
IA = 1 if Seeker chooses A, = 0 otherwise
IB = 1 if Seeker chooses B, = 0 otherwise
Hider’s score for accuracy of 2nd-order beliefs (most-likely-value forecast)
SH (m, PS ) =
2
0
if PS = m
otherwise
m = Hider’s 2nd-order beliefs (most-likely-value forecast)
PS = Seeker’s 1st-order beliefs that Hider chooses A
Hider’s score for accuracy of 2nd-order beliefs (probabilistic forecast)

SH (p, I ) = 2 − 
6 X
j=1

I[x ,y ] − p[x ,y ]
j j
j j
2

p[x ,y ] = Hider’s 2nd-order beliefs that Seeker’s 1st-order beliefs lie in [xj , yj ]
j j
I[x ,y ] = 1 if Seeker’s 1st-order beliefs lie in [xj , yj ], = 0 otherwise
j j
First- and Second-Order Subjective Expectations
26
Elicitation of beliefs
2nd-order beliefs
Comparison between point and probabilistic 2nd-order beliefs
if the answer in terms of ‘most likely value’ and the answer in terms of
probabilities over intervals were not coherent, the feasibility of eliciting
2nd-order beliefs probabilistically would be undermined
coherence: the ‘most likely value’ as a measure of central tendency for the
subjective distribution that is characterized by the probabilities over intervals
mode?
we only elicited probabilities within certain intervals
the mode is a local concept and we cannot tell which interval it is in
assuming that the mode is in the interval assigned with the highest probability is not a good
assumption, because the intervals have different widths
assuming that the density is uniform within each interval is not a good assumption, because the
middle intervals have a large width
we don’t make any inference on the mode
mean? median?
back
First- and Second-Order Subjective Expectations
27
Elicitation of beliefs
2nd-order beliefs
Comparison between point and probabilistic 2nd-order beliefs
more
Parametric analysis
the elicited ‘most likely value’ matches closely the mean and the median of the fitted
subjective distribution
Empirical distribution
25th perc.
50th perc.
75th perc.
most likely value - fitted mean
-5
0
6.85
most likely value - fitted median
-5.96
0
7.16
Note: percentage points
more
Nonparametric analysis
the elicited ‘most likely value’ lies within the bounds for the mean and median of the
subjective distribution respectively in 69% and 79% of the observations
Evidence suggests that subjects are able to state 2nd-order beliefs probabilistically
back
First- and Second-Order Subjective Expectations
28
Parametric analysis
Fitted subjective distribution
Use the probabilities assigned to each interval [0,5]%, (5,20]%, (20,50]%,
(50,80]%, (80,95]% and (95,100]% to fit the subjective distribution which
represents 2nd-order beliefs
Assumptions about the shape of the subjective distribution:
1
if positive probability is placed over 1 interval (approx.
an isosceles triangle (whose support is the interval)
10% of obs.)
2
if positive probability is placed over 2 intervals (approx.
20% of obs.)
adjacent: an isosceles triangle
non adjacent: 4% of obs., excluded
3
if positive probability is placed over 3 or more intervals (approx.
Beta distribution
70% of obs.)
back
First- and Second-Order Subjective Expectations
29
Parametric analysis
Fitted subjective distribution
from the knowledge of the probabilities assigned to each interval, we can infer
the values of the subjective CDF at the right endpoints of the intervals
denote the right endpoints of the six intervals as r1 , ..., r6
denote the values of the CDF at the points r1 , ..., r6 as Q(r1 ), ..., Q(r6 )
denote the left and right endpoints of the intervals where positive probability is
place as l and r respectively
assume that the function Q is distributed according to the Beta distribution
let Beta(x, α, β, l, r ) denote the CDF of a Beta distribution with shape
parameters α and β and location parameters l and r evaluated at x
for each subject i and period t, use the elicited Qi,t (rj ), l and r to find the
parameters α and β that solve the least-squares problem:


6
X
2 
min
Beta(rj , α, β, l, r ) − Qi,t (rj )

α,β 
j=1
back
First- and Second-Order Subjective Expectations
30
Nonparametric analysis
Bounds on the subjective median and mean
Example:
a subject assigns 30% to (20,50], 60% to (50-80], 5% to (80-95] and 5% to (95-100]
bounds on the subjective median
[50 − 80]
bounds on the subjective mean
placing all of each interval’s probability mass at the interval’s lower and upper
endpoints respectively
lower bound = 0.30 × 20 + 0.60 × 50 + 0.05 × 80 + 0.05 × 95 = 44.75
upper bound = 0.30 × 50 + 0.60 × 80 + 0.05 × 95 + 0.05 × 100 = 72.75
[44.75 − 72.75]
back
First- and Second-Order Subjective Expectations
31
Choice and beliefs
Consistency and coherence: empirical frequencies vs. theoretical probabilities
(i)
choice
and
1storder
beliefs
(ii)
choice
and
2ndorder
beliefs
(iii)
(i),(ii),(iii)
1st- and 2ndorder beliefs
choice,
1st- and 2ndorder beliefs
consistency
consistency
obs.
%
%
strong coherence
0%
5%
10%
%
%
%
weak
coherence
%
strong coherence
0%
5%
10%
%
%
%
weak
coherence
%
all (N=456)
empirical
theoretical
89
73
75
71
34
26
40
34
52
41
83
81
33
26
38
29
46
32
68
54
P 6= 0.5, Q(0.5) =
6 0.5 (N=173)
empirical
81
57
theoretical
50
50
19
0
28
9.75
37
19
55
50
17
0
25
2.44
32
4.75
47
25
Note: theoretical probabilities under the assumption that participants’ choice, 1st- and 2nd-order beliefs are
submitted randomly and independently the one of the others P(A) = 0.5, P ∼ U[0, 100], Q(0.5) ∼ U[0, 100]
0%-strong coherence is satisfied if PH = 1 − QH (0.5) or PS = QS (0.5)
5%-strong coherence is satisfied if |PH − (1 − QH (0.5))| ≤ 0.05 or |PS − QS (0.5)| ≤ 0.05
10%-strong coherence is satisfied if |PH − (1 − QH (0.5))| ≤ 0.10 or |PS − QS (0.5)| ≤ 0.10.
back
First- and Second-Order Subjective Expectations
32
Heterogeneity
Choice and beliefs
How often choice coincides
with optimal response to ...
Frequency
always
(subject is always indifferent)
fraction of periods
(when subject is not indifferent)
0/1
1/1
0/2
1/2
2/2
0/3
1/3
2/3
3/3
0/4
1/4
2/4
3/4
4/4
all subjects
... 1st-order beliefs
... 2nd-order beliefs
No. subjects
No. subjects
38
26
3
3
4
3
15
18
5
10
4
5
5
1
6
4
3
4
6
13
15
7
114
114
4
8
1
5
12
back
First- and Second-Order Subjective Expectations
33
Heterogeneity
Rounding probabilistic beliefs
Responses to elicitation questions. 114 participants. round 1.
Question
1st-order beliefs
2nd-order beliefs:
2nd-order beliefs:
2nd-order beliefs:
2nd-order beliefs:
2nd-order beliefs:
2nd-order beliefs:
2nd-order beliefs:
‘most likely value’
prob. over [0,5]%
prob. over (5,20]%
prob. over (20,50]%
prob. over (50,80]%
prob. over (80,95]%
prob. over (95,100]%
0
1–4
50
0.09
0.06
0.57
0.39
0.17
0.12
0.29
0.43
0.00
0.01
0.03
0.00
0.00
0.00
0.00
0.04
0.44
0.50
0.02
0.00
0.14
0.19
0.00
0.04
Fraction of responses
96–99
100
M10
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.11
0.16
0.03
0.00
0.02
0.04
0.00
0.04
0.24
0.19
0.13
0.37
0.45
0.44
0.38
0.16
M5
other
0.12
0.07
0.23
0.21
0.19
0.18
0.30
0.28
0.00
0.01
0.00
0.03
0.04
0.03
0.04
0.02
back
First- and Second-Order Subjective Expectations
34
Heterogeneity
Rounding probabilistic beliefs
Patterns in the responses to elicitation questions. 114 participants. all 4 round.
Response pattern
Question
1st-order beliefs
2nd-order beliefs:
2nd-order beliefs:
2nd-order beliefs:
2nd-order beliefs:
2nd-order beliefs:
2nd-order beliefs:
2nd-order beliefs:
‘most likely value’
prob. over [0,5]%
prob. over (5,20]%
prob. over (20,50]%
prob. over (50,80]%
prob. over (80,95]%
prob. over (95,100]%
All 0
or 100
All 0, 50
or 100
some M10
some M5
some 1-4
or 96-99
some
other
0.12
0.06
0.49
0.27
0.11
0.11
0.22
0.33
0.43
0.46
0.04
0.00
0.11
0.10
0.00
0.05
0.25
0.29
0.11
0.23
0.35
0.33
0.21
0.13
0.17
0.15
0.26
0.42
0.36
0.39
0.46
0.36
0.00
0.01
0.07
0.03
0.01
0.01
0.03
0.09
0.03
0.04
0.04
0.05
0.06
0.06
0.08
0.04
back
First- and Second-Order Subjective Expectations
35
Observed choices
sample frequency of choice A:
60% of Hider-observations, 41% of Seeker-observations
sample frequency of choice outcomes (Hider’s choice, Seeker’s choice):
(A,A)
25%
(A,B)
36%
(B,A)
17%
(B,B)
23%
Seekers win the game slightly less often than Hiders
the frequency with which Seekers choose alternative A increases over the
course of the four rounds
First- and Second-Order Subjective Expectations
36
2nd-order beliefs
Epistemic uncertainty
Consider the situation that can be equivalently described by any of the following:
Q(0.5) = 0.5
a subject believes her opponent’s 1st-order beliefs are equally likely to be smaller or larger than 0.5
a subject best responding to her 2nd-order beliefs is indifferent between A and B
Is a subject who reports such 2nd-order beliefs expressing epistemic uncertainty?
Expressing epistemic uncertainty would probably lead to place positive probability over more intervals.
Evidence:
never only 1 interval
most often over 2, 4 or 6 intervals
use of 2 intervals ∼ use of 4 intervals ∼ use of 6 intervals
all obs.
only obs.
with
Q(0.5) 6= 0.5
only obs.
with
Q(0.5) = 0.5
%
%
%
1
2
3
4
5
6
10
18
11
26
7
28
16
10
16
24
11
23
0
30
5
29
1
35
all
100
100
100
Number of intervals
back
First- and Second-Order Subjective Expectations
37
Heterogeneous updating of 1st- and 2nd-order beliefs
Subjects i = 8, 21, 82, 103, 112 play as Seeker in period 1 and 2. In period 1 they
report the same 1st-order beliefs (25%), they make the same choice (B) and they
observe the same choice by their opponent (A).
25
21
75
50
21
21
103
8
82
103
82
112
0
82
8
112
25
50
8
103
Subjective most likely value of opponent's beliefs
75
112
0
Subjective prob.of opponent choosing A
100
Subjects update
their 2nd-order beliefs
from period 1 to period 2
100
Subjects update
their 1st-order beliefs
from period 1 to period 2
1
2
Period
1
2
Period
back
First- and Second-Order Subjective Expectations
38
Heterogeneity
Rounding probabilistic beliefs
more
Is there evidence of elicited beliefs being rounded?
Yes, generally at least to a multiple of 5 percent.
Do participants differ in their rounding rules?
Yes, the most frequent rules being:
‘all 0 or 100’: always 0 or 100 percent
‘all 0, 50 or 100’: always 0, 50 or 100 percent, at least once 50 percent
‘some M10’: always a multiple of 10 percent, at least once different from 0, 50 and 100 percent
‘some M5’: always a multiple of 5 percent, at least once different from 10 percent
assuming that participants don’t change their rounding rules across rounds.
Do participants use different rounding rules when answering different questions?
Yes.
For most participants there is less rounding in 2nd-order than 1st-order beliefs.
Are the results on strong and weak coherence of 1st- and 2nd-order beliefs
affected by the rounding rules?
No.
First- and Second-Order Subjective Expectations
39