Technical Appendix
1. Instructions for Participants.
See below a sample of the instructions provided to participants in Study 2 (the ABB’ Matrix
of Set 2, see Table 2). The examples remained fixed, but the matrix presented under rules of the
game was changed according to the matrix subjects were supposed to play.
We invite you to participate in a decision making experiment concerning competition
between two players. You will be asked to make several decisions, and then you will be paid
according to your decisions and the decisions of your competitors.
The structure of the decision task is simple. You will be assigned the role of either Column
player or Row player. Consider the following payoff matrix.
Column Player
Left
Right
6,4
1,5
Row
Up
Player Middle 5,3
2,6
4,2
Down 3,6
In the matrix above, if the Row player chooses Up and the Column player chooses Left,
then the Row player receives 6 francs and the Column player receives 4 francs. On the other
hand, if the Row player chooses Up but the Column player chooses Right, then the Row player
receives 1 franc while the Column player receives 5 francs.
To help you in correctly reading the payoff matrix, we present below four examples.
Example 1: Consider the case where Row player chooses Down and the Column player chooses
Left. In this example, what would be the payoff for Row player? Please circle the appropriate
answer.
a) 1
b) 2
c) 6
d) 0
e) 3
The correct answer is (e). The numbers in the cell corresponding to Row player choosing Down
and the Column player choosing Left is (3,6). Note the Row player’s payoff precedes the
comma, while the Column player’s corresponding payoff follows the comma. Therefore the
ROW player’s payoff is 3.
Example 2: Now suppose the Row player chooses Middle and the column player chooses Right.
In this case, what is the payoff for the Column player?
a) 1
b) 2
c) 6
d) 0
e) 3
The correct answer is (c). The payoff corresponding to Row player choosing Middle and the
column player choosing Right is (2,6). As the Column player’s payoff is featured after the
comma, it is 6.
Example 3: Consider the case where Row player chooses Down and the Column player earned 6.
In this example, what did the Column player choose?
a) Left
b) Right
The correct answer is Left. Note if the Row player’s payoff chooses Down then the Column
player earns 6 or 2 depending on whether the player chooses Left or Right respectively. As the
Column player’s payoff is 6, the Column player must have chosen Left.
Example 4: If the Column player chooses LEFT and the ROW player earned 5, then what did the
ROW player choose?
a) Up
b) Middle
c) Down
The correct answer is Middle. If the Column player chooses Left, then Row player can earn 6, 5
and 3 by choosing Up, Middle and Down respectively. So the correct answer is Middle where
the Row player earns 5.
Rules of the game. You would be assigned the role of either Row or Column player. The
experiment involves 40 trials. In each trial, you will compete with a different person. At the end
of every trial you will be informed about the choice made by your competitor, and your earnings
for that trial.
Note that you will not know the identity of your competitor in any of the trials. Similarly,
your competitors will not know your identity. Thus, players remain anonymous in this game.
Further, you will compete with a different player in each trial. The payoff matrix for the game
that you will play today is:
Column Player
Left
Right
Row Up
14,14 18,26
Player Middle 26,18 10,10
Down 18,22 10,22
At the end of every trial you will be informed about your opponent’s decision, your
decision, and your earnings of that trial.
Finally, at the end of 40 trials your cumulative earnings will be converted to US dollars at
the rate of 100 francs=$2
If you have any questions about the Instructions, please raise your hand so that the
supervisor can answer your questions. If you do not have any questions, please go ahead and
enter your decision for the first trial of the game.
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2. Individual Differences Observed in Study 2.
One implication of the symmetric mixed strategy equilibrium is that all participants
should play alike. However, we observed variations in the behavior of individual participants,
suggesting that perhaps the beliefs of participants were not mutually consistent. For example,
individual Row players’ probability of choosing the dominating choice A, p( A |{ A, B, A'}) ,
ranged from 0.15 to 1 (See Figure 1). The distribution of p( A |{ A, B, A'}) had two modes, one in
the interval 0.3-0.4 and the other at 0.8-0.9. When A was not the dominating choice, the
corresponding conditional probability p( A |{ A, B, B'}) ranged from zero to 0.55, with the mode
in the probability class 0.1-0.2. Next we examined the conditional probability of individual Row
players choosing B. The individual-level p( B |{ A, B, B'}) ranged from 0.45 to 1, with the mode
in the interval 0.9-1. Similarly, p( B |{ A, B, A'}) ranged from zero to 0.875, with the mode in the
probability class 0.1-0.2.
Consistent with the pure strategy equilibrium in the direction of the asymmetric
dominance effect, Column players chose Left more often when they played ABB’ matrices. That
is, p( Left |{ A, B, B'})  p( Left |{ A, B, A ') . The two modes for p( Left |{ A, B, B'}) were in the
intervals 0.7-0.8 and 0.9-1. On the other hand, the mode for p( Left |{ A, B, A'}) fell in the
interval 0-0.1.
In the control condition we also observed substantial variation in the behavior of
individual participants. In set 5, where the mixed strategy prediction for choosing A was 40%,
we found that p( A |{ A, B}) ranged from zero to 0.75, with six participants in the probability
class 0.3-0.5. For set 6, note that the mixed strategy prediction p( A |{ A, B})  0.6 . In this case,
the actual probabilities varied from 0.175 to 0.825, with a mode in the class 0.7-0.8. In the case
of Column players, the probability of choosing Left varied from zero to 0.875 in set 5 and ranged
from 0.25 to 0.975 in set 6.
3
Figure A1: Individual Differences in the ABA’ and ABB’ matrices of Study 2
8
6
ABA' matrices
4
ABB' matrices
2
0.9-1.0
0.8-0.9
0.7-0.8
0.6-0.7
0.5-0.6
0.4-0.5
0.3-0.4
0.2-0.3
0.1-0.2
0
0-0.1
Numbers of Participants
Distribution of Row Players Choosing A
Probability Class
8
6
ABA' matrices
4
ABB' matrices
2
0.9-1.0
0.8-0.9
0.7-0.8
0.6-0.7
0.5-0.6
0.4-0.5
0.3-0.4
0.2-0.3
0.1-0.2
0
0-0.1
Number of Participants
Distribution of Row Players Choosing B
Probability Class
8
6
ABA' matrices
4
ABB' matrices
2
Probability Class
4
0.9-1.0
0.8-0.9
0.7-0.8
0.6-0.7
0.5-0.6
0.4-0.5
0.3-0.4
0.2-0.3
0.1-0.2
0
0-0.1
Number of Participants
Distribution of Column Players Choosing Left
3. Examination of Heterogeneity in Learning in Study 2.
Allowing for heterogeneity in the form of latent classes substantially improved model fit.
For Column players, we find that allowing for heterogeneity increased the size of the estimated
. However, the size of the segment with the larger  is quite small (17.98%). This finding that
estimates of delta could be biased downward in the absence allowing for heterogeneity is consistent with
Wilcox (2006). However, we do not see such an effect in the case of Row players, where the first
segment is large (76.15%). A potential explanation for this is that we allowed for heterogeneity
in all the model parameters. As noted by Camerer and Ho (1998) and also highlighted by Wilcox
(2006), there is a need for more research to understand how allowing for heterogeneity in
multiple parameters may cause the values of those parameters to interact.
Row Players
Reward
Parameter
Segment 1
Calibration
Sample
Validation
Sample
Column Players
Segment 2
Segment 1
Segment 2
ρ
φ

λ
Log-Likelihood
AIC
BIC
Pseudo-ρ2
0199***
0.126***
0.970***
0.990***
0.216***
0.001***
0.01***
0.01***
-213.762
-229.762
-268.697
0.797
0.990***
0.000
0.568***
0.565***
0.900***
0.113***
0.209***
3.000***
-102.842
-118.842
-157.777
0.902
Log-Likelihood
-77.4964
-70.885
AIC
BIC
Pseudo- ρ2
-93.496
-126.886
0.853
-86.885
-120.276
0.866
Note: *** Significant at 0.01 level
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4. Estimation of Learning in ABA’ and ABB’ matrices in Study 2.
On estimating a separate set of parameters for each type of matrix, we note that the
model provides a better account of the behavior of participants in ABB’ (see the fit statistics
below). The estimated values of λ and ρ are lower for the ABB’ matrices, implying that Row players
were less sensitive to payoff and were giving less weight to past choices than was the case for the ABA’
matrices. Note that in both the ABA’ and ABB’ matrices the values for δ imply that players followed
neither pure reinforcement learning nor pure belief learning.
ABA'
Reward
Calibration
Sample
Validation
Sample
ABB'
Parameter
ρ
0.974***
0.072***
φ
0.938***
0.873***

λ
LogLikelihood
0.342***
0.607***
0.415***
0.058***
-344.3049
-171.358
AIC
-360.305
-187.358
BIC
-393.695
-220.748
Pseudo-ρ2
LogLikelihood
0.343
0.667
-174.4872
-81.460
AIC
-190.4872
-97.4598
-218.332
-125.305
0.338
0.691
BIC
Pseudo- ρ
2
Note: *** Significant at 0.01 level
6