ISSN (Print) 0473-453X

Discussion Paper No. 1200

ISSN (Online) 2435-0982

AN INDIVIDUAL EVOLUTIONARYLEARNING MODEL MEETS COURNOT
Jasmina Arifovic
Liang Diao
Nobuyuki Hanaki

January 2023

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan

An individual evolutionary learning model meets
Cournot∗
Jasmina Arifovic†

Liang Diao‡

Nobuyuki Hanaki§

January 3, 2023

Abstract
In this paper, we extend the individual evolutionary learning model by incorporating other-regarding considerations and apply the model to some Cournot
games. Using the model fitted to the experimental data of a repeated 3-player
Cournot game (with nonlinear cost and demand functions), we construct outof-sample predictions regarding the “feedback effects” and “number effects”
and test these using data gathered via newly conducted experiments. The prediction regarding the feedback effect is only partially confirmed, being observedfor 3- and 4-player games but not the 2-player game. The prediction regarding
the number effect is also partially confirmed in that while the model predicts
the number effect to be observed with detailed and not aggregate feedback, theeffect is observed with both types of feedback.
Keywords: Individual Evolutionary Learning, Oligopoly, Experiment
JEL Classification: D43, D83, D90∗

We sincerely acknowledge the contribution of the late Jasmina Arifovic, who sadly passed away
before her final editing of this then unfinished paper. Comments and suggestions from Nick Vriend
are gratefully acknowledged. Part of this research was conducted when Hanaki was affiliated with the
Université Côte d’Azur. This research has benefited from the financial support of (a) the Joint Usage/Research Center, the Institute of Social and Economic Research, Osaka University, (b) Grant-inaid for Scientific Research Nos. 18K19954 and 20H05631 from the Japan Society for the Promotion of
Science, and (c) the French National Research Agency (ANR) under an ORA-Plus project “BEAM”(ANR-15-ORAR-0004), and was performed within the framework of Programme Investissements
d’Avenir under and UCAJEDI (ANR-15-IDEX-01) of Université Côte d’Azur (project UCAinACTION).
†
Simon Fraser University, Canada.
‡
Simon Fraser University, Canada. E-mail: liang diao@sfu.ca§
Institute of Social and Economic Research, Osaka University, Japan.
E-mail:nobuyuki.hanaki@iser.osaka-u.ac.jp1

1

Introduction

Cournot games have long been one of the workhorse models of industrial organization.
One of their interesting features is that different learning models lead to sharply contrasting outcomes. Learning models that rely on best replies converge to the Cournot–
Nash equilibrium, while those based on imitating the most successful player (which
includes evolutionary selection) converge to the competitive (Walrasian) equilibrium(see, among others, Vega-Redondo, 1997; Vriend, 2000; Vallée and Yildizoğlu, 2009).
In addition, simple and plausible trial-and-error learning converges to the collusive
outcome (Huck et al., 2003, 2004a).
These theoretical predictions have led to extensive experimental analyses of thesegames. Huck et al. (1999, 2000), Offerman et al. (2002), and Apesteguia et al. (2010)
experimentally examine “feedback effects,” i.e., the impact of differences in the information provided to participants after each play of the game (feedback) on the outcomes. These studies demonstrate that outputs are higher (thus, more competitive)
in settings where participants receive detailed feedback (price, aggregate quantity,quantities chosen, and profits obtained by each of the other participants in the same
market) after each play of the game, compared with those settings where they receive
only aggregate information as feedback (price and aggregate quantity, apart fromtheir own choice of quantity and the resulting profits).
Note that imitating what successful others did is easier with detailed feedback
than with only aggregate feedback, although, as Bigoni and Fort (2013) show, theextent to which each piece of information is used when making decisions may differ
across participants. Recently, Friedman et al. (2015) showed that previously reported
feedback effects are only a temporary phenomenon (observed in the first 50 periods oftheir 1,200-period experiment with 2 or 3 players per market), and that after a couple2

of hundreds of periods, even with detailed feedback, participants learn to collude.
Bosch-Domènech and Vriend (2003) varied the difficulty to perform best reply (bymanipulating the information required to compute payoffs in the instructions as wellas the decision time) to investigate whether participants imitate successful others
when undertaking the best reply is more difficult. They did not, however, observe
outputs much higher in the environment more conducive to imitation.
Another well-known result from the experimental analysis of Cournot competition
is “number effects,” (Selten, 1973) which concerns the number of players (firms)
interacting and the resulting degree of competition. In an experiment with aggregate
feedback, Huck et al. (2004b) show that while some collusion can be observed when
only two players interact in a market, there is near Cournot–Nash equilibrium withthree players interacting, and that outcomes are never collusive and typically above
the Cournot–Nash equilibrium with four or more players interacting (see Horstmann
et al., 2018, for a recent survey as well as additional experimental evidence). In fact,
Oechssler et al. (2016) shows that this is also the case in near-continuous 1,200-period
settings similar to that of Friedman et al. (2015) and thus, unlike feedback effects, it
is robust against increasing the number of periods.
In this paper, we extend the individual evolutionary learning model (first applied
to Cournot games by Arifovic, 1994) by incorporating other-regarding considerations
as done by Arifovic and Ledyard (2012) and apply it to Cournot games. We first fit themodel to the experimental data of a repeated 3-player Cournot game (with nonlinear
cost and demand functions) by Offerman et al. (2002) to fix the parameter values. We
then simulate the model in n-player games (with n ∈ {2, 3, 4}) to make out-of-sample
predictions regarding both the feedback effects and the number effects.
We then conduct our own experiment to test these out-of-sample predictions of
the model. The experimental data partially confirm the prediction regarding the3feedback effect; namely, the treatment with detailed feedback resulted in competitive
outcomes significantly more frequently than the treatment with only the aggregate
feedback for n ∈ {3, 4} but not for n = 2. Furthermore, the prediction regarding
the number effect is also partially confirmed; namely, while the model predicts thenumber effect to be observed only with the detailed feedback and not the aggregate
feedback, it is observed with both the detailed and the aggregate feedback treatments.
The remainder of the paper is organized as follows. Section 2 presents the extended
individual evolutionary learning model, along with its fit to existing experimental dataand out-of-sample predictions. The experimental design to test the out-of-sample prediction of the model is presented in Section 3, followed by the results of the experimentin Section 4. Section 5 concludes.

2

Individual Evolutionary Learning Model

We extend the individual evolutionary learning (IEL) model that incorporates theother-regarding consideration proposed by Arifovic and Ledyard (2012), in which thefollowing utility function represents each agent’s preference.

ui (qi , Q−i ) = π i (qi , Q−i ) + β i π̄(Q) − γ i max{0, π̄(q) − π i (qi , Q−i )}(1)where π i (qi , Q−i ) is the profit for agent i when i produces qi and the other agentsP
produce Q−i = j̸=i qj in total. π̄(Q) is the average profit obtained by the agentsP
when the total output is Q = i qi . The β i ≥ 0 captures the degree to which i cares
about the average profit. Because β i ≥ 0, the higher the average profit is, the higher
the i’s utility is. The term max{0, π̄(q) − π i (qi , Q−i )} captures how much less profit
i makes compared with the average profit, and γ i ≥ 0 the degree to which i cares4about making less profit than the average. As we can see from the negative sign infront of the third term, the less profit i makes compared with the average, the lower
i’s utility is.
The profit for firm i is defined as follows.

p
π i (qi , Q−i ) = (a − b Q)qi − cqid

(2)√
In this paper, we assume {a, b, c, d} = {45, 3, 1, 3/2} because these are the values
used in the experiment of Offerman et al. (2002).
The existence of other-regarding preferences is often posited as an explanation forseemingly irrational behavior in experimental settings. For example, in the context ofCournot games, Iriş and Santos-Pinto (2014) have used other-regarding preferences
to explain several regular deviations from theoretical predictions.
As the superscript i on β i and γ i indicates, we assume agents are heterogeneous.
Andreoni and Miller (2002) report that the vast majority of behavior in a standard
dictator game can be explained by the presence of a range of heterogeneous otherregarding preferences. Blanco et al. (2011) find further evidence for a range of heterogeneous other-regarding preferences among the experimental population. However,they also find that these preferences are unstable or manifest differently in different
experimental settings.
Furthermore, as Bigoni and Fort (2013) show in their experiment where participants can choose which information to consider, not all the participants utilize thesame feedback information, which may be a result of participants belonging to different types, as Rassenti et al. (2000) suggest. We, therefore, assume that there are thefollowing five types of agents in the population:• Individualistic: β i = 0, γ i = 0
5• Cooperative: β i ∼ U (0, B), γ i = 0 where B is the upper bound• Retaliatory: β i = 0, γ i ∼ U (0, G) where G is the upper bound• Mixed: β i ∼ U (0, B),γ i ∼ U (0, G)• Flexible: A flexible agent starts as an Individualistic agent. At the start of eachturn, the agent updates their γti (βti ), entering the utility function in equation(a), with a fixed probability ρ as follows:

γti =

−i
i
π̄t−1
− πt−1ii−ii(G − γt−1) × ωf + γt−1if π̄t−1≥ πt−1
−i
π̄t−1

βti =

−i
i
− π̄t−1
πt−1
i
i
−i
i
(B − βt−1) × ωf + βt−1
if π̄t−1< πt−1iπt−1−iwhere π̄t−1
is the average profit of other agents in the previous turn, and ωfmeasures how sensitive the Flexible type agent is to others’ profits.

2.1

Dynamics of IEL

In IEL, each agent i, regardless of type, has a set of J actions (quantities), Sti with
|Sti | = J, to choose from in each period t. The initial set of actions, S0i , is randomly
generated from the set of possible actions for player i, S i , and the initial action, q0i ,is randomly selected from S0i . Learning in IEL takes the form of updating Sti at theend of each period by the following two steps: Experimentation and Replication.
Experimentation introduces new actions into Sti that otherwise might never havea chance to be tried. This ensures that a certain amount of diversity is maintainediin Sti . For each qj,t
∈ Sti (j = 1, ..., J), with probability µ, a new action, q, is selectedirandomly according to N (qj,t, σ|S i ) and replaces it. This means that the ‘new’ action6

ithat is replacing qj,t, on average, has the same value. µ and σ are parameters of themodel. Let Seti be the set of actions after experimentation.
Replication reinforces, after experimentation, actions that would have been good
choices in period t. This allows potentially better-performing actions to replaceiworse-performing actions. We assume that the performance of qj,t
∈ Seti is evaluatedaccording to the type-dependent utility function stated above, conditional on the ini|I i (xt )) =formation i received regarding the outcome in period t, I i (xt ). Let v i (qj,tiiui (qj,t, Q−i |I i (xt )) be the computed performance level of qj,tholding constant the out-

put levels of other firms. This measures the utility i that would have been obtainediin period t holding everything else constant. The element of the new
had i played qj,ti
i
, is chosen as follows. For each j = 1, · · · , J, select two members, qj,t+1
action set St+1iiand ql,t, uniformly randomly with replacement. Then,of Seti , qm,ti
qj,t+1=i qm,twith probabilityeiv i (qm,t|I i (xt ))iv i (qm,t|I i (xt ))ev i (q i |I i (xt ))
l,t

+e

(3)



i
 ql,totherwise.
i
Once St+1
is generated, in period t + 1, the action of agent i is chosen probabilis-

tically according toi iieλv (qk,t+1 |I (xt ))i
ψk,t+1 = PJi
λv i (qj,t+1|I i (xt ))
j=1 e

(4)

where λ ≥ 0 is a parameter of the model that governs the sensitivity of action choice
to their performance level.

7

2.2

Model fitting

To fix values of the parameters of the IEL model, we fit the model to the datareported in Offerman et al. (2002). In Offerman et al. (2002), participants play a 3player repeated Cournot game as specified above for 100 periods. Three treatments,
Q, Qq, and Qqπ, are considered by varying the information participants receive after
each play of the game (feedback after each period) across treatments. In Treatment
Q, participants are informed of the realized price and aggregate quantity (in addition
to their own quantity and profit). In Treatment Qq, in addition to the information
provided in treatment Q, participants are informed of the quantity chosen by each of
the other participants in the group. In Treatment Qqπ, participants are informed, in
addition to those provided in Qq, profits obtained by each of the other participantsin the group.
These variations in feedback after each period across the three treatments influence
the simulations of our model as follows.

In Treatment Q , agents only know the aggregate output. They cannot calculate
the average profit without information on the individual output level because thecost function is nonlinear. Therefore, we assume that all agents are Individualistic
agents in our simulation. This restriction means that we have three parameters—µ, σ, and λ—in our model.

We have conducted a grid search in the space ofµ ∈ {0.03, 0.07, 0.1, 0.3, 0.6}, σ ∈ {0.3, 0.5, 0.7, 2, 5, 10, 20}, and λ ∈ {6 × 10−4 , 8 ×
10−4 , · · · , 24 × 10−4 } (in step size of 2 × 10−4 ). For each combination of parametervalues, we run 30 simulations. Based on the measure of fit described in Appendix A,
we select µ = 0.03, σ = 2, and λ = 0.002 as the best parameter values.

8In Treatment Qq and Qqπ, in addition to the aggregate output, agents also know

the individual output of others in the group in Qq. In Qqπ, they are further informedof the profits of others in the group. Although it is hard to say how well agentscompute others’ profits from their outputs given the nonlinearity of the cost function,
it is possible. Therefore, we simulate the model with the full set of parameters. Wevary the proportions of the five types of agents in the population (fI , fC , fR , fF , andfM for Individualistic, Cooperative, Retaliatory, Flexible, and Mixed, respectively,
P
while maintaining x fx = 1 where x ∈ I, C, R, F, M ), B (the upper bound of β i ), G
(the upper bound of γ i ), ρ and ω f (probability of updating and sensibility to others’
profit for Flexible type), and fix the values of µ, σ, and λ to those we selected forTreatment Q.
Because of the large parameter space that needs to be investigated, we conduct
a grid search in two steps, as detailed in Appendix B: First, a coarse grid search,and second, a finer grid search for the subregion of the parameter space that resulted
in the highest fit under the coarse grid search. For each combination of parametervalues considered, 50 simulations are run. We find, in addition to µ = 0.03, σ = 2,
and λ = 0.002 that were fixed, the sets of parameter values reported in Table 1 to be
the best ones in fitting the model to the experimental data according to the criterion
reported in Appendix A.1
Comparing the best set of parameter values for treatment Qq and Qqπ reported
in Table 1 reveals an interesting difference. Our simulations show that Qq can be best
described by the interactions between Individualistic and Cooperative types, althoughmore than two-thirds of the agents are the former. In contrast, for Qqπ, there is
no Individualistic type. Instead, it is best described by the interactions between
1

There was another set of parameter values for Qq that was equally good, but we have used theone reported in Table 1 for our out-of-sample simulation to be reported in the next subsection. See
Appendix B.

9

Table 1: The best set of parameter values for the three treatments.

Q
Qq
Qqπ

µ
0.03
0.03
0.03

σ
2.0
2.0
2.0

λ
0.002
0.002
0.002

fI
1.0
0.8
0.0

fC
0.2
0.2

fR
0.0
0.4

fF fM
0.0 0.0
0.4 0.0

B
80
60

G
0
50

ρ
0
0.1

ωf
0
1.0

Cooperative, Retaliatory, and Flexible, with the Cooperative share being the lowest.
This difference in composition, particularly the prevalence of the Individualistic typein Treatment Qq, may reflect the difficulties participants (in the experiment) have incomputing others’ profits from the quantity information.
Figure 1 depicts the distributions of group outputs in three treatments from theexperiment of Offerman et al. (2002) (Top) and the IEL model (Bottom). The model
does a good job of capturing the distributions from the experiment in Treatment Q
that is centered around the Cournot–Nash equilibrium. While the model does notexactly replicate the distributions from the experiment in Treatments Qq and Qqπ, itdoes capture the qualitative differences across treatments; namely, in Treatment Qqπ,
collusive and competitive outcomes are much more frequently observed comparedwith treatments Qq and Q. Furthermore, in Treatment Qq, the output distribution
is skewed toward collusive ones compared with Treatment Q.
Thus, the simulation of the fitted model demonstrates the feedback effect. However, this is because the model is fitted to the data that demonstrate the effect.
We conduct out-of-sample simulations by varying the number of players in the next
subsection to generate a set of hypotheses to be tested using our experiments.

10Treatment Q

Results of Offerman et al. (2002)
Treatment Qq

Treatment QqπTreatment Q

Results of IEL simulation
Treatment Qq

Treatment QqπFigure 1: TOP: Distributions of group outputs in the three treatments of Offerman
et al. (2002). Regenerated from the original experimental data. BOTTOM: Distributions of 15,000 simulated group outputs (50 simulations by 100 periods in 3
treatments) from the best set of parameter values.

2.3

Out-of-sample simulations: Varying number of players

Figure 2 presents the results of simulations in which we vary the number of players,n ∈ {2, 3, 4}, in each of the three information treatments, Q, Qq, and Qqπ.2 The
n = 3 case is identical to that shown in Figure 1 but replicated to facilitate thecomparison.
Figure 2 shows that in Treatment Q, the outcomes are distributed around Cournot–
Nash regardless of the number of players. Thus, the model predicts no number effect2

Participants in our experiment (Section 3) could choose from integer values between 40 and 170
in 2-player treatments, 40 and 113 in 3-player treatments, and 40 and 85 in 4-player treatments,
inclusive, as a quantity in each period. In simulating the model, we introduced the same constraints
in quantities.

11

Treatment Q

n=2
Treatment Qq

Treatment QqπTreatment Q

n=3
Treatment Qq

Treatment QqπTreatment Q

n=4
Treatment Qq

Treatment QqπFigure 2: Distribution of the 45,000 simulated group outputs (50 simulations by 100
periods in 9 treatments) from the best set of parameter values.

12under Treatment Q.3 In Treatment Qqπ, on the other hand, when n = 2, collusive andcompetitive outcomes are more likely to be observed than Cournot–Nash outcomes.
For n = 3, the collusive outcome becomes less likely (compared with n = 2), while theCournot–Nash and collusive outcomes instead become more prevalent. The output
distribution is centered around the Cournot–Nash for n = 4, thereby demonstrating
the number effect.
These differences in the simulated outcomes between treatment Q and Qqπ constitute our main hypotheses to be tested in the experiment.
Hypothesis 1 The feedback effects (the more detailed the feedback is, the more competitive outcomes become) exist. Furthermore, when the number of players is 2 or 3,
it is not just that there are more competitive outcomes, but also that we observe morecollusive outcomes, with more detailed than aggregate feedback.
Hypothesis 2 The number effect (the larger the number of players, the less collusive
the outcome) arises with detailed feedback (Treatment Qqπ) but not with aggregate
feedback (Treatment Q).
Note that while Horstmann et al. (2018) reports the number effect with detailed
feedback, which is consistent with our Hypothesis 2, Huck et al. (2004b) details
the number effect with aggregate feedback, which is inconsistent with our second
hypothesis. However, there are several differences in setting that make comparison of
our simulation results and these existing experimental results difficult. First, and mostimportantly, these two experiments are based on linear demand and cost functions(in the case of Horstmann et al. (2018) with zero cost), while our simulation is basedon nonlinear cost and demand functions. Second, there are 25 and 60 periods in
3

However, the relative locations of the collusive, Cournot–Nash, and competitive outcomes change
depending on the number of players. Namely, Cournot–Nash is closer to collusive than to competitivein n = 2, while the opposite is the case for n = 4.

13the experiments in Huck et al. (2004b) and Horstmann et al. (2018), respectively,
while our simulation is based on 100 periods. The results in Huck et al. (2004b),
which are based on 25 periods, therefore can arise because of insufficient learning byparticipants. Because we are unaware of experimental analysis concerning the number
effect in the Cournot game using the nonlinear demand and cost functions consideredby Offerman et al. (2002) under varying feedback conditions, we have conducted a
new set of experiments of our own to test these hypotheses.

3

Experiment Design

Table 2 presents the treatments in our two-by-three design: we vary feedback information (Partial vs. Full) and the number of players in a market (n ∈ {2, 3, 4}). In
Partial-Info treatments, the players received feedback information on the total quantity, the price, and their own individual output and profit at the end of each period(thus, identical to treatment Q above). In Full-Info treatments, the players received
additional information on the individual outputs and profits of others in the samegroup (thus, identical to treatment Qqπ). When they made decisions, they could
always review the full history of the individual outputs in a graph and a table (seeAppendix D.3). Both information treatments were run in 2-, 3-, and 4-player Cournot
games.
We repeated the game for 100 periods. In each period, firms must decide simultaneously how much to produce. They can only choose integer values between 40 and
170 in 2-player treatments, 40 and 113 in 3-player treatments, and 40 and 85 in 4player treatments, inclusive. Note all firms face the same cost and demand functions.
All firms in the industry receive the same price for each commodity produced. Boththe relationship between own production and costs and aggregate production and14

Table 2: Summary of treatments.

Number of players
2 players
3 players
4 players

Feedback information
Partial-Info (πi ,Q,P ) Full-Info(πi ,Q,P ,qj ,πj )Partial2
Full2
Partial3
Full3
Partial4
Full4

Notes: All the feedback information concerns the previous period. Subscript i refers
to the firm itself; subscript j concern other firms in the industry; π denotes profit; Q
denotes aggregate output; and P denotes price.

price were illustrated in interactive graphs. Players needed to correctly answer questions about the cost and demand functions before proceeding with the experiment.
The introduction, along with the interactive graphs, were made available throughout
the experiment (see Appendix D.2 for an example of the on-screen instruction.)
Unlike Offerman et al. (2002) but as in Bigoni and Fort (2013), we provided anon-screen profit calculator to help players figure out their best responses. They could
enter their own hypothetical output and the aggregate output of others in the market
to get their own hypothetical profit. Players used the calculator on average threetimes per period, with 72 percent of players using it more than once per period.
During the experiment, players earned experimental points according to the profits
they obtained. Participants were rewarded based on the sum of the points earned
during 100 periods. At the end of the experiment, the experimental points were
exchanged for Canadian dollars at an exchange rate of 5,000 experimental points =1 Canadian dollar.
The experiment is programmed in oTree (Chen et al., 2016). The Cournot gamewas preceded by a part of the advanced version of Raven’s Progressive Matrix test
(Raven, 1998). For the Raven test, we selected 16 questions from the original test15

Table 3: Summary of the number of groups and average earnings (in Canadian dollars).
Treatment
Partial2
Partial3
Partial4
Full2
Full3
Full4

Num. of Groups
6
6
6
9
7
6

Cournot game Raven test
$20.7
$2.7
$14.3
$2.3
$11.0
$2.5
$19.4
$2.6
$12.7
$3.1
$9.8
$2.5

Total earning
$30.4
$23.6
$20.5
$29.0
$22.8
$19.3

Notes: Total earnings equal to the sum of the earnings in Cournot game, Raven test,
and $7 participation fee.

to be finished within 10 minutes.4 In each question, players analyzed a geometric pattern and identified a missing part to complete the series. For each question
answered correctly, players were paid 0.25 Canadian dollars (see Appendix D.1 forthe Raven test instructions). The Raven test measures “fluid intelligence”; that is,“the capacity to think logically, analyze and solve novel problems, independent ofbackground knowledge” (Mullainathan and Shafir, 2013, p.48). Recently, Proto et al.
(2019) reported that participants with higher scores on the Raven test are more likely
to achieve and sustain cooperation in an infinitely repeated prisoners’ dilemma gamethan those with lower scores. Because our experiment on Cournot games involves
similar trade-offs between short-run temptation and long-run gain, we decided togather this information to control its effect, if any, when analyzing the data.
Laboratory experiments were conducted between September 2019 and January
2020. Participants were recruited at Simon Fraser University. A total of 117 subjects
participated. An experiment lasted between 1.5 and 2 hours. Average earnings were23.5 Canadian dollars (see Table 3 for the number of groups and earnings under
different treatments). The number of participants differs across treatments because
4

The original test consists of 48 questions to be answered in 30 to 40 minutes. We selected 1 inevery 3 questions while retaining the order of questions.

16

of variation in the rate of show up or appearance.

4

Results of the Experiment

Figure 3 shows the distribution of group outputs observed for each of our six treatments. In the three Partial-Info treatments, total group outputs are distributed
around the Cournot–Nash equilibrium, just as we observed in the simulations of theextended IEL shown in Figure 2. In contrast, in the three Full-Info treatments, morecompetitive group outputs are observed compared with the Partial-Info treatment
for each group size. While the frequency of collusive outcomes is low for the Full2
treatment, it is much higher in Full3 than in the Full4 treatment.
Let us define “collusive outcomes” to be outcomes such that Qg ≤ 12 (QN ash + QColl )and “competitive outcomes” to be outcomes such that Qg ≥ 21 (QN ash + QComp ) where
Qg is the total output of firms, and QN ash , QComp , and QColl are the corresponding
Nash, competitive, and collusive levels of total outputs, respectively. We also define
remaining outcomes that are around QN ash to be “Nash outcomes.”
Figure 4 shows the distribution of the relative frequencies of “Nash”, “Collusive”,
and “Competitive” outcomes for each group across the various treatments. In eachplot, filled marks represent groups in the Partial-Info treatment while unfilled marks
represent the Full-Info treatment. The point at the very top of the triangle corresponds to (“N ash′′ , “Collusive′′ , “Competitive′′ ) = (100, 0, 0); that is, the group’s
output was Nash outcomes 100% of the time. The left and the right bottom apexes of
the triangle corresponds to (“N ash′′ , “Collusive′′ , “Competitive′′ ) = (0, 100, 0) and(0, 0, 100), respectively. That is, the group’s output was collusive or competitive outcomes 100% of the time. The p values beneath the plot are for the Mann–Whitney
test (two-tailed) comparing the frequencies of collusive and competitive outcomes17

Partial2

Full2

Partial3

Full3

Partial4

Full4

Figure 3: Distribution of group outputs across the six treatments.

18

20

llu
Co

sh

60

60

Na

siv

e

80

40

40

80

20

100

0
0

n=2
0

n=3
0

100

20

20

20

40
60
Competitive

80

100

p=0.23 (Collusive)p=0.81 (Competitive)

siv
llu
Co

siv
llu
Co

siv
llu
Co

0
0

60

60

40

40

80

20

20

100

100

100

80

40

sh

60

80

20

100

Na

40

sh

sh

80

100

e

80

e

e

0

60

40

80

n=4

Na

Na

60

60

40
60
Competitive

100

20

80

40

20

0
0

20

40
60
Competitive

80

100

p=0.94 (Collusive)p=0.01 (Competitive)
0

0
0

20

40
60
Competitive

80

100

p=0.56 (Collusive)2
p=0.02 (Competitive)

100

Figure 4: Distribution of the frequencies of80 “Nash”, “Collusive”, and “Competi40
tive” outcomes. Filled (unfilled) marks
represent
partial (full) info treatment. The60
p values beneath the plot are from 60a Mann–Whitney test (two-tailed) comparing the40
frequencies of collusive and competitive
outcomes between the Partial and Full-Info80
20
treatments.
100
Co

sh

llu

Na

siv

e

20

0

0

20

40
60
Competitive

80

100

between the Partial-Info and Full-Info treatments.
For n = 3 (middle panel), we can see2 that while the filled dots are mostly dis1

tributed around the top apex (which means the outcomes are mostly “Nash” in mostof the groups), the unfilled dots are distributed either toward the bottom left or rightapexes of the triangle (which means that “Collusive” or “Competitive” outcomes are
more frequently observed than “Nash”). In particular, we observe that the frequencies of competitive outcomes are significantly higher under the Full-Info than thePartial-Info treatment for n ∈ {3, 4}. We therefore make the following observation.
Observation 1 The feedback effect exists in the experiment. Full-Info treatments result in competitive outcomes significantly more frequently than the Partial-Info treatments for n ∈ {3, 4}.
Observation 1 is consistent with the first part of Hypothesis 1. However, thesecond part of Hypothesis 1 (about collusive outcomes) is not confirmed in our data.

19

Table 4: Tacit collusion.
Dependent variable: Tacit Collusion

FID
Agent Number
Agent Number times FID
Constant
Group FE
Observations
standard deviations in parentheses∗
p < 0.10, ∗∗ p < 0.05, ∗∗∗ p < 0.01

(1)
none
-1.231∗∗∗(0.013)-0.666∗∗∗(0.002)0.313∗∗∗(0.004)5.352∗∗∗(0.005)
Yes
4000

Notes: FID is a dummy variable: 1 if under Full-Info treatments, 0 otherwise.

We now investigate our Hypothesis 2 about the number effect. We do not, however, observe any significant differences across the three group sizes in terms of thefrequency of collusive outcomes, neither in the Partial-Info (p=0.40, Kruskal–Wallistest) nor Full-Info treatment (p=0.71, Kruskal–Wallis test).
To check the number effect more closely, following Horstmann et al. (2018), Engel(2007), and Suetens and Potters (2007), we measure the degree of tacit collusion by
N ash

pt −p
5
ϕE
We then run a group fixed effectst ≡ pJP M −pN ash where pt is the market price.

regression to test the impact of the number of agents on this measure.6 Table 4 shows
the result.
Note that the degree of tacit collusion is lower in the Full-Info treatment as demonstrated by the negative and significant coefficient for the Full-Info treatment dummy(FID). This is consistent with our analyses above on the feedback effect summarized
5

Replacing pN ash by pW alras yields similar results.
We use group fixed effects regression because there is substantial heterogeneity in terms of the
distribution of outcomes.
620

in Observation 1. The negative and significant coefficient of Agent Number in Table 4
indicates a lower degree of collusion for larger markets, demonstrating the existenceof the number effect in the Partial-Info treatment which is contrary to our Hypothesis 2. In fact, the positive and significant coefficient of Agent Number × FID showsthat the number effect is weaker under the Full-Info treatment compared with thePartial-Info treatment. We, therefore, make the following observation.
Observation 2 The number effect exists in both the Full and Partial-Info treatments.
The number effect is weaker, however, under the Full-Info treatment, contrary to
Hypothesis 2.

4.1

Group behavior

Why are collusive outcomes rare in the Full2 treatment compared to its n = 3 counterpart, and unlike what the simulated IEL predicts? One possible reason is thedifferences in average cognitive ability among participants across treatments. While
we have randomized participants across treatments, it is possible that the distributionof participants in terms of their cognitive ability was not the same across the treatments. Because it has been shown that participants with higher cognitive ability tend
to achieve a higher degree of cooperation (Proto et al., 2019), variation in cognitive
ability across sessions (or more importantly, across treatments) can result in participants achieving more cooperative (collusive) outcomes in some sessions (treatments)
than in others.
As a first check, we run the following simple linear regression:

Average Outputij = βAverage Raven scoreij + T reatmentj + ϵij(5)where Average Outputij and Average Raven scoreij are the average individual out21

Table 5: Group output and the average score of the Raven test.
Dependent variable: Average Individual Output(1)
Full Sample
Average Raven Score-1.635∗(0.873)Treatment FE
Yes
Observations
40
standard deviations in parentheses∗
p < 0.10, ∗∗ p < 0.05, ∗∗∗ p < 0.01

(2)Partial-Info-0.461(0.711)
Yes
18(3)
Full-Info-2.996∗(1.540)Yes
22

Notes: Column 1 shows the result from the full sample. Column 2 uses the subsampleof three Partial-Info treatments. Column 3 uses the subsample of three Full-Info treatments.

puts and the average score of the Raven test for participants in group i with treatment
j, respectively, and T reatmentj is the treatment fixed effect.
Table 5 details the results. The negative coefficient of Average Raven Score in
Column 1 means that groups with a higher average score on the Raven test tend to
produce less and are more likely to reach the collusive outcome. On average, a onepoint increase in the average Raven score leads to a 1.63 decrease in average individual
output. This finding is consistent with Proto et al. (2019). In our experiment, thissignificant correlation between the Raven test score and the average output is driven
by the groups in the Full-Info treatments, as shown in Column 3. This correlation is
not significant in groups under the Partial-Info treatments.
Figure 5 compares the average group Raven scores from all the treatments. The
average Raven score for the Full3 treatment is much higher than in the other treatments. This might indeed be the reason we observe more collusive outcomes in the
Full3 treatment than in the Full2 treatment, contrary to what our simulation predicted.
To better understand the mechanism of the observed feedback effect in our exper-

22

Figure 5: The average group Raven score across treatments.
iment, we analyze the data at the individual level in the next subsection.

4.2

Individual behavior

At the individual level, participants exhibit a large degree of heterogeneity. To quantify their behavior, we divide the output levels into three categories: Individualistic,
Cooperative, and Retaliatory. We define an individual output decision to be Cooperative if the output is 10 units lower than the best response output level basedon the total output of others in the previous round; Retaliatory if the output is 10
units higher than the best response output level based on the total output of othersin the previous round; and Individualistic otherwise. In addition to the above three
types, we also define an action to be Flexible if the agent increases (or decreases)
their output by at least 10 units, immediately following an increase (or decrease) in
the aggregate output of the others in the group in the previous round.

23

We use these definitions because the total output of others is available in alltreatments, and participants can easily look for their profit-maximizing output level,
conditional on others’ total outputs, by using the calculators provided in the experiment.7 In fact, an average agent uses the profit calculator 313 times throughout theexperiment.8
Figure 6 illustrates the composition of the various decision types for the two
information treatments. The participants under Full-Info treatments make more Retaliatory decisions and fewer Individualistic decisions than those under Partial-Infotreatments. Given aggregate output levels are provided to the participants in all treatments, the significantly higher percentage of Retaliatory decisions under the Full-Infotreatments must be driven by the extra information (namely, individual quantities
and profits) provided.
To understand the effect of various action types on average output, the following
regression is conducted:

Average Outputij = β1 Cooperativeij + β2 Retaliatory ij + β3 F lexibleij + Groupj + ϵij(6)where Average Outputij is the average output per round for individual i in the groupj; Cooperativeij , Retaliatory ij , and F lexibleij are the number of Cooperative, Retaliatory, and Flexible decisions (across 100 periods), respectively, for individual i ingroup j; Groupj is the group fixed effect; and ϵij is the error term.9
Figure 7 provides the estimated coefficients of the various decision types. Not
surprisingly, an increase in the number of Cooperative decisions significantly decreases
7

Although the information available is the total group output, participants can easily deduct
their own output from the total group output to compute the total output of other group members.
8
81 percent of participants use it at least 50 times throughout the experiment.
9
Individualistic decisions are dropped given collinearity.

24Figure 6: Number of various types of individual action by information type.
average individual output, while an increase in the number of Retaliatory decisions
significantly increases average individual output. Flexible decisions have no significant
effect on average individual output. Importantly, these effects are similar across thedifferent information treatments and not statistically significantly different at the 5%
level (see, Table 6, in Appendix C), suggesting that the different distributions of
outputs across treatments we saw in Figure 3 are not driven by the relative strengthsof the various types of actions.

5

Summary and Discussion

In this paper, we extend the individual evolutionary learning model by incorporatingother-regarding considerations and apply the model to some Cournot games. Based on
the model fitted to the experimental data of a repeated 3-player Cournot game (withnonlinear cost and demand functions), we make out-of-sample predictions regarding
25

Figure 7: The estimated coefficients of various decision types for the two treatments.
the feedback and number effects and test them using the data gathered via newly
conducted experiments. The prediction regarding the feedback effect is partiallyconfirmed. Namely, we observe it in the 3- and 4-player games but not in the 2-player
game. The prediction regarding the number effect is also partially confirmed in thatwhile the model predicts the number effect to be observed with detailed feedback,and not under aggregate feedback, the effect is observed with both types of feedback.
We investigate the role of cognitive ability and individual behavior to better understand the experimental outcomes. The data suggest that the differences in the
average cognitive ability of participants across treatments may account for a higher
frequency of collusive outcomes observed in the 3-player game with detailed feedback than the 2-player game with detailed feedback, although the model predicts the
opposite.
The individual level analyses reveal that participants are making Retaliatory de-26cisions (that is to produce much more than the myopic best response level) much
more frequently with detailed feedback than with aggregate feedback, which resultsin more competitive outcomes in the former than the latter.
We believe the approach taken in the paper—namely, to calibrate the parameter values of a boundedly rational behavioral model by fitting the model to an
existing experimental data set and then testing the out-of-sample predictions of thecalibrated model with newly corrected data—can be a powerful tool for advancing research in behavioral and experimental economics. A well-known problem in working
with boundedly rational behavioral models is the so-called “wilderness” of bounded
rationality (Sims, 1980; Hommes, 2013); that is, because of the large degree of freedom modelers have in constructing these models, systematic investigation becomes
extremely difficult. By restricting our attention to those models with a solid empirical (experimental) foundation, and further testing their out-of-sample prediction
with newly gathered data, we can narrow down the set of models to advance one’s
investigation.

References
Andreoni, J. and J. Miller (2002): “Giving according to GARP: An experimental test of the consistency of preferences for altruism,” Econometrica, 70, 737–753.
Apesteguia, J., S. Huck, J. Oechssler, and S. Weidenholzer (2010): “Imitation and the evolution of Walrasian behavior: Theoretically fragile but behaviorally robust,” Journal of Economic Theory, 145, 1603–1617.
Arifovic, J. (1994): “Genetic algorithm learning and the cobweb model,” Journalof Economic Dynamics and Control, 18, 3–28.

27

Arifovic, J. and J. Ledyard (2012): “Individual evolutionary learning, otherregarding preferences, and the voluntary contributions mechanism,” Journal of
Public Economics, 96, 808–823.
Bigoni, M. and M. Fort (2013): “Information and learning in oligopoly: An
experiment,” Games and Economic Behavior, 81, 192–214.
Blanco, M., D. Engelmann, and H. T. Normann (2011): “A within-subject
analysis of other-regarding preferences,” Games and Economic Behavior, 72, 321–
328.
Bosch-Domènech, A. and N. J. Vriend (2003): “Imitation of Successful Behaviour in Cournot Market,” Economic Journal, 113, 495–524.
Chen, D. L., M. Schonger, and C. Wickens (2016): “oTree – An open-source
platform for laboratory, online, and field experiments,” Journal of Behavioral andExperimental Finance, 9, 88–97.
Engel, C. (2007): “How much collusion? A meta-analysis of oligopoly experiments,”
Journal of Competition Law and Economics, 4, 491–549.
Friedman, D., S. Huck, R. Oprea, and S. Weidenholzer (2015): “From imitation to collusion: Long-run learning in a low-information environment,” Journalof Economic Theory, 155, 185–205.
Hommes, C. (2013): Behavioral Rationality and Heterogeneous Expectations in Complex Economic Systems, Cambridge, UK: Cambridge University Press.
Horstmann, N., J. Krämer, and D. Schnurr (2018): “Number effects and tacit
collusion in experimental oligopolies,” Journal of Industrial Economics, LXVI, 650–
700.
28

Huck, S., H.-T. Normann, and J. Oechssler (1999): “Learning in Cournot
Oligopoly - An Experiment,” Economic Journal, 109, C80–C95.
——— (2000): “Does information about competitors’ actions increase or decrease
competition in experimental oligopoly markets?” International Journal of Industrial Organization, 18, 39–57.
——— (2003): “Zero-knowledge cooperation in dilemma games,” Journal of Theoretical Biology, 220, 47–54.
——— (2004a): “Through trial and error to collusion,” International Economic Review, 45, 205–224.
——— (2004b): “Two are few and four are many: number effects in experimental
oligopolies,” Journal of Economic Behavior and Organization, 53, 435–446.
Iriş, D. and L. Santos-Pinto (2014): “Experimental Cournot oligopoly and inequity aversion,” Theory and Decision, 76, 31–45.
Mullainathan, S. and E. Shafir (2013): Scarcity: Why Having Too Little Means
So Much, New York, NY: Times Books, Henry Holt and Company, LLC.
Oechssler, J., A. Roomets, and S. Roth (2016): “From imitation to collusion:a replication,” Journal of Economic Science Association, 2, 13–21.
Offerman, T., J. Potters, and J. Sonnemans (2002): “Imitation and belief
learning in an oligopoly experiment,” Review of Economic Studies, 69, 973–997.
Proto, E., A. Rustichini, and A. Sofianos (2019): “Intelligence, Personality and Gains from Cooperation in Repeated Interactions,” Journal of Political
Economy, 127, 1351–1390.
29

Rassenti, S., S. S. Reynolds, V. L. Smith, and F. Szidarovsky (2000):“Adaptation and convergence of behavior in repeated experimental Counot games,”
Journal of Economic Behavior and Organization, 41, 117–146.
Raven, J. C. (1998): Raven’s Advanced Progressive Matrices (APM), San Antonio,
TX: Pearson, 2003 ed.
Selten, R. (1973): “A simple model of imperfect competition, where 4 are few and
6 are many,” International Journal of Game Theory, 2, 141–201.
Sims, C. A. (1980): “Macroeconomics and Reality,” Econometrica, 48, 1–48.
Suetens, S. and J. Potters (2007): “Bertrand colludes more than Cournot,”Experimental Economics, 10, 71–77.
Vallée, T. and M. Yildizoğlu (2009): “Convergence in the finite Cournot
oligopoly with social and individual learning,” Journal of Economic Behavior andOrganization, 72, 670–690.
Vega-Redondo, F. (1997): “The Evolution of Walrasian Behavior,” Econometrica,
65, 375–384.
Vriend, N. J. (2000): “An illustration of the essential difference between individual
and social learning, and its consequences for computational analyses,” Journal of
Economic Dynamics and Control, 24, 1–19.

30

A

Measure of Model Fit to the Experimental Data

The model’s fit to the experimental data is measured based on the distribution ofgroup-level total outputs. The unit of observations is the level of aggregate output
P
in one period (Qt = i qti ). Let Fsim,n be the empirical distribution function basedon the n observations generated by the simulation, and Fexpr,m be the empirical
distribution function based on the m observations from the experiment.
nFsim,n (x) =1X
I[−∞,x] (Xi )
n i=1
m

1 X
I[−∞,x] (Xi )Fexpr,m (x) =m i=1
where I[−∞,x] (Xi ) is the indicator function, equal to 1 if Xi ≤ x and equal to 0 otherwise, with Xi ∈ {100, 105, 110, · · · , 395, 400}. Distance between the two distributions
is measured by
Dn,m =

X(Fsim,n (x) − Fexpr,m (x))6 .
x

We also considered two other measures of distance

D2n,m =

X
(Fsim,n (x) − Fexpr,m (x))2xD3n,m = supx |Fsim,n (x) − Fexpr,m (x)|but the results are similar. These results are available from the authors upon request.
Note that D3n,m places all the weight on minimizing the largest difference between
the two distributions while sacrificing the distance in other parts. Dn,m and D2n,m ,in contrast, take distances for all the bins into account.

31

B

Two-step Grid Search

B.1

Coarse grid search

Grid searching is done for all possible combinations of shares of the five types in the
population (fI , fC , fR , fF , and fM for Individualistic, Cooperative, Retaliatory, FlexP
ible, and Mixed, respectively) such that fx ∈ {0, 0.2, 0.4, 0.6, 0.8, 1} while x fx = 1
where x ∈ I, C, R, F, M . For the other parameters, we consider the space of B ∈{0, 20, 40, 60}. G ∈ {0, 20, 40, 60}, ρ ∈ {0.1, 0.3, 0.7, 0.9}, and ω f ∈ {0.1, 0.3, 0.7, 1}.
The same exercise is carried out across treatments.

B.2

Finer grid search

B.2.1

Treatment Qq

The coarse grid search led to restricting our attention to the following subspace. For
the shares of the five types fI ∈ {0.5, 0.6, 0.7, 0.8, 0.9}, fC ∈ {0, 0.1, 0.2, 0.3, 0.4},
fR ∈ {0, 0.1, 0.2, 0.3} fF ∈ {0, 0.1, 0.2, 0.3}, and fM = 1 − (fI + fC + fR + fF ). For theother parameters, B ∈ {30, 40, 50, 60, 80}, G ∈ {0, 20, 40, 60}, ρ ∈ {0.1, 0.3, 0.7, 0.9}
and ω f ∈ {0.1, 0.3, 0.7, 1}.
As a result, we obtain the following two sets of best parameter values.

B.2.2

fI

fC

fR

fF

fM

B

G

ρ

ωf

Qq (a)

0.8

0.2

0.0

0.0

0.0

80

0

0

0

Qq (b)

0.7

0.3

0.0

0.0

0.0

30

0

0

0

Treatment Qqπ

The coarse grid search led to restricting our attention to the following subspace. For
the shares of the five types fI ∈ {0.0, 0.1, 0.2, 0.3, 0.4}, fC ∈ {0.0, 0.2, 0.3, 0.4, 0.5},32

fR ∈ {0.0, 0.3, 0.4, 0.5, 0.6} fF ∈ {0.0, 0.2, 0.3, 0.4, 0.5, 0.6}, and fM = 1 − (fI + fC +fR +fF ). For the other parameters, B ∈ {20, 30, 40, 50, 60, 80}, G ∈ {20, 30, 40, 50, 60},
ρ ∈ {0.1, 0.3, 0.7, 0.9} and ω f ∈ {0.1, 0.3, 0.7, 1}.
As a result, we obtain the following set of best parameter values.

Qqπ

fI

fC

fR

fF

fM

B

G

ρ

ωf

0.0

0.2

0.4

0.4

0.0

60

50

0.1

1.0

33

C

Supplementary Materials

Table 6: Effects of decision types on individual output.
Dependent variable: Average Individual Output(1)
Partial-Info
Cooperative-.292∗∗∗(.0298)Retaliatory.231∗∗∗(.0244)
Flexible.223(.171)
Cooperative × FID(2)
Full-Info-.26∗∗∗(.0378).234∗∗∗(.0349)-.261(.203)

Retaliatory × FID
Flexible × FID
FID
70.8∗∗∗(1.32)
Yes
54

Constant
Group FE
Observations
standard deviations in parentheses∗
p < 0.10, ∗∗ p < 0.05, ∗∗∗ p < 0.01

99.8∗∗∗(1.55)
Yes
63(3)
Full Sample-.292∗∗∗(.0284).231∗∗∗(.0233).223(.163).0323(.0487).00321(.0433)-.484∗(.268)-9.59∗∗(3.63)
70.8∗∗∗(1.26)
Yes
117

Notes: Column 1 shows the statistics from the Partial-Info treatments. Column 2 shows
the statistics from the Partial-Info treatments. Column 3 shows the statistics from theFull sample, where FID is a dummy variable: 1 if it is under Full-Info treatments, 0
otherwise.

34

D

Experiment Introduction

D.1

Raven test

We do not display the decision screen for the Raven test because of issues withcopyright.

D.2

Cournot introduction

Here we show the on-screen Introduction for the 2-agent Full-Info treatment as an
example. The players in the Partial-Info treatments read the same introduction but
do not see anything about historical information.

35

36

37

D.3

Cournot decision page

Once players answer the question correctly, they will see the decision page, where they
can move the slide bar to calculate the hypothetical payoff. The same instructions
are available at the bottom.

In the Full-Info treatments (but not the Partial-Info treatments), they can also
click the “Show History” button to see the outputs of the other players.

38

D.4

Cournot result page

Once everyone in the group has made their decisions, they see the result page. The
players in the Partial-Info treatment do not see the information of other firms.

39