ISSN (Print) 0473-453X

Discussion Paper No. 1131

ISSN (Online) 2435-0982

MARKET CONCENTRATION
AND INCENTIVESTO COLLUDE
IN COURNOT OLIGOPOLY
EXPERIMENTS
Nobuyuki Hanaki
Aidas Masiliūnas

April 2021

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

Market Concentration and Incentives to Collude in
Cournot Oligopoly Experiments∗
Nobuyuki Hanaki†

Aidas Masiliūnas‡

April 19, 2021

Abstract
Multiple Cournot oligopoly experiments found more collusive behavior in markets
with fewer firms (Huck et al., 2004; Horstmann et al., 2018). This result could be
explained by a higher difficulty to coordinate or by lower incentives to collude inmarkets with more firms. We show that the Quantal Response Equilibrium can
explain how the change in incentives alone could result in more collusive output insmaller markets. We propose a new method to manipulate the group size while
keeping constant the locations of key outcomes, payoffs at these outcomes and the
incentives to collude. Experiments using this normalized payoff function find thatthe number of firms has no direct effect on the average output or profit. We conclude
that higher rates of aggregate collusion in markets with fewer firms are driven by
the changes in incentives or focality rather than purely the number of firms. Thesefindings imply that antitrust policies aimed at preventing collusion should focus on
incentives rather than on the market concentration.
Keywords: experiment, oligopoly, collusion, group size, Quantal Response Equilibrium
JEL classification: C72 C91 D43 D83∗
We would like to thank participants at the Second Wuhan Cherry Blossom Workshop in Experimental Economics, 2020 ESA Global Virtual Conference and Vilnius Winter Meeting 2020. We gratefully acknowledge financial support from the Aix-Marseille School of Economics, Joint Usage/ResearchCenter at ISER, Osaka University, Japan Society for the Promotion of Science (18K19954, 20H05631),
and the French government-managed l’Agence Nationale de la Recherche under Investissements d’Avenir
U CAJEDI (ANR-15-IDEX-01).
†
Institute of Social and Economic Research, Osaka University, 6-1 Mihogaoka, Ibaraki, Osaka 567-0047,
Japan. Email: nobuyuki.hanaki@iser.osaka-u.ac.jp‡
Global Asia Institute, National University of Singapore, 10 Lower Kent Ridge Road, Singapore 119076.
Email: aidas.masiliunas@gmail.com

1

1

Introduction

Cournot oligopoly experiments consistently find more tacit collusion in markets with fewer
competing firms, a result termed the “number effect” (see Horstmann et al., 2018, for a
recent overview and meta-analysis). The mechanism behind this effect has never beenstudied. If it is driven by an increased difficulty to coordinate with many participants,
collusion could be prevented by regulating the market concentration. On the other hand,the number effect could be driven by changes in the incentives to collude, which would
suggest focusing on the gains from collusion instead of market concentration. We showthat both the Friedman index of collusion and the Quantal Response Equilibrium (QRE)
predict more collusive behaviour in smaller markets in the standard setting considered inprevious experiments. We then introduce a method to manipulate the market size while
keeping constant the incentives to collude and the locations of the collusive, competitive,
and Nash equilibrium outcomes. Our experiments replicate the standard result of morecollusion in smaller markets using the standard treatments, but find no number effect when
the incentives are held constant. We conclude that if the incentives are held constant, thenumber of competing firms does not affect the collusion rates.
Understanding how the number of interacting parties affects collusion rates is importantfor multiple reasons. First, it is important to know which behavioral regularities found insmall scale experiments are likely to generalize to real life situations with a large numberof participants. Previous research identified systematic differences between small and large
groups in various games (Diederich et al., 2016, Choi et al., 2020, Hommes et al., 2020).
We extend this line of research by studying both the consequences and the mechanism
through which the group size affects choices. Second, it is useful to understand which
factors influence collusion rates to identify the potentially collusive markets and assess theconsequences of antitrust policies (Levenstein and Suslow, 2006). Historically, regulators2

focused on market concentration, measured either by the number of significant competitors
in the market or by the Herfindahl-Hirschman Index.1 Recently, the focus shifted towards
the economic factors that affect the incentives to collude (Shapiro, 2010) and it has been
debated whether an indicator of upward pricing pressure should replace the concentrationbased methods (Farrell and Shapiro, 2010; Jaffe and Weyl, 2013). Whether mergers should
be screened based on market concentration or on the incentives to increase prices depends
on the nature of the number effect. We show that the results from the previous Cournot
experiments (e.g. Huck et al., 2004) could be driven either by the reduced market size orby the accompanying increase in the incentives to collude. When we manipulate only themarket size, aggregate collusion rates do not change, suggesting that market concentration
alone does not have a large effect on the collusion rates and regulators might be right tofocus on the incentives rather than market concentration.
We find that the market size has no direct effect on the average output or profits, but it
affects the distribution of output. Specifically, we find higher variance in smaller markets,
therefore more choices are classified as collusive – but also as competitive. This result could
be explained by the difference in how the market size affects the variance of the payoffrelevant statistic, either the mean or the sum of output. The variance of average output
produced by other firms decreases in larger markets, therefore when payoffs are a functionof the average output, extremely high or low output is unlikely to maximize profits. The
variance of total output does not decrease in larger markets, therefore this effect does
not apply to standard Cournot treatments. We formalize the intuition of the varianceeffect using a modification of the QRE, in which choice probabilities are derived from thelikelihood that each action is the best response. We show that this solution concept fits
data better than QRE and can explain why variance is higher in smaller markets.
1

See Davies et al. (2011), Farrell and Shapiro (1990) and U.S. Department of Justice & Federal Trade Commission, Horizontal Merger Guidelines (2010), available at
http://www.justice.gov/atr/public/guidelines/hmg-2010.pdf

3Our study extends the findings from the previous literature that studied the relationship between collusion and market size in Cournot oligopoly experiments. Fouraker andSiegel (1963) found slightly more collusive behavior in Cournot duopolies than in triopolies.
Huck et al. (2004) found that average output is more collusive and more markets are classified as collusive in duopolies than quadropolies. Roux and Thöni (2015) replicated theseresults in baseline treatments without punishment. Waichman et al. (2014) found a higher
frequency of collusion counts in duopolies than in triopolies when communication was notpossible, although the effect was not significant in the manager sample. Similarly, Fonseca
et al. (2018) found an increase in the collusiveness of output when the number of firms
decreases from six to four and from four to two, in treatments without communication. In
quadropolies without a forward market, output was more competitive than the equilibrium
prediction, but more collusive than equilibrium in duopolies (Le Coq and Orzen, 2006).
Van Koten and Ortmann (2013) also ran baseline treatments without a forward market
and replicated the result of competitive output in quadropolies but output in triopolies andduopolies was not significantly different from the equilibrium prediction. Horstmann et al.
(2018) found that duopolies were more collusive than triopolies, which in turn were morecollusive than quadropolies, as measured using two collusion indexes. Horstmann et al.
(2018) also performed a meta-analysis of previously published results and found higher
rates of tacit collusion in duopolies than in triopolies or quadropolies.2
Similar results were obtained in Bertand oligopoly experiments. Fouraker and Siegel(1963), Dufwenberg and Gneezy (2000), Orzen (2008), Davis (2009) and Fonseca andNormann (2012) found less competitive behavior and higher profits in Bertrand duopolies,
compared to triopolies or quadropolies. A meta-analysis and additional experiments in
2

A slightly different design was used by Friedman et al. (2015), who compared Cournot duopolies totriopolies with a unit elastic demand function in a low information environment and 1200 4-second rounds.
By the end of the game, output converged to the collusive outcome in the duopoly, but not in the triopoly.
Oechssler et al. (2016) replicated the study using a standard linear demand function and found morecollusive behavior in duopolies than in quadropolies.

4Horstmann et al. (2018) corroborate these results.
The origins of the group size effect have been studied in other games. Isaac and Walker(1988) noted that in public goods games, an increase in group size would decrease themarginal per capita return of contributions to a public good. A change in the group size
could therefore be divided into a “pure” group size effect and the part of the effect driven
by the changes in incentives to contribute. Isaac and Walker (1988) observed a significant
decrease in contributions when the group size was increased from 4 to 10. However, theeffect disappeared when the payoff function was corrected to keep MPCR constant acrossgroup sizes.3 Similarly, we find a decrease in the group size effect in Cournot oligopoly
when the payoff function is normalized to keep the incentives comparable.

2

Experimental Design

2.1

Payoff Function

We study a symmetric n-firm Cournot oligopoly. Each firm i P N simultaneously chooses
output qi . Price pi is determined by a linear inverse demand function:

pi pqi , qi q  max 0, 81  qi¸

θjPpN ziqqj(1)

where qi denotes the outputs of firms other than i.
The sole difference compared to the standard Cournot oligopoly implementation (e.g.
Bigoni and Fort, 2013, Huck et al., 2004) is the addition of θ, which could be interpreted
as the degree of product differentiation (Vives, 1984, Horstmann et al., 2018). If θ  1, as
is commonly assumed, products are homogeneous and the market price is common for all
3
Follow-up studies found that the group size may have a significant effect, although it depends on thevalue of MPCR and the nature of the game; for example, see Isaac et al. (1994), Barcelo and Capraro(2015), Nosenzo et al. (2015), Zelmer (2003).

5

firms. Otherwise, prices are different for each firm.
We assume that the marginal cost of production is equal to one, so that the cost function
is C pqi q  qi . Then profits obtained by i are
πi pqi , qi q  ppi pqi , qi qqi  qi qs  F C
The s parameter scales the payoffs to equalize equilibrium payoffs across games witha different number of firms. In previous research, a change in equilibrium payoffs due
to a different group size was corrected using different exchange rates (e.g. Huck et al.,2004, Bosch-Domènech and Vriend, 2003). Instead, we use an explicit scaling parameter toincrease transparency and maintain the same order of magnitude of payoffs across treatments, preventing any treatment effects due to the salience of payoffs. We set the fixed
cost F C  130, providing a subsidy that prevents negative payoffs and generates positive
payoffs in the Walrasian equilibrium.4
Three key outcomes are typically studied in Cournot oligopoly games: collusive outcome, at which the sum of payoffs earned by all players is maximized; Nash equilibrium,
at which all players best-respond to the action profile of all other players and Walrasianequilibrium, at which all players maximize their relative profits. We calculate the symmetric outcomes of interest using the standard procedure. In a symmetric Nash equilibrium,
80
C
qiN  2 θ80
pn1q . In a symmetric collusive outcome, qi  2 2θpn1q . In a symmetric Wal-

rasian equilibrium,5 qiW  1 θ80
pn1q . There sometimes are asymmetric collusive outcomes
and asymmetric Nash equilibria; in all asymmetric Nash equilibria, the total output is
4

The choice of the scaling parameters and the strategy space also ensure that in the normalized treatments all the payoffs are three-digit numbers (i.e. within the range 100-999), therefore no part of thepayoff space is particularly salient.
5With the parameters used in the experiment, the Walrasian equilibrium is at the point where priceequals marginal cost. In general, the symmetric relative payoff maximization point could be different forsmall values of θ, i.e. if θpn  1q 1.

6

equal to the total output produced in the symmetric equilibrium.6

2.2

Treatments

We aim to understand why output is more collusive in markets with fewer firms. Basedon the findings from previous experiments and the insights from competition policy, we
identify three potential explanations for this effect:
1. In smaller markets, it is easier to coordinate. In competition policy, market
concentration is controlled in part due to a belief that “the presence of many competitors tends to make it more difficult to sustain coordination (...)”.7 This belief
is supported by the results from economic experiments. Smaller groups manage tocoordinate on higher prices and receive higher profits in Bertrand oligopoly (Dufwenberg and Gneezy, 2000; Davis, 2009; Fonseca and Normann, 2012), coordinate on theefficient equilibrium in a minimum effort game (Van Huyck et al., 1990) and sustain
cooperation in voluntary contribution mechanism games (Nosenzo et al., 2015). The
success of smaller groups could be explained by the use of “language of coordination” (Davis, 2009), which allows participants to signal the intentions to collude andidentify deviations from collusive agreements (Masiliūnas, 2017). We will call this
explanation the “pure number effect” (following Isaac and Walker, 1988), as it isdriven purely by the group size, rather than by the change in incentives or otherelements of the game.
6
In the normalized treatments with the parameters used in the experiments and a discrete strategy
space, there are 4 asymmetric equilibria in markets with two firms: (7,9), (9,7), (0,16), (16,0). There are
18 asymmetric equilibria in markets with three firms and 18 in markets with four firms. The average output
is the same in all equilibria. Multiplicity of equilibria is common in Cournot oligopoly: for example, inthe standard treatments, there are 2 asymmetric equilibria in two-firm markets, 6 equilibria in three-firm
markets and 18 equilibria in four-firm markets.
7
U.S. Department of Justice & Federal Trade Commission, Commentary on the Horizontal Merger
Guidelines (2006), available at http://www.justice.gov/atr/public/guidelines/215247.pdf. Also see Ivaldi
et al. (2007).

7

2. In smaller markets, there are more incentives to collude or less incentives
to deviate from collusive agreements. In competition policy, mergers are expected to increase the individual incentives to increase prices, commonly measured
using the value of diverted sales or an index of upward pricing pressure (Shapiro,
2010, Farrell and Shapiro, 2010). This “unilateral effect” is typically explained by
the merged firm recapturing some of the sales lost due to a price increase by selling
more substitute products (Jaffe and Weyl, 2013). The stability of collusive agreements in infinitely repeated Cournot experiments depends on the incentives to makea collusive agreement and the incentives to deviate from it. In standard Cournot
oligopoly, firms have more incentives to collude, but also more incentives to deviate
from the collusive agreement when there are more firms in the market. Overall, larger
markets require a higher minimum discount factor to sustain collusion in an infinitely
repeated game (Friedman, 1971). Our experiments follow the previous literature anduse finitely repeated games, therefore collusive equilibria do not exist; however, we
will show that changes in incentives predict more collusive output in smaller markets.
3. In smaller markets, more choices are classified as collusive because thecollusive outcome is closer to the focal options. Preferences and beliefs dependon the choice set and item’s location in the set; for example, there is a preference
for options in the middle (Valenzuela and Raghubir, 2009; Chang and Liu, 2008) orfor the multiples of 10 or 100. In standard Cournot oligopoly, the focality of the
key outcomes, their location on the strategy space and the distance between themchange as a result of a different market size. These changes could affect the behaviorof boundedly rational participants and therefore more choices might be misclassified
as collusive in smaller markets.
We designed the “normalized” treatments to keep the incentives and the focality of8

actions the same across markets of different size to identify the pure number effect. We
also ran experiments with the “standard” design to compare the number effect between thetwo designs using the participants from the same subject pool and identical experimental
procedures.
Table 1 summarizes the key differences between the five treatments. The difference
between standard and normalized treatments lies in how the output of the opponents isaggregated, which is determined by the θ parameter. In standard treatments, we set θ  1,
regardless of the market size, as is common in the previous literature (e.g. Huck et al.,2004, Roux and Thöni, 2015, Horstmann et al., 2018, Oechssler et al., 2016). As a result,a change in the market size does not affect the profit of firm i if the output of firm i andthe sum of output of all other firms are held constant.8 In normalized treatments, we setθ  n1 1 , therefore market size does not affect the profits of i if the output of i and theaverage output of all other firms are held constant. This small difference in the aggregationof opponents’ output has important consequences on how the incentive structure responds
to changes in market size.
In the standard treatments, market size affects the output and payoffs in the three
key outcomes: Nash equilibrium, collusive outcome and Walrasian equilibrium. First twocolumns of Table 1 summarize these differences in markets with 2 and 4 firms (S2 andS4). Nash equilibrium payoffs are held constant using a scaling parameter (s  1 in 4-firm
markets, s  0.36 in a 2-firm markets), but the payoffs in the collusive outcome and themost profitable deviation from it are different. A common measure of the incentives to
π pN qcollude is the Friedman index (Friedman, 1971), defined as F  ππppCDq
qπpC q , where π pDq isthe payoff in the most profitable unilateral deviation from the collusive outcome. Table 1
shows that in the standard treatments, the incentives to collude decrease in the group size.9
8Note than in the experiment, we set s to different values in 2 and 4 player markets, in line withthe previous literature, therefore the profits in one treatment will be multiplied by a scalar; however,multiplication does not affect the best response.
9Friedman index values in our experiment are identical to the values in standard experiments that9

Table 1: Parameter values and key outcomes in each treatment.
TreatmentnθsStrategy space
qiC
qiN
qiW
π pC qπ pN qπ pW qFriedman index# participants# markets

S2
2
1
0.36
0-50
20
26.67
40
418
386
130
0.89
36
18

S4
4
1
1
0-50
10
16
20
530
386
130
0.64
60
15

N2
2
3
1
8-24
10
16
20
530
386
130
0.64
96
48

N3
3
1.5
1
8-24
10
16
20
530
386
130
0.64
72
24

N4
4
1
1
8-24
10
16
20
530
386
130
0.64
96
24

In the normalized treatments, the locations of the three key outcomes and the payoffs areinvariant to the market size.
The second difference between standard and normalized treatments is in the strategy
space. In standard, firms could choose output from 0 to 50. In normalized, the strategy
space was restricted to 8-24, for several reasons. First, we aimed to increase the accuracy of
categorization by having the three key outcomes evenly spaced across the strategy space. If
all three outcomes were located at the bottom of the strategy space (e.g. as in S4), there ismore scope for exceeding the equilibrium prediction and the average output might appear
to indicate competitiveness. Instead, in normalized treatments the Nash equilibrium wasplaced in the middle of the strategy space and the other two outcomes were located at asimilar distance from it. However, we did not want any outcomes to be located at either endof the strategy space, so as not to misclassify extreme choices as either perfectly collusive
or competitive. It was also necessary to bound the strategy space from below to decrease
the attractiveness of collusion by an alternating play of asymmetric outcomes.10
manipulated group size in Cournot oligopoly, e.g. Huck et al. (2004).
10
In N2, the symmetric collusive outcome is (10,10), generating a payoff of 530 ECU per firm. However,total payoffs would be maximized in the output profile (8, 16), with an average payoff of 546 EU per firm.

10Finally, the strategy space in normalized treatments was labeled such that the key outcomes would not be focal. For example, in S4 treatment, which uses the same parameters
as the previous literature, the collusive and competitive outcomes are at focal locations(respectively 10 and 20), and it is found that these outcomes are commonly chosen (Bigoni
and Fort, 2013). It is unclear whether these values are chosen because they were focal,
or because participants converge to the collusive or competitive outcome. To simplify the
explanation of the game and separate the key outcomes from focal values, we re-labeled thestrategy space, using strategy space from 0 to 16 to represent output 8-24. The mapping
was performed in two ways: in “increasing” treatments, higher output was represented byhigher numbers in the strategy space (i.e. output of 8 was labeled as “0” and 24 was labeled
as “16”). In “decreasing” treatments, higher output was represented by lower numbers (i.e.
output of 8 was labeled as “16” and 24 was labeled as “0”), reversing the strategy space.
This labeling ensures that the key outcomes are never at focal locations (in increasing
treatments, key outcomes are at 2, 8 and 12; in decreasing treatments they are at 4, 8 and
14). Running some treatments with a reversed strategy space helps to further evaluate
the importance of focality and to eliminate it’s effect when classifying choices (for moredetails, see Appendix A).11 In the data analysis section, we map the choices made by participants back into the output of the original Cournot payoff function and pool increasing
and decreasing treatments.
In N3, the asymmetric collusive outcome is (8, 8, 16), with a profit of 535.33 EU per firm. In N4, all output
profiles in which total output equals 40 generate the same payoff of 530 ECU per person. If the strategy
space was not bounded from below, the payoff difference between symmetric and asymmetric collusive
outcomes would be much larger, and cause different patterns of behaviour in 2 and 4 firm markets. For
example, the asymmetric collusive outcome in an unbounded 2-firm treatment is (4, 24), with a payoff of
810 ECU per person. The lower limit of 8 makes the asymmetric collusive outcome less attractive, while
keeping the symmetric collusive outcome in the interior of the strategy space. To completely eliminate theasymmetric collusive outcome, the lowest available output would have to be 10. In the data, we do not
observe any successful attempts of asymmetric collusion.
11
We find evidence that focal outcomes are more commonly chosen. For example, action labeled as “10”is the most commonly chosen action in N3I (increasing treatment with a market size of 3), second mostcommon in N4I and N3D, third most common in N2I, N2D and N4D. When the same action is labeled
as “6”, it is chosen less than half of the times. If we hadn’t run the treatments with a reversed strategy
space, the frequency of competitive outcomes would likely be overestimated.

11

2.3

Other Design Details

The stage game was repeated 20 times under partner matching. An alternative random
matching protocol would have made it very difficult to study collusion. Participants had
30 seconds to make their decision in each round (as in Bigoni and Fort, 2013). If nodecision was made within the time limit, the output chosen in the previous round was
implemented. If no decision was made in round 1, output was drawn at random from a
uniform distribution.12
Original instructions were in French; Appendix D provides a complete English translation. Instructions were identical in all treatments and framed using neutral language.13
Participants could learn about the incentive structure using either a payoff table or a payoff
calculator (examples of the information seen by the participants are displayed in figures F.1
and F.2 in Appendix F). The payoff table listed participant’s payoffs for some combinationsof chosen output and the average output of the opponents: 441 combinations in standard
treatments (21x21 action profiles, 0-50 in increments of 2.5) and 289 combinations in normalized treatments (17x17 action profiles, 0-16 in increments of 1). The payoff calculator
could be used to compute the payoff for any combination of own output and the average
output of the opponents.
In each round other than the first one, participants had access to two additional tools.
The “output-payoff graph” visually displayed the previous round output-payoff combinations of all the firms in the market (see Friedman et al., 2015 for a similar design). This
information is needed to imitate the best, which could lead to convergence to the Walrasianequilibrium (Vega-Redondo et al., 1997). The second tool was a table that listed the history of chosen output, payoff and average output of the opponents in the previous rounds.
12
The decision was not made within the time limit 3.5% of the time, primarily in the first two rounds. In
the analysis part, we use all the decisions, although excluding the decisions that were not made explicitly
does not change the overall results.
13
For a discussion about how the use of neutral language rather than the more commonly used economic
framing affects preferences and beliefs in Cournot oligopoly, see Masiliūnas and Nax (2020).

12

This information would be used to make decisions from experience (e.g. reinforcement
or belief learning). Participants could switch between the four tools at any time, and we
tracked how much time was spent using each tool, just as in Bigoni and Fort (2013). This
process data provides additional insight into how the decision making process is affectedby the market size.
After 20 rounds, participants continued the experiment with different games for another20 or 40 rounds (depending on the treatment). This data was collected to study learningtransfer and is used in a separate paper. Table C.1 in Appendix C shows the structure of theentire experiment. In this paper, we use only choices from the first 20 rounds. Participants
were aware that the experiment will contain multiple parts, but did not know how many
parts there will be and what type of games will be played. In each part, participants werematched with different opponents, therefore they never interacted with their opponents
from the first part again. One round from each part was randomly selected at the endof the experiment, and the earnings from these rounds were added up and paid to theparticipants in cash.14
Additional information was collected at the end of the experiment. We elicited social
preferences using the Social Value Orientation slider measure (Murphy et al., 2011, using
the z-Tree implementation by Crosetto et al., 2019). We also measured the cognitive
abilities using a part of the advanced version of Raven’s Progressive Matrices task (Ravenand Court, 1998). In this test, participants had 10 minutes to solve 16 tasks. After thesetasks, we collected the age, gender and year of study of the participants.
In total, 360 participants took part in experiments. The numbers of participants andmarkets in each treatment are shown in Table 1. In normalized treatments, exactly one
half of the participants in each treatment took part in “increasing” treatments and theother half in “decreasing” treatments. We collected more observations for the normalized
14

Earnings were denominated in ECU and exchanged to cash using rate 150 ECU = 1 euro.

13treatments because the standard treatments have already been studied in the previous
literature.
All experiments were run in the LEEN laboratory of the Université Côte d’Azur in
May and October 2018. The experiments took on average 75 minutes and participants on
average received 14.2 euros. Participants were recruited using ORSEE (Greiner, 2015) andexperiments were programmed using z-Tree (Fischbacher, 2007).

3

Quantal Response Equilibrium Predictions

We have shown that the market size interacts with the incentive scheme differently instandard and in normalized treatments. This section quantifies how these differences inincentives are predicted to affect the collusion rates under bounded rationality, as modeledby the Quantal Response Equilibrium (McKelvey and Palfrey, 1995).
QRE requires consistency between actions and beliefs, but the responses to beliefs arenoisy, therefore all actions have a positive probability to be played. In a game with nplayers, a set of players N and a set of pure strategies Qi  tq1 , . . . , qm u, denote theset of all probability measures on Qi by ∆i  tp1 , . . . , pm u and the set of all probability
measures on i Qi by ∆  t∆1 , . . . , ∆n u. We will use shorthand notation p  ppi , pi q for
any p P ∆, where pi is the mixed strategy of player i and pi is the mixed strategy profile
of all other players. The probability with which player i chooses action qk is pi pqk q. The
expected payoff that i obtains by choosing qk is denoted by πi pqk , pi q. Nash equilibrium
assumes that each player chooses the action with the highest expected payoff. Instead,
QRE assumes that participants are maximizing their decision utility ui pqk , pi q, equal to
the sum of the expected payoff and the noise term:
ui pqk , pi q  π pqk , pi q

14

εik

(2)

If each of the stochastic terms εik is independently drawn from a type-I extreme value
distribution with parameter λ (McFadden, 1981) and each player chooses the action thatgenerates the highest decision utility, the probability that i will play qk can be calculatedusing a noisy best-response function σi pqk , pi q, defined as:σi pqk , pi q  °eλπi pqk ,pi q
λπi pqj ,pi q
qj PQi e(3)

The logit QRE is a probability distribution p P ∆ that satisfies pi pqk q  σi pqk , pi q,
for all i P N and qk P Qi (McKelvey and Palfrey, 1995). In other words, QRE requires
the mixed strategy of each player to be a noisy best-response to the mixed strategy profile
used by all other players.
Parameter λ measures precision, or sensitivity to expected payoff differences. If λ  0,
all actions are chosen with equal probabilities. A positive λ indicates that actions thatgenerate higher expected payoffs are chosen more often. If λ Ñ 8, there is no error andplayers always choose the action with the highest expected payoff, therefore QRE reduces
to the Nash equilibrium.
Since the closed-form expressions of logit QRE are generally unknown, we calculate
QRE using the tracing procedure from Turocy (2005), implemented using Gambit software(McKelvey et al., 2015). The calculations are performed using a discretized strategy spaceof 51 strategies (0, 1, . . . , 50) in standard treatments and 17 strategies (8, 9, . . . , 24) innormalized treatments. Payoffs used in the calculations are converted into monetary euro
amounts (in experiments, the exchange rate was 150 ECU = 1 euro).

3.1

QRE in Standard Treatments

We start by evaluating how QRE predictions respond to changes in market size in standard
treatments and move to the normalized treatments in subsection 3.2. First, we illustrate

15

0.04

QRE in S2 and S4, λ = 1.5

0.02
0.00

0.01

Density

0.03

S2
S4

0

5

10

15

20

25

30

35

40

45

50

Quantity

Figure 1: QRE distribution with λ  1.5 in standard treatments. Vertical dashed lines
indicate Nash equilibrium in S4 (16) and S2 (26.7).
the QRE predictions by calculating the choice probabilities for a specific value of the noise
parameter (λ  1.5, very close to the value in the best-fitting QRE model estimated in
section 5.1) and then show how the predictions respond to changes in λ.
Figure 1 shows the calculated QRE probability distributions in treatments S2 andS4. Dashed lines mark the corresponding Nash equilibrium output (26.7 and 16). Both
distributions are centered around the Nash equilibrium, but more choices exceed the Nashequilibrium prediction in S4 than in S2 (59%, compared to 51%), largely because in S4
the Nash equilibrium is located at the bottom of the strategy space. Consequently, theaverage QRE output is very close to the Nash equilibrium in S2 (26.8, compared to 26.7),
but exceeds the Nash equilibrium in S4 (20.6, compared to 16).
Next, we introduce two measures to quantify the degree of collusion and compare themacross a range of parameter values. The first measure is the ratio of average output to Nashequilibrium output (Huck et al., 2004), calculated as r  q̂qpNλq , where q̂ pλq is the average
output in a QRE with parameter value λ, and q N is the Nash equilibrium prediction.
Figure 2 plots r in S2 and S4 for values λ P r0, 10s. If the sensitivity to payoff differences ishigh, QRE approaches Nash equilibrium and thus r Ñ 1. If the sensitivity is low, average
16

1.1

1.3

1.5

S2
S4

0.9

Average choice/NE prediction

Ratio of average choice to NE

0

2

4

6

8

10

λ

Figure 2: Ratio of average QRE output to Nash equilibrium output in standard treatmentswith 2 and 4 firm markets.
output is below Nash equilibrium in two-firm markets and above it in four-firm markets,
predicting more collusive behavior in smaller markets.
The second measure is the frequency of collusive or competitive choices. It is common
to classify choices by identifying the outcome that is closest to the chosen output (see Hucket al., 2004). We partition the strategy space into actions that are closest to either collusive
outcome, Nash equilibrium or Walrasian equilibrium, and compare the QRE predictions
about the frequency of each category across treatments. We find that smaller markets are
predicted to have more collusive and less competitive choices, for a wide range of λ values(Figure 3).
Overall, QRE predicts more collusive behaviour in smaller markets when the market
size is manipulated using the standard Cournot payoff function. This finding indicatesthat the experimental result of lower collusion rates in smaller markets could in principle
be driven by changes in the incentive structure or the locations of the key outcomes.

17

0.2

0.4

0.6

S2
S4

0.0

0.2

0.4

Walrasian choices

0.6

S2
S4

0.0

Collusive choices

Choices classified as JPM in QRE Choices classified as RPM in QRE

0

2

4

6

8

10

0

2

λ

4

6

8

10

λ

(a) Collusive output(b) Competitive output

Figure 3: Fraction of choices classified as collusive (closer to the collusive outcome than tothe other two outcomes) or competitive (closer to the Walrasian equilibrium than to theother two outcomes) in standard treatments.

3.2

QRE in Normalized Treatments

Next, we calculate QRE in normalized treatments. The normalized payoff function was
designed to keep the incentives similar across different market sizes. We therefore expect
smaller differences in predicted collusion rates between normalized treatments.
Figure 4 illustrates the QRE distribution for λ  1.5. The markets with 3 and 4 firms
are similar, but the 2-firm market is somewhat shifted towards more competitive output.
This shift is caused by a difference in how the distribution of beliefs about average output
is constructed. In N2, beliefs coincide with the QRE distribution, since players face only
one opponent. In N3 and N4, the distribution of beliefs about the average output will havea lower variance because extreme values of average output are less likely in larger markets.
Higher variance of the belief distribution in N2 makes higher output more attractive becausesuch output is on average more profitable when the average output chosen by the opponents
is more extreme (see the payoffs in Table E.1, Appendix E).
18

0.05

0.06

N2
N3
N4

0.04

Density

0.07

QRE in S2 and S4, λ = 1.5

8

10

12

14

16

18

20

22

24

Quantity

Figure 4: QRE distribution in normalized treatments at λ  1.5. Vertical dashed line
indicates Nash equilibrium output.
We quantify collusion using the two measures introduced in the previous section: aratio of average output to Nash equilibrium output and the frequency of output classified
as either collusive or competitive. Figures 5 and 6 show how these measures are predicted
to depend on the market size.
Figure 5 plots the ratio of average QRE output to Nash equilibrium output. In contrast
to the standard treatments (Figure 2), we find that smaller markets are predicted to beslightly less collusive. However, the treatment difference is much smaller compared to thestandard treatments: the ratio differs by at most 5%, in contrast to the differences of up
to 60% in the standard treatments.
Figure 6 shows that the slightly less collusive output in markets with two firms isprimarily driven by a higher frequency of competitive choices in this treatment. The
predicted fraction of collusive choices is nearly identical across the different market sizes,
and much smaller than the differences in the standard treatments.
Overall, the QRE prediction that smaller markets will be more collusive holds onlyin the standard but not in the normalized treatments. The comparison of the group size
effect under both designs can therefore identify whether the results found in the Cournot

19

1.1

1.3

1.5

N2
N3
N4

0.9

Average choice/NE prediction

Ratio of average choice to NE

0

2

4

6

8

10

λ

Figure 5: Ratio of average QRE output to Nash equilibrium output in normalized treatments with 2, 3 and 4 firm markets.

0.2

0.4

0.6

N2
N3
N4

0.0

0.2

0.4

Walrasian choices

0.6

N2
N3
N4

0.0

Collusive choices

Choices classified as JPM in QRE Choices classified as RPM in QRE

0

2

4

6

8

10

0

λ

2

4

6

8

10

λ

(a) Collusive output(b) Competitive output

Figure 6: Fraction of choices classified as collusive (closer to the collusive outcome than tothe other two outcomes) or competitive (closer to the Walrasian equilibrium than to theother two outcomes) in normalized treatments.

20

literature are primarily driven by the pure number effect, or by the changes in the incentive
structure or the focality of key outcomes.

4

Results

First, we will compare the two designs (standard and normalized) in whether they alter theeffect of the market size on the aggregate levels of collusion. Afterwards, we will compare
the distributions of output and classify behavior to identify how the market size affects thefrequency of collusive output.

4.1

Aggregate Output

In the standard treatments, theoretical predictions change with the market size, therefore
average output (q̄i ) needs to be normalized before treatments can be compared. We do sousing two measures of collusion: the ratio of actual to predicted output and a collusionindex.
Ratio of actual to predicted output. A simple way to normalize output is tocalculate the ratio of chosen output to Nash equilibrium output: r  q̄i {qiN . Values below
1 indicate that output is more collusive than the equilibrium prediction, values above 1
indicate competitive output. Figure 7 shows the dynamics of the across markets average r.
To compare across treatments, we calculate r separately for each market, aggregated either
across all rounds (1-20) and only the last five rounds (15-20), following the convention in
the literature (e.g. Huck et al., 2004). We use a Mann-Whitney U (MWU) test to test
whether the ratio of output is significantly affected by market size.
In standard treatments, there is a significant difference between the markets with 2
and 4 firms (MWU p

0.0001 if all rounds are used, p  0.001 if last 5 rounds are used).

This result replicates the finding from the literature that smaller markets tend to be more
21

1.75

1.75

1.5

Ratio of average output to NE

Ratio of average output to NE

S2
S4

1.25

1

.75

.5

N2
N3
N4

1.5

1.25

1

.75

.5
0

5

10

15

20

0

5

Period101520PeriodFigure 7: Ratio of average output to equilibrium output by treatment over time.
collusive (Huck et al., 2004). In the normalized treatments, there is no difference between
markets with 2, 3 or 4 firms (MWU p-values for the pairwise comparison are at least 0.5826
if all rounds are used and 0.3123 if the last 5 rounds are used).
Collusion index. Used in Horstmann et al. (2018), Engel (2007) and Suetens andPotters (2007), the collusion index measures where the market output falls in the range
between the collusive outcome and the Nash equilibrium or the Walrasian equilibrium:
ϕN  pq̄i  qiN q{pqiC  qiN q and ϕW  pq̄i  qiW q{pqiC  qiW q. Both indexes would beequal to 1 if market output was equal to the collusive output; the first index would beequal to 0 if output was equal to the Nash equilibrium, the second would be equal to 0 if
output was equal to the Walrasian equilibrium. Note that in the normalized treatments,
the treatment comparison is identical for all three indexes because the key outcomes areinvariant to market size. We find that in the standard treatments, 2-firm markets are
more collusive than 4-firm markets, as measured by ϕN (MWU p  0.0001 for all rounds,p  0.0103 for last 5 rounds) or by ϕW (MWU p

0.0001 for all or only the last 5 rounds).

In the normalized treatments, there is no difference between the markets with 2, 3 or 4
firms (MWU p ¡ 0.5786 for all rounds and p ¡ 0.3123 for the last 5 rounds).
From a policy perspective, it is important to know whether firms in more concentrated markets receive higher profits. Nash equilibrium predicts identical profits in all
22

Table 2: Three measures of collusion and profits across treatments. First number is theaverage in rounds 1-20, the number in brackets is the average in rounds 15-20.
Index
r
ϕN
ϕW
πi

S2
1.02
[1.05]-0.10[-0.20]0.63[0.60]
340.0[338.8]S4
1.30[1.23]-0.80[-0.62]-0.08[0.03]
230.1[243.9]N2
0.98[0.99]
0.05[0.02]
0.43[0.41]
387.3[381.9]N3
0.99[1.02]
0.02[-0.06]0.41[0.36]
371.1[348.0]N4
1.00[1.02]-0.002[-0.04]0.40[0.37]
372.4[361.9]five treatments. Table 2 shows that in the standard treatments, profits are significantly
higher in 2-firm markets, both overall (MWU p

0.0001) and in the last 5 rounds (MWUp  0.0001). There is, on the other hand, no significant difference among three market
sizes in the normalized treatments (the lowest MWU p-value is 0.2323 in all rounds and
0.1450 in the last 5 rounds).
Additionally, we evaluate the treatment effects using a panel data GLS regression witha random effect on the market level, taking into account the inter-temporal dependence
of decisions as well as the dependence among the outputs of firms in the same market.
Standard errors are clustered on the market level. As dependent variables, we use thethree collusion indexes (r, ϕN , ϕW ). The main independent variables are dummy variables
indicating market size. For the normalized treatments, we also include an indicator of thestrategy space labeling. A variable equal to the inverse of a round is included to capturechanges in collusion due to experience. Table 3 shows that, in the standard treatments,
markets with four firms are significantly less collusive than markets with two firms, for all
three indexes. The coefficients of the inverse round variable indicate increasing collusion
over time. In the normalized treatments, the 3-firm and 4-firm markets are not different
from the 2-firm markets. The “decreasing labels” variable indicates treatments in whichhigher output was labeled with lower numbers. The coefficient of this variable is signifi-

23

Table 3: Random effects GLS regression. Standard errors clustered on the market level.

3-firm market
4-firm market
1/RoundDecreasinglabelsConstant
N

Normalized
DV: r
DV: ϕ
DV: r
DV: ϕN DV: ϕW––
0.0109-0.0291-0.0175(0.42)(-0.42)(-0.42)0.277*** -0.705*** -0.716***0.0189-0.0504-0.0303(6.37)(-4.96)(-11.02)(1.04)(-1.04)(-1.04)0.206**-0.501* -0.340*** -0.0382*
0.102*
0.0611*(2.49)(-1.96)(-2.68)(-1.75)(1.75)(1.75)-0.0903*** 0.241*** 0.144***(-5.19)(5.19)(5.19)
0.988*** -0.00840 0.695***
1.034*** -0.0907* 0.346***(32.33)(-0.07)(16.45)(58.88)(-1.94)(12.30)
1920
1920
1920
5280
5280
5280

t statistics in parentheses* p 0.10, ** p 0.05, *** pStandard
DV: ϕN–W

0.01

cantly different from 0, indicating the importance of labeling. The effect is largely driven
by action “10” being focal; in the increasing treatments, this action is more competitive
than the Nash equilibrium but in decreasing treatments it is more collusive. We note thatthe labeling of the strategy space has a significant and consistent effect on collusion rates,
which is even stronger than the effect of market size.15
These results are robust to different specifications. Table C.3 in Appendix C shows theestimated treatment effects using data only from the last 5 rounds, when participants would
have accumulated experience. The treatment effects remain qualitatively the same. Results
also do not change if we remove decisions that were implemented because participants failto make a decision within the time limit. In the normalized treatments, we ran regressions
separately for the decreasing and increasing strategy space and found no significant number
effects (Table C.2 in Appendix C). Including age and gender does not change the results,
and coefficients for these two variables are not significantly different from zero.
Result 1. In standard treatments, aggregate output is more collusive in smaller markets.
15

Further details on the effect of strategy space labeling are in Appendix A.

24

.2

S2
S4

.06

C
C

N

N

C

W

N

W

N2
N3
N4

W

.04

DensityDensity

.15

.02

.1

.05

0

0
0

5

10

15

20

25

30

35

40

45

50

8

Output

10

12

14

16

18

20

22

24

OutputFigure 8: Kernel density of individual output, pooled across rounds.
In normalized treatments, market size has no effect on aggregate output.

4.2

Distribution of Output

Comparing treatments in terms of aggregate outcomes is important to compare our results
to the previous literature and for policy reasons, as regulators care about the average
profits and prices in the markets. But we are also interested in how and why the market
size affects the distribution of output and the frequency of collusive output.
Figure 8 shows the kernel density estimate of individually chosen output in the standard
and normalized treatments. In standard treatments, a larger market size decreases output
but the adjustment is not as strong as predicted by Nash equilibrium. Consequently, theaverage output is close to the equilibrium prediction in 2-firm markets, but exceeds it
in 4-firm markets. In normalized treatments, distributions are centered around the Nashequilibrium but the variance is notably lower in larger markets: the standard deviation ofchosen output is 4.9 in N2, 4.2 in N3 and 3.7 in N4. We evaluate the statistical significanceof these differences by calculating the standard deviation for each market and comparing
their distribution across treatments. In normalized treatments, we find that the standard
deviation is significantly higher in two-firm markets, compared to the markets of larger
size (Mann-Whitney U test p  0.0041 comparing N2 vs N3 and p  0.0017 comparing
25

14

5.5

Standard deviation of output

Standard deviation of output

S2
S4

12

10

8

N2
N3
N4

5

4.5

4

3.5

3
0

5

10

15

20

0

Period

5

10

15

20

PeriodFigure 9: Standard deviation of choices by treatment over time.
N2 vs N4). In standard treatments, market size has no significant effect on the standard
deviation. Figure 9 plots the evolution if the standard deviation, calculated using all choices
in each round. In standard treatments, there is no difference between the two market sizes.
In normalized, there is no difference at the start of the game, but a gap between the threetreatments appears and grows over time.
Differences in the choice distributions affect the fraction of choices classified as collusive.
We use two methods to identify the frequency of collusion. First, we use collusion counts(Waichman et al., 2014), defined as the number of rounds in which quantities are in thecollusive region, that is closer to the collusive outcome than to the Nash equilibrium. Unlike
Waichman et al. (2014), we perform the classification using individual rather than total
market output, because aggregating choices over a larger number of firms decreases thelikelihood that the group would be classified as collusive.16 The average number of rounds
in which individual output is classified as collusive is 5.1 in S4 and 5.2 in S2, a difference
that is not significant (MWU p  0.49). In normalized treatments, collusion counts go up
from 3.0 in N4 to 4.1 in N3 and 5.8 in N2; there is significantly more collusion in 2-firm
16

Classification using the market output replicates the usual finding of more collusive choices in smallermarkets: in standard treatments, the average collusion counts go up from 5.1 to 10.3; in normalized
treatments, from 3.7 to 8.1 to 9.0. The difference between 2 and 4-firm markets is significant in bothstandard (MWU p  0.0084) and normalized treatments (MWU p  0.0009). The 3-firm market is not
significantly different from the other two.

26

100

Percent

80

60

40

20
CollusionNash Equilibrium
Walrasian Equilibrium
0

N2

N3

N4

S2

S4

Figure 10: Classification of choices according to which outcome they are closest to. Data
from last 5 rounds.
markets than in 3-firm markets (MWU p  0.0299) or 4-firm markets (MWU p  0.0018).
The comparison of collusion counts indicates a higher incidence of collusion in smallermarkets, in contrast to the results based on aggregate output. The reason behind thisdifference can be seen by inspecting the output distributions. In standard treatments, adecrease in market size primarily lowers the frequency of above-equilibrium output, lowering the average output but not increasing the share of collusive choices. In normalized
treatments, an increased variance in smaller markets does not change the average output
but increases the frequency of choices that are close to the collusive output.
Next, we classify individual output based on the outcome it is closest to. We follow Hucket al. (2004), but use output from individual firms rather than the total market output.
We aim to identify the outcome to which choices converge, therefore we use the average
output of each firm in the last 5 rounds. Figure 10 shows that in standard treatments,
a larger market size increases the fraction of choices classified as Walrasian (competitive)
but has no effect on the frequency of collusive choices. In normalized treatments, a larger

27

market size decreases the frequency of collusive choices but has a non-monotonic effect on
the frequency of Walrasian or Nash choices.
Result 2. In normalized treatments, the frequency of collusive output and the variance of
output are higher in smaller markets. In standard treatments, market size has no effect on
the frequency of collusion or the variance of output.

5

Model Estimation

Our standard treatments replicate the previous literature, finding more collusion in smallermarkets. In the normalized treatments, we find no effect of the market size on the aggregate
output or profits. However, in these treatments smaller markets tend to have a higher
variance of output, an effect that is not found in the standard treatments. This section
will test whether models of bounded rationality can explain these data patterns.

5.1

Quantal Response Equilibrium

First, we test whether the data patterns can be explained by QRE. In section 3, we showedthat QRE correctly predicts that the number of competitors makes output less collusive
in the standard treatments, but does not affect it in the normalized treatments. But QRE
also makes a prediction about the entire distribution of choices, therefore we can test
whether it can explain the differences in variance observed in the normalized treatments.
We estimate the logit QRE using a method adapted from Bajari and Hortacsu (2005):
we calculate the noisy best-response to the empirical choice distribution, and use a gridsearch procedure to find a precision parameter (λ) that maximizes the likelihood of datain the experiment. This method produces an unbiased estimate of the precision parameter
under the assumption that QRE is correct because in QRE beliefs would coincide with theempirical distribution. In practice, QRE will never fit the data perfectly, so there will be
28

Table 4: Goodness of fit and estimated parameter values in QRE.
Treatment
S2
S4
N2
N3
N4
Total

Separate estimation
λ̂
LL
1.395 -2739.97
1.535 -4561.56-0.292 -5436.98
2.507 -4014.05
4.686 -5161.69-21914.25

Combined estimationλ̂
LL
1.497 -2740.34
1.497 -4561.65
1.497 -5531.25
1.497 -4024.05
1.497 -5281.54-22138.83

a discrepancy between this method and the tracing procedure we used to compute QRE
in section 3.17
When estimating QRE, we either fit a separate model for each treatment, or fit a single
model, combining data from all five treatments. In the literature, it is common to estimate
a separate model for each game (e.g. Lim et al., 2014), but it implicitly assumes thatthe sensitivity to payoff differences is changing with market size. Since the normalized
treatments were designed to be comparable with each other, allowing the noise parameterto vary with the market size would lead to overfitting. If very different values of λ areneeded to explain the number effect, results would be driven not by the elements in QRE,
but by some other unmodeled factors. The second method therefore tests whether all
the treatment differences could be explained by a model with a single value of the noise
parameter.18
Table 4 provides an overview of the estimated parameter values and goodness of fit (log17There are two main benefits of using the estimation procedure from Bajari and Hortacsu (2005). First,
it gives the flexibility to perform a combined estimation using multiple treatments with different payoff
function, which is not possible with the standard tracing procedure. Second, the computational complexityof the estimation procedure is greatly reduced, since it is no longer necessary to compute the fixed point
for a large number of parameter values. Computational complexity is especially problematic for the gameswith many participants and a large strategy space, such as our S4 treatment (51 strategies for each of the
4 participants).
18
We also estimated QRE separately for normalized and standard treatments, requiring λ to be the sameacross different market sizes but allowing them to differ between standard and normalized treatments. Thisis justified by the important differences between standard and normalized treatments, such as the rangeof payoffs or the size of the strategy space, which affect the QRE predictions. In practice, the estimatesare very close to the combined estimation. We report these results and goodness of fit in Appendix C.

29

0.04

Density

0.00

0.02

Experiment
QRE

0.02

0.04

Experiment
QRE

0.00
0

10

20

30

40

50

0

10

20

QuantityGame N2, λ = −0.29

24

0.05
0.00

0.05
20

Experiment
QRE

0.15
Density

0.15

Experiment
QRE

0.00
16
Quantity

50

Game N4, λ = 4.69

0.10

Density

0.15
Density

0.10
0.05
0.00

12

40

Games N3, λ = 2.51
Experiment
QRE

8

30
Quantity

0.10

Density

0.06

Game S4, λ = 1.54

0.06

Game S2, λ = 1.39

8

12

16
Quantity

20

24

8

12

16

20

24

Quantity

Figure 11: Best-fitting QRE estimated separately for each treatment (vs kernel density).
likelihood) for each method. When a separate model is fit for each game, the estimatedvalues of the noise parameter are similar in the two standard treatments (λ̂  1.4 andλ̂  1.5), but very different in the three normalized treatments. That indicates that QRE
can explain the number effect in standard treatments (result 1), but cannot explain theincreased variance in smaller markets of the normalized treatments (result 2), unless it isassumed that sensitivity to expected payoff differences significantly decreases in smallermarkets. In fact, the estimated value of λ is negative in N2, indicating that the output
distribution in this treatment could be explained by participants choosing actions withlower expected payoffs.
Figure 11 illustrates the goodness of fit by comparing the kernel density estimation ofexperimental data to the choice probabilities predicted by the best-fitting QRE. In thestandard treatments, QRE can explain the shift in the choice distribution that resultsfrom a different market size. In the normalized treatments, the fit is less good as QRE
underestimates the frequency of focal outcomes, such as the Nash equilibrium and theendpoints of the strategy space.

30

0.04

Density

0.00

0.02

Experiment
QRE

0.02

0.04

Experiment
QRE

0.00
0

10

20

30

40

50

0

10

20

QuantityGame N2, λ = 1.50

24

0.05
0.00

0.05
20

Experiment
QRE

0.15
Density

0.15

Experiment
QRE

0.00
16
Quantity

50

Game N4, λ = 1.50

0.10

Density

0.15
Density

0.10
0.05
0.00

12

40

Game N3, λ = 1.50
Experiment
QRE

8

30
Quantity

0.10

Density

0.06

Game S4, λ = 1.50

0.06

Game S2, λ = 1.50

8

12

16

20

Quantity

24

8

12

16

20

24

Quantity

Figure 12: Best-fitting QRE estimated for all treatments (compared to kernel density).
The additional requirement for the noise parameter to be constant across treatmentsin the second method hardly changes the fit in the standard treatments, but decreases
the fit in the normalized treatments, especially in N2 and N4. Figure 12 shows the choiceprobabilities estimated by a model with a single value of λ̂  1.497. The fit remains goodin the standard treatments, but QRE fails to predict any difference in the distribution of
choices among the three normalized treatments.
Overall, QRE can explain why output is on aggregate more collusive in smaller markets in standard treatments and why there is no difference in the normalized treatments(result 1). QRE can also explain the overall change in the shape of the choice distribution
in standard treatments, but it cannot explain why variance decreases in market size in thenormalized treatments (result 2).

5.2

Frequent Response Equilibrium

The finding that variance is decreasing in market size only in the normalized treatments
could be explained by the difference in how the output of the opponents is aggregated.

31

In the standard treatments, the sum of opponents’ output is mapped into the same bestresponse, regardless of the market size. In the normalized treatments, the average of
opponents’ output is mapped into the same best-response, regardless of the market size.
An increase in market size increases the variance of the sum of output chosen by theother firms, but decreases the variance of the average output. Therefore, for a given bestresponse correspondence, the variance of the best-response distribution would increase inmarket size in the standard treatments, but decrease in normalized treatments (for moredetails and an illustration, see Appendix B).
We can test these predictions by comparing the incentives to respond to the observed
feedback in experiments. First, we test the prediction that the variance of total output of
the opponents is increasing in market size, but the variance of average output is decreasing.
For each participant, we calculate the standard deviation of either the total or the mean
opponents’ output across all 20 rounds and then compute the standard deviation in eachtreatment. As predicted, the standard deviation of the total opponents’ output increases
from 10 in S2 to 19 in S4 in standard treatments. In normalized treatments, the standard
deviation of the average opponents’ output decreases from 4.2 in N2 to 2.3 in N3 and 1.9
in N4.
These differences affect the shape of the best-response distribution. Figure 13 shows
the distribution of best responses to the output of the opponents observed in that round,aggregated across all participants and all rounds. These distributions of ex-post rational
output would have been observed if participants always had correct beliefs and optimally
responded to them. Standard deviation of the best-response distribution is increasing inmarket size in the standard treatments (5.7 in S2 and 8.1 in S4) but decreasing in thenormalized treatments (5.8 in N2, 4.4 in N3 and 3.1 in N4). This result shows that theobservation of variance decreasing in market size only in the normalized treatments could
be explained by participants responding to observed feedback.
32

.15

.2

C

S2
S4

N

W

N2
N3
N4

.15

DensityDensity

.1

.1

.05
.05

0

0
0

10

20

30

40

50

8

Output

10

12

14

16

18

20

22

24

OutputFigure 13: Kernel density of ex-post rational actions, pooled across all rounds.
We use the results about the difference in observed feedback to propose a modificationof QRE, which we will refer to as the Frequent Response Equilibrium (FRE). QRE assumes
that each player chooses a noisy best response to the choices that everyone else is expected
to make; therefore, the choice probabilities are a function of expected payoffs. FRE assumes
that each player (noisily) chooses the action that is most likely to be the best response, so
the choice probabilities are a function of the expected likelihood to be the best response.19
QRE could emerge as the long-run outcome of logit response dynamics (Alós-Ferrer andNetzer, 2010; Cason et al., 2021), as players form beliefs from observed history and choose
stochastic best-responses. Instead, FRE could emerge as the long-run outcome if players
favor actions that are frequent best-responses to the observed history, in a manner similar toprobability matching (Vulkan, 2000). It has been shown that when decisions are made fromexperience, participants are not very sensitive to the average payoff an action generates,
but are sensitive to how frequently an action provides the highest payoff (Erev and Barron,
19
There is a difference between the action that generates the highest expected payoff and an action thatis most likely to provide the highest payoff, because the former does not take the magnitude of payoffs
into account. For example, consider our treatment N2 and suppose that players expect the opponent todraw their action from a uniform distribution. Compare the attractiveness of the equilibrium output of
16 to the lowest possible output of 8. Producing 16 provides a higher expected payoff than 8 because it
generates high profits when the opponent is choosing a low or an intermediate output level (Table E.1).
However, 16 is rarely the exact best-response – it is the best response only if the other player chooses 16
as well, whereas 8 is the best response to any output above 21. When evaluated based on the likelihood
to be the best response, output of 8 would therefore outperform 16, while 16 would outperform 8 basedon the expected payoff.

33

2005; Yechiam and Busemeyer, 2006).
We wanted FRE to differ from QRE only in the way the attractions are determined(based on the likelihood of being a best-response rather than expected payoff), therefore
we retained the assumption that attractions are mapped into choice probabilities using asoftmax function with a λ parameter. If λ Ñ 8, FRE approaches Nash equilibrium because
the action that has the highest likelihood to be the best response must also provide thehighest expected payoff.
We define FRE formally by extending the definition of QRE from section 3. Let qi tq1, . . . , qi1, qi 1, . . . , qnu be the pure strategy profile of all players other than i. Thenpi P ∆i is the mixed strategy profile of all others players, where ∆i is the the setof all possible mixed strategy profiles. The likelihood that qi will be played in pi ispi pqi q. Denote the set of strategies that are best responses to a pure strategy profile qiby B pqi q  tqk P Qi : @qj P Qi : πi pqk , qi q ¥ πi pqj , qi qu. Then the likelihood that qk is
the best-response to qi is calculated by bpqk , qi q, defined as:$''& |Bpq1 q| if qk P B pqi q
ibpqk , qi q ''%0
if qk R B pqi q
The likelihood that action qk is the best-response conditional on probabilistic beliefpi is calculated by rpqk , pi q °p

q p qqi b qk , q i p i q i .

FRE assumes that player are

maximizing the probability that the action will be the best response; therefore, the utility
function defined in equation (2) is replaced by:ui pqk , pi q  rpqk , pi q

εik

(4)

The definition of the solution concept follows the one of QRE, detailed in section 3,
with the choice probabilities determined by equation (3). We evaluate the fit of FRE and
34

0.4

FRE in N2, N3 and N4, λ = 8
N2
N3
N4

0.2
0.1

0.05

Density

0.10

0.3

S2
S4

0.0

0.00

Density

0.15

FRE in S2 and S4, λ = 8

0

10

20

30

40

50

8

Output

12

16

20

24

Output(a) Standard treatments(b) Normalized treatments

Figure 14: FRE distribution with λ  8. Vertical dashed lines indicate symmetric Nashequilibria.
compare it to QRE by calculating the FRE choice distribution at various values of λ andthen evaluating the goodness of fit by fitting FRE to the experimental data.
FRE is calculated using the same tracing procedure used for QRE, but instead ofusing the game’s payoffs πi pqk , qi q, we use the likelihood that an action is a best response,
calculated by bpqk , qi q. Note that this modification leads to a significant loss of information
about the incentives faced by the players.20
Figure 14 shows the FRE choice probabilities for λ  8. In standard treatments, S4
stands out due to the high predicted frequency of producing nothing, which is the best
response when the total output of the other firms exceeds 80. In normalized treatments,
the FRE choice distribution has a higher variance in smaller markets. N2 is notable due
to a high predicted frequency of the two most extreme output levels. This prediction is
explained by a higher likelihood of observing extreme average output in N2 than in N3 or
N4, and a best-response function that makes the output of 8 optimal when the average20

It should be possible to develop a hybrid model of QRE and FRE, in which attractions are a convex
combination of the monetary payoff and the best-response likelihood. However, such a model is beyond
the scope of this paper.

35

0

5

10

15

10
8
6
4
2
0

Standard deviation of output

20
15
10
5
0

Standard deviation of output

S2
S4

N2
N3
N4

0

λ

5

10

15

λ

(a) Standard treatments(b) Normalized treatments

Figure 15: Standard deviation of output in estimated FRE distributions for λ P r0, 15s.
output of the other firms exceeds 21, while 24 is optimal when it falls below 11. Overall,
the direction of change in FRE choice distributions in normalized treatments reflects theempirical pattern.
We further explore the differences in variance across market sizes by calculating thestandard deviation of the choice distribution for λ values between 0 and 15. Figure 15
shows that FRE correctly predicts higher standard deviation in smaller markets for all λvalues in this range, but only in the normalized treatments.
Next, we fit FRE to the data using the method originally developed by Bajari andHortacsu (2005). First, for each action, we calculate the expected likelihood of being thebest response, assuming that the strategy profile of other players is generated by each player
independently drawing their strategy from the empirically observed output distribution.
The expected likelihoods are mapped into choice probabilities using the softmax function
with parameter λ. We fit the model by estimating the value of λ that maximizes thelikelihood of the empirical choice distribution.21
21

Note that the value of λ estimated in FRE cannot be compared to the λ values in QRE because theattractions in FRE are measured in likelihood (ranging from 0 to 1) while attractions in QRE represent
the expected monetary earnings.

36

Table 5: Goodness of fit and estimated parameter values in QRE and FRE.

S2
S4
N2
N3
N4
Total

QRE
Separate
Combined
λ̂
LL
λ̂
LL
1.395 -2739.97 1.497 -2740.34
1.535 -4561.56 1.497 -4561.65-0.292 -5436.98 1.497 -5531.25
2.507 -4014.05 1.497 -4024.05
4.686 -5161.69 1.497 -5281.54-21914.25-22138.83

FRE
Separate
Combined
λ̂
LL
λ̂
LL
13.63 -2665.76 7.84 -2701.296
4.42 -4701.504 7.84 -4715.934
4.98 -5324.712 7.84 -5369.582
14.68 -4001.647 7.84 -4018.677
15.32 -5045.288 7.84 -5136.418-21738.91 7.84 -21941.91
Game S4, λ = 4.42
0.06

Game S2, λ = 13.63

0.04

Density

Experiment
Ex−post

0.00

0.02

0.08
0.04
0.00

Density

Experiment
Ex−post

0

10

20

30

40

50

0

10

20

Quantity

Games N3, λ = 14.68

20

Experiment
Ex−post0.15
Density

0.15

0.00

0.05

0.10

Density

0.00
16

50

Game N4, λ = 15.32
Experiment
Ex−post

0.05

0.10
0.00

0.05

Density

0.15

Experiment
Ex−post

12

40

0.10

Game N2, λ = 4.98

830
Quantity

24

8

12

16

Quantity

20

24

8

12

Quantity

16

20

24

Quantity

Figure 16: Best-fitting FRE estimated separately for each treatment.

Experiment
Ex−postDensity

0.04

0.08

Experiment
Ex−post

0.00
0

10

20

30

40

50

0

10

20

QuantityGame N2, λ = 7.84

24

0.00

0.05
20

Experiment
Ex−post0.20
Density

0.15

Experiment
Ex−post0.00
16
Quantity

50

Game N4, λ = 7.84

0.10

Density

0.20
Density

0.10
0.00

12

40

Game N3, λ = 7.84
Experiment
Ex−post830
Quantity

0.10

Density

Game S4, λ = 7.84

0.00 0.02 0.04 0.06 0.08

Game S2, λ = 7.84

8

12

16
Quantity

20

24

8

12

16
Quantity

Figure 17: Best-fitting FRE estimated for all treatments.
37

20

24

Table 5 extends the results from Table 4, comparing the estimated values and goodnessof fit for QRE and FRE. Figures 16 and 17 illustrate the fit by comparing the choiceprobabilities in the best-fitting FRE to the kernel density estimates of the experimental
data. Overall, FRE has a higher total log-likelihood than QRE, both when a separate modelis estimated for each treatment and when a single model is estimated for all treatments.
FRE also fits better in each individual game, except for S4. FRE fails to explain choices in
S4 because it overestimates the frequency of choosing 0. In the experiment, participants
would have maximized their round earnings by producing 0 each time the total output
by other participants exceeded 80 (which happened in over 20% of the rounds), yet 0
was chosen only about 2% of the time. In the normalized treatments, predicted choiceprobabilities are close to the empirical data, although the estimated λ value is much lower
in N2 than in N3 or N4. The higher estimated level of noise in N2 is again caused by FRE
overestimating the frequency of the most extreme output levels. When λ is required to be
the same in all five games, FRE can explain the increased variance in smaller markets of
the normalized treatments, although the higher noise level needed to reduce the predictedfrequency of the extreme choices in S4 and N2 reduces the goodness of fit in the otherthree treatments.
Overall, we conclude that the increased variance in smaller markets, found in the normalized treatments, can be explained by a static solution concept based on the frequency
with which each action is the best response. This model also provides a better fit than
QRE, although it overestimates the frequency of extreme output levels in N2 and S4.

6

Concluding Remarks

Previous Cournot oligopoly experiments found higher rates of tacit collusion in smallermarkets. We demonstrate that this result could be explained by changes in the payoff

38

structure that occur when the market size is manipulated using the standard Cournot
payoff function. We propose an alternative way to manipulate the market size, whichmakes markets with a different number of competitors more comparable. This normalized
design is used to identify whether differences in collusion rates are driven by the market
size itself, or by the changes in incentives or the focality of key outcomes. We replicate
the finding of more collusive output in smaller markets using the standard treatments, but
find no effect of market size in the normalized treatments. These results suggest that the
number effect is largely driven by the changes in incentives or focality, instead of purely
the number of interacting firms.
In the normalized treatments, we also find that the variance of individual output isdecreasing in market size. This effect cannot be explained by the Quantal Response Equilibrium. Instead, it could be explained by the difference in feedback caused by the implemented aggregation rule: total output in standard treatments, average output in normalized treatments. The variance of average output decreases when the market size goesup, making extreme responses more costly. Instead, the variance of total output increases
in larger markets. This difference has consequences beyond the average rates of collusion.
Interestingly, increased variance in smaller markets in the normalized treatments provides
some support for the original result of higher frequency of collusion in smaller markets,
although the frequency of competitive choices goes up as well.
A better understanding of how the number of competitors affects collusion rates could
improve the design of competition policy. When deciding whether to approve a merger, regulators evaluate whether the resulting increase in market concentration would significantly
lessen competition. The practices used by the Department of Justice and the Federal Trade
Commission, two agencies in charge of enforcing antitrust law in the U.S., are explained in
the Horizontal Merger Guidelines.22 The guidelines identify two channels through which
22

U.S. Department of Justice & Federal Trade Commission, Horizontal Merger Guidelines (2010), avail-39

mergers could enhance market power: “unilateral effects” and “coordinated effects”.23 Unilateral effects refer to the higher incentives to increase prices in more concentrated markets;for example, a merger of firms that sell similar products increases the incentive to raise
prices because the lost sales that would have been diverted to the competitor’s products are
now diverted to the products sold by other divisions of the same firm. Coordinated effects
refer to the increased likelihood of implicit or explicit coordination on higher prices in moreconcentrated markets. Successful coordination requires the ability to detect and punish
the firms that deviate from collusive agreements,24 which is easier when there are fewer
firms and the behavior of rivals is more predictable. These two channels correspond to thetwo mechanisms studied in this paper: individual incentives and the pure number effect.
Just as the real firms, participants in Cournot oligopoly could collude more in smaller markets because of higher incentives or because implicit coordination is easier when there arefewer participants. Identifying the mechanism is critical to select the appropriate strategy
for regulating mergers. If collusion is driven primarily by coordinated effects, regulators
should focus on market concentration, as was advocated in the 1968 and 1982 guidelines(Shapiro, 2010). But if, instead, it is primarily driven by the unilateral effects, regulators
should instead estimate the value of diverted sales by evaluating the degree of product differentiation, market elasticity of demand or costs of output suppression. Recent versionsof the guidelines advocate this view, introducing the concept of unilateral effects in the
1992 guidelines and accentuating it in 2010 (Shapiro, 2010). Consequently, the guidelines
have focused more on the economic factors and techniques to estimate the value of diverted
sales rather than the market concentration.25 The results of our experiments support thisable at http://www.justice.gov/atr/public/guidelines/hmg-2010.pdf
23
The European Commission provides similar guidelines, naming the two channels “coordinated effects”
and “non-coordinated effects”, see Guidelines on the Assessment of Horizontal Mergers under the Council
Regulation on the Control of Concentrations between Undertakings, 2004 O.J. (L 24) 1 (EC), available at
http://eur-lex.europa.eu/LexUriServ/LexUriServ.do?uri=OJ:C:2004:031:0005:0018:EN:PDF
24
U.S. Department of Justice & Federal Trade Commission, Commentary on the Horizontal Merger
Guidelines (2006), available at http://www.justice.gov/atr/public/guidelines/215247.pdf
25
Section 4 of the 2010 Guidelines: “The measurement of market shares and market concentration is40

shift, providing evidence that collusion in small markets occurs not because of a smaller
number of competitors, but because of the increased incentives to collude. Mergers that do
not create additional incentives to collude (e.g. in markets with a low diversion ratio andlow margins, Farrell and Shapiro, 2010) might not need to be blocked even if they increase
market concentration.
Our study has several limitations that would be interesting to address in future research.
We focused only on implicit collusion, but it would be interesting to compare our results toa framework in which explicit collusion is possible, perhaps by adding communication. On
one hand, coordination problem might be more important when communication is possible;on the other hand, evidence shows that with communication, even large groups manage tocollude, therefore the number effect is not found even with the standard Cournot payoff
function (Waichman et al., 2014; Fonseca et al., 2018). It would also be interesting to extend
the setup to other games, for example, Bertrand competition, where a similar number effecthas been observed for both tacit and explicit collusion (Fonseca and Normann, 2012).

not an end in itself, but is useful to the extent it illuminates the merger’s likely competitive effects”.

41

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Appendix
A

Strategy Space Labeling

Figure A.18 shows the distribution of choices in the normalized treatment, comparing thetwo different labeling schemes. In the increasing scheme, quantities from 8 to 24 werelabeled from 0 to 24. In the decreasing scheme, the labeling was reversed, with the choiceof 0 corresponding to the output of 24 and the choice of 16 corresponding to the output of 8.
Figure A.18 shows that labeling has some effect on the distribution of choices. The originallabels seen by participants are displayed on the two axes. Data shows that participants
make competitive choices more often in the increasing treatments, for all market sizes.
In part, this is explained by action “10” being more focal and frequently chosen in alltreatments. In increasing treatments, the action labeled as “10” is midway between the
Nash and Walrasian equilibria, but in decreasing treatments it is more collusive than the
Nash equilibrium prediction. However, there are other differences as well. With all threemarket sizes, action 16 is chosen more often than action 0, regardless of whether it ismapped into output 8 or 24. There is some evidence that the distribution of choices isshifted towards the actions with higher labels, especially in the 3-firm market. Thus it
seems that the effect of labeling is driven both by the focality of action “10” and by apreference for choosing actions labeled with higher numbers.
The right column of Figure A.18 shows the distribution of choices only in the last 5
rounds. The effect of labeling persists in 2 and 3 firm markets, although the magnitude ofthe effect is lower.
We evaluate the effect of labeling formally by comparing the ratio of average output
to Nash equilibrium prediction. Table A.6 shows that on average, the chosen output
is above NE prediction in increasing treatments, but below it in decreasing treatments.

48

14

12

10

N2D labels
8

4

N

2

0

16

14

W

12

10

N2D labels
8

C

6

4

N

2

0

14

16

2

0

14

16

2

0

14

16

W

Percent
10
0

0

5

5

Percent
10

15

15

C

6

20

16

0

2

4

6

8
N2I labels

10

N2I

12

14

16

0

2

4

N2D

14

12

10

N3D labels
8

10

12
N2D

(b) N2, rounds 15-20
4

0

16

14

W

12

10

N3D labels
8

C

6

4

N

W

Percent
10
5
0

0

5

Percent
10

15

N

2

15

C

6

8
N2I labels
N2I(a) N2, rounds 1-20
16

6

0

2

4

6

8
N3I labels

10

N3I

12

14

16

0

2

4

N3D

12

10

N4D labels
8

4

N3D

2

0

16

14

12

10

N4D labels
8

C

W

6

4

N

W

5
0

0

5

Percent
10
15

N

12

(d) N3, rounds 15-20

Percent
10
15

20

C

6

10

20

14

8
N3I labels
N3I(c) N3, rounds 1-20
16

6

0

2

4

6

8
N4I labels
N4I

10

12

14

16

0

N4D

2

4

6

8
N4I labels
N4I(e) N4, rounds 1-20

10

12
N4D

(f) N4, rounds 15-20

Figure A.18: Output distributions in normalized treatments for the increasing and deceasing labeling schemes.

49

Table A.6: Comparison of the ratio of average output to NE prediction and average profit
per round. The number in brackets shows the values averaged across rounds 15-20.

{

qi qiNπi

N2I
1.03[1.02]
366.4[377.5]N3I
1.07[1.10]
322.8[299.5]N4I
1.03[1.02]
354.2[363.4]

N2D
0.94[0.97]
408.3[386.3]N3D
0.92[0.95]
419.5[396.6]

N4D
0.98[1.02]
390.6[360.3]Mann-Whitney U test shows that choices are significantly more collusive in the decreasing
treatments (MWU p  0.0111 in 4-firm treatments, p  0.0085 in 3-firm treatments andp  0.0296 in 2-firm treatments). If we look only at the last 5 rounds, the differenceis significant only in 3-firm market (p  0.0178). A similar result is found in termsof the generated profits, which are significantly higher in decreasing treatments (MWUp  0.0153 in 4-firm treatments, p  0.0056 in 3-firm treatments and p  0.0392 in 2-firm
treatments). In the last 5 rounds, the difference is significant only in 3-firm treatments(MWU p  0.0243).
The manipulation of labels allows us to more accurately assess the baseline aggregate
levels of collusion. Had we run only the increasing treatments, as is done in all the previous
literature, we would likely conclude that there is a tendency to behave more competitively
than predicted by the Nash equilibrium, especially in treatments with the market size of 3
and 4. But the comparison to the decreasing treatments reveals that this tendency is driven
in part by the labeling rather than the incentive scheme. By combining the data from two
different labeling schemes, we can evaluate and eliminate the effect of focal locations.

50

B

Response to Feedback

The finding that variance is decreasing in market size only in the normalized treatments
could be explained by the difference in how the output of the opponents is aggregated in
the two designs. In the standard treatments, the sum of opponents’ output is mapped
into the same best-response, regardless of the market size. In the normalized treatments,
the average of opponents’ output is mapped into the same best-response, regardless of themarket size. The distribution of the total and average output varies with the market size,
affecting the variance of the best-response distribution. As an illustration, suppose thateach opponent chooses their output qi from a normal distribution N pµ, σ 2 q. Then the sumof the total output chosen by pn  1q opponents is the sum of pn  1q random variables
drawn from normal distributions. The probability distribution of the total output is anormal distribution with the following parameters:


¸

n 1



qj  N ppn  1qµ, pn  1qσ 2 q

j 1

The probability distribution of the average output chosen by pn  1q opponents is:
1


¸

n 1

n  1 j 1

qj  N pµ,

1

n1

σ2q

Note that as the market size increases, the variance of the total output goes up, but thevariance of the average output goes down. This difference is subsequently translated intoa difference in the variance of the best-response distribution, as illustrated in Figures B.1
and B.2. The curves inside the figures display the probability distributions of the payoffrelevant statistics – total output for standard treatments and average output for normalized
treatments. The distributions are plotted assuming that the mean of the distribution isequal to the Nash equilibrium and the standard deviation is equal to 8 in standard and 351

50Best response

0

10

20

30

40

S2
S4

0

30

60

90

120

150

Sum of output of the opponents

Figure B.1: Best-response curve, probability distribution of average output of the opponents and the resulting probability distribution of the best response (on the left). Standard
deviation equals to 8.
in normalized treatments (close to the standard deviations seen in experiments, as shownin Figure 9). The variance of the total opponents’ output is increasing in market size
(Figure B.1), but the variance of the mean output is decreasing in market size (Figure B.2).
The black line in each figure shows how the total or mean output chosen by opponents(which is on the x-axis) is mapped into the best response. The curves on the left side of the
figure show the resulting distribution of best-responses to the corresponding distributionof either the sum or the mean output of the opponents. As one can observe, a higher
variance of the latter distribution translates into a higher variance of the best-response
distribution. We conclude that the variance of the best-response distribution would be
predicted to increase in market size in standard treatments, but decrease in market size innormalized treatments.

52

24Best response

8

12

16

20

N2
N4

8

12

16

20

24

Average output of the opponents

Figure B.2: Best-response curve, probability distribution of average output of the opponents and the resulting probability distribution of the best response (on the left). Standard
deviation equals to 3.

53

C

Additional Results

Table C.1: Structure of all treatments that were run. This paper uses data only from thefirst block of 20 rounds. In games marked with I, the strategy space 8-24 was mapped into
0-16. In games marked with D, the strategy space was reversed before mapping into 0-16.
Games marked with S use a standard Cournot oligopoly incentive structure with a strategy
space 0-50. SC3 is a 3-person game of strategic complements, with either a strategy space
0-16 (SC3) or 0-50 (SC’).
Treatment
Rounds 1-20
Rounds 21-40
Rounds 41-60# Sessions# Participants# Markets in B1

1
2
N2I N4IN3I N3I
SC3 SC3
4
4
48
48
24
12

3
N3I
SC3
3
36
12

4
5
6
N2D N4D N3D
N3D N3D SC3
SC3 SC3 4
4
3
48
48
36
24
12
12

7
S2
S3
SC3’
3
36
18

8
S4
S3
SC3’
5
60
15

Table C.2: Random effects GLS regression. Standard errors clustered on the group level.
Data from all rounds.

3-firm market
4-firm market
1/RoundConstantN

Increasing
DV: r
DV: ϕN
DV: ϕW
0.0412-0.110-0.0659(1.61)(-1.61)(-1.61)0.000326 -0.000868 -0.000521(0.01)(-0.01)(-0.01)0.00697-0.0186-0.0112(0.23)(-0.23)(-0.23)1.024*** -0.0651
0.361***(49.51)(-1.18)(10.90)
2640
2640
2640

t statistics in parentheses* p 0.10, ** p 0.05, *** p

0.01

54

DV: r
-0.0193(-0.44)0.0375(1.33)-0.0834***(-2.82)
0.953***(34.52)2640

Decreasing
DV: ϕN DV: ϕW
0.05160.0309(0.44)(0.44)-0.1000-0.0600(-1.33)(-1.33)0.222*** 0.133***(2.82)(2.82)
0.124* 0.475***(1.69)(10.74)2640
2640

Table C.3: Random effects GLS regression. Standard errors clustered on the group level.
Data from rounds 15-20.

3-firm market
4-firm market
1/Round

DV: r–Standard
DV: ϕN DV: ϕW––

0.182***(3.79)
1.861(0.43)-0.418**(-2.56)-6.749(-0.53)-0.571***(-8.07)-2.619(-0.39)0.943***(3.64)
576

0.187(0.24)
576

0.750*(1.87)
576

D Treatments
ConstantNt statistics in parentheses* p 0.10, ** p 0.05, *** p

DV: r0.0322(0.85)
0.0256(0.93)-1.574(-1.15)-0.0612**(-2.40)1.112***(13.82)
1584

Normalized
DV: ϕN DV: ϕW-0.0858 -0.0515(-0.85)(-0.85)-0.0683 -0.0410(-0.93)(-0.93)
4.197
2.518(1.15)(1.15)
0.163** 0.0979**(2.40)(2.40)-0.3000.220*(-1.40)(1.71)
1584
1584

0.01

Table C.4: Goodness of fit and estimated parameter values in QRE and FRE, separately
for standard and normalized treatments.
Treatment
S2
S4
N2
N3
N4
Total

QRE
λ̂
LL
1.482 -2740.24
1.482 -4561.73
1.526 -5534.16
1.526 -4023.46
1.526 -5279.18-22138.77

55

FRE
λ̂
LL
8.28 -2696.34
8.28 -4720.376
7.70 -5365.159
7.70 -4019.388
7.70 -5139.97-21941.23

0.04

Density

0.00

0.02

Experiment
QRE

0.02

0.04

Experiment
QRE

0.00
0

10

20

30

40

50

0

10

20

QuantityGame N2, λ = 1.53

16

20

24

0.15

Experiment
QRE

0.00

0.05

0.10

Density

0.15

Experiment
QRE

0.00

0.00

12

50

Game N4, λ = 1.53

0.05

0.10

Density

0.15

Experiment
QRE

8

40

Game N3, λ = 1.53

0.05

Density

30
Quantity

0.10

Density

0.06

Game S4, λ = 1.48

0.06

Game S2, λ = 1.48

8

12

16

Quantity

20

24

8

12

Quantity

16

20

24

Quantity

Figure C.1: Best-fitting QRE estimated separately for standard and normalized treatments(compared to kernel density).

Game S4, λ = 8.28

0.08
Density

0.00

0.02

Experiment
Ex−post

0.04

0.04

Experiment
Ex−post

0.00

10

20

30

40

50

0

10

20

QuantityGame N2, λ = 7.70

Density

0.15

Experiment
Ex−post20

24

Experiment
Ex−post

0.00

0.05
0.00
16
Quantity

50

Game N4, λ = 7.70

0.10

Density

0.20
Density

0.10
0.00

12

40

Game N3, λ = 7.70
Experiment
Ex−post830
Quantity

0.20

0

0.10

Density

0.06

Game S2, λ = 8.28

8

12

16
Quantity

20

24

8

12

16

20

24

Quantity

Figure C.2: Best-fitting FRE estimated separately for standard and normalized treatments(compared to kernel density).

56

D

Instructions

The original French version of instructions is available on request. We provide an English
translation below. We reproduce only the instructions for the treatments with the “normalized” design. The only difference in the instructions for the “standard” design is the
cardinality of the strategy space: number “16” in highlighted places was replaced by “50”.

GENERAL INSTRUCTIONS

Welcome to the Laboratory of Experimental Economics of Nice (LEEN – Nice Lab).
By agreeing to participate in this experiment, you agree with the regulations of the laboratory, which are available on our website or on request.
In this experiment your decisions will be anonymous and will partly determine your final
payment, therefore read the following instructions carefully. The participation fee of 5 EUR
is included in the payoff function. Your earnings will be paid to you individually andconfidentially private once you complete a short questionnaire at the end of the experiment.
In this experiment you can earn money. During the experiment we will refer to ECU(Experimental Currency Unit) instead of EUR. The total amount of ECU that you will
have earned during the experiment will be converted into cash and paid individually atthe end of the experiment. The conversion rate used to convert your ECU into your cash
payment will be 150 ECU = 1 EUR.
We ask you not to communicate or to disturb the other participants. We also ask you toturn off your mobile phones and not use them during the experiment.
In these rules are not followed, the experiment may be stopped and all payments canceled.
57

If you encounter a technical problem, we ask you to raise your hand silently and wait for
the experimenter.
All the participants with whom you interact during this experiment will receive the same
instructions and participate in the same experiment.

DESCRIPTION OF THE EXPERIMENTThe experiment will have several parts. Each part will consist of 20 rounds. At the endof the experiment one round from each part will be randomly selected for payment. All
rounds have an equal chance to be selected. Your earnings from the selected rounds will
be added up, converted into cash and paid to you in private.
In each round you will be matched with other participants. At the start of each part you
will be informed about how many participants you will be playing with. You will choose
a number (between 0 and 16), which we will call “your action”. Every other participant
will choose an action at the same time. Your payoff will depend on your action and on the
average action of all other participants with whom you were matched. In each round, youwill have 30 seconds to make a decision. To make a choice, you must enter your action
into the field at the top and click “OK” before the time runs out. The participants withwhom you will interact will face the same task as you and will have the same informationand payoff function. The task, the payoff function and the participants with whom you
will interact will be the same in each round of one part. In each new part, you will playagainst participants with whom you did not interact in previous parts.
The exact way of how your payoff depends on your action and on the average action ofother participants will be explained using a payoff table and a payoff calculator, which will
be available on the computer screen when you will be making your decision.
58• The payoff table shows your payoffs for some combinations of your action and theaverage action of other participants.
• The payoff calculator allows you to enter any action for yourself and an average
action of other participants, and displays a payoff that you would receive in thatcase.

Starting from round 2, you will also be informed about your payoff in the previous round.
Furthermore, you will have an option to view the following additional information:

1. Average choices and their history. This option gives you information about the
average choice of other participants and your payoff in the previous round, as well as
in all earlier rounds of that part.
2. Individual choices and payoffs. This option gives you information about the
choices and payoffs of each member in your group, including yourself.

You will be able to switch between these options using buttons on your computer screen.
In addition, after the first round of each part we will ask you to guess the average actionof other participants in that round. The closer your guess is to the average choice ofother participants, the higher will be your payment. If your guess is G and the actual
average action of other participants is D, your payment will be higher the smaller is theabsolute difference between G and D (denoted |G  D|). In particular, your payoff will



be:

D |
1  |G16

 100 ECU. Notice that if your guess is exactly equal to the average choice(G  D  0), you will receive 100 ECU. At the end of the experiment one of these tasks will
be randomly chosen. The payment from the chosen task will be added to your earnings.

59

At the end of all parts you will be informed about your payoff in ECU from the rounds
that were randomly selected for payment. Payoff from these rounds will be summed up,
converted into EUR and paid in private once you complete a short questionnaire. In thequestionnaire you will have a chance to make additional income which will be added toyour earnings. Please stay seated until we ask you to come to receive the earnings.
If you have any further questions, please raise your hand now. The experiment will start
once everyone has finished reading the instructions.

60

E

Payoff Tables
Table E.1: Payoffs in N2, N3 and N4 treatments.
Output
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24

8
514
553
590
625
658
689
718
745
770
793
814
833
850
865
878
889
898

9
10
490 466
526 499560 530
592 559622 586
650 611676 634
700 655
722 674742 691
760 706
776 719790 730
802 739812 746820 751826 754

11
442
472
500
526
550
572
592
610
626
640
652
662
670
676
680
682
682

12
418
445
470
493
514
533
550
565
578
589
598
605
610
613
614
613
610

Average output chosen by opponents
13 14 15 16 17 18 19 20
394 370 346 322 298 274 250 226
418 391 364 337 310 283 256 229
440 410 380 350 320 290 260 230
460 427 394 361 328 295 262 229
478 442 406 370 334 298 262 226
494 455 416 377 338 299 260 221
508 466 424 382 340 298 256 214
520 475 430 385 340 295 250 205
530 482 434 386 338 290 242 194
538 487 436 385 334 283 232 181
544 490 436 382 328 274 220 166
548 491 434 377 320 263 206 149
550 490 430 370 310 250 190 130
550 487 424 361 298 235 172 109
548 482 416 350 284 218 152 108
544 475 406 337 268 199 130 107
538 466 394 322 250 178 106 106

21
202
202
200
196
190
182
172
160
146
130
112
111
110
109
108
107
106

22
178
175
170
163
154
143
130
115
114
113
112
111
110
109
108
107
106

23
154
148
140
130
118
117
116
115
114
113
112
111
110
109
108
107
106

24
130
121
120
119
118
117
116
115
114
113
112
111
110
109
108
107
106

Table E.2: Payoffs in S2 treatment.
Output
0
2.5
5
7.5
10
12.5
15
17.5
20
22.5
25
27.5
30
32.5
35
37.5
40
42.5
45
47.5
50

0
2.5 5.0
130 130 130
200 197 195
265 260 256
326 319 312
382 373 364
434 422 411
481 467 454
524 508 492
562 544 526
596 575 555
625 602 580
650 625 600
670 643 616
686 656 627
697 665 634
704 670 636
706 670 634
704 665 627
697 656 616
686 643 600
670 625 580

7.5
130
193
251
305
355
400
440
476
508
535
557
575
589
598
602
602
598
589
575
557
535

10.0
130
191
247
299
346
389
427
461
490
515
535
551
562
569
571
569
562
551
535
515
490

12.5
130
188
242
292
337
377
413
445
472
494
512
526
535
539
539
535
526
512
494
472
445

15.0
130
186
238
285
328
366
400
429
454
474
490
501
508
510
508
501
490
474
454
429
400

Average output chosen by opponents
17.5 20.0 22.5 25.0 27.5 30.0 32.5 35.0
130 130 130 130 130 130 130 130
184 182 179 177 175 173 170 168
233 229 224 220 215 211 206 202
278 272 265 258 251 245 238 231
319 310 301 292 283 274 265 256
355 344 332 321 310 299 287 276
386 373 359 346 332 319 305 292
413 398 382 366 350 335 319 303
436 418 400 382 364 346 328 310
454 434 413 393 373 353 332 312
467 445 422 400 377 355 332 310
476 452 427 402 377 353 328 303
481 454 427 400 373 346 319 292
481 452 422 393 364 335 305 276
476 445 413 382 350 319 287 256
467 434 400 366 332 299 265 231
454 418 382 346 310 274 238 202
436 398 359 321 283 245 206 168
413 373 332 292 251 211 170 130
386 344 301 258 215 173 130 113
355 310 265 220 175 130 112 112

61

37.5
130
166
197
224
247
265
278
287
292
292
287
278
265
247
224
197
166
130
114
113
112

40.0
130
164
193
218
238
254
265
272
274
272
265
254
238
218
193
164
130
115
114
113
112

42.5
130
161
188
211
229
242
251
256
256
251
242
229
211
188
161
130
116
115
114
113
112

45.0 47.5 50.0
130 130 130
159 157 155
184 179 175
204 197 191
220 211 202
231 220 209
238 224 211
240 224 209
238 220 202
231 211 191
220 197 175
204 179 155
184 157 130
159 130 118
130 117 117
117 117 117
116 116 116
115 115 115
114 114 114
113 113 113
112 112 112

Table E.3: Payoffs in S4 treatment.
Output
0
2
4
6
8
10
12
14
16
18
20
22
24
26
28
30
32
34
36
38
40
42
44
46
48
50

0
130
286
434
574
706
830
946
1054
1154
1246
1330
1406
1474
1534
1586
1630
1666
1694
1714
1726
1730
1726
1714
1694
1666
1630

2
130
274
410
538
658
770
874
970
1058
1138
1210
1274
1330
1378
1418
1450
1474
1490
1498
1498
1490
1474
1450
1418
1378
1330

4
130
262
386
502
610
710
802
886
962
1030
1090
1142
1186
1222
1250
1270
1282
1286
1282
1270
1250
1222
1186
1142
1090
1030

6
130
250
362
466
562
650
730
802
866
922
970
1010
1042
1066
1082
1090
1090
1082
1066
1042
1010
970
922
866
802
730

8
130
238
338
430
514
590
658
718
770
814
850
878
898
910
914
910
898
878
850
814
770
718
658
590
514
430

10
130
226
314
394
466
530
586
634
674
706
730
746
754
754
746
730
706
674
634
586
530
466
394
314
226
130

12
130
214
290
358
418
470
514
550
578
598
610
614
610
598
578
550
514
470
418
358
290
214
130
84
82
80

14
130
202
266
322
370
410
442
466
482
490
490
482
466
442
410
370
322
266
202
130
90
88
86
84
82
80

16
130
190
242
286
322
350
370
382
386
382
370
350
322
286
242
190
130
96
94
92
90
88
86
84
82
80

Average output chosen by opponents
18 20
22 24 26 28 30 32
130 130 130 130 130 130 130 130
178 166 154 142 130 128 128 128
218 194 170 146 126 126 126 126
250 214 178 142 124 124 124 124
274 226 178 130 122 122 122 122
290 230 170 120 120 120 120 120
298 226 154 118 118 118 118 118
298 214 130 116 116 116 116 116
290 194 114 114 114 114 114 114
274 166 112 112 112 112 112 112
250 130 110 110 110 110 110 110
218 108 108 108 108 108 108 108
178 106 106 106 106 106 106 106
130 104 104 104 104 104 104 104
102 102 102 102 102 102 102 102
100 100 100 100 100 100 100 100
98 98
98 98 98 98 98 98
96 96
96 96 96 96 96 96
94 94
94 94 94 94 94 94
92 92
92 92 92 92 92 92
90 90
90 90 90 90 90 90
88 88
88 88 88 88 88 88
86 86
86 86 86 86 86 86
84 84
84 84 84 84 84 84
82 82
82 82 82 82 82 82
80 80
80 80 80 80 80 80

62

34
130
128
126
124
122
120
118
116
114
112
110
108
106
104
102
100
98
96
94
92
90
88
86
84
82
80

36
130
128
126
124
122
120
118
116
114
112
110
108
106
104
102
100
98
96
94
92
90
88
86
84
82
80

38
130
128
126
124
122
120
118
116
114
112
110
108
106
104
102
100
98
96
94
92
90
88
86
84
82
80

40
130
128
126
124
122
120
118
116
114
112
110
108
106
104
102
100
98
96
94
92
90
88
86
84
82
80

42
130
128
126
124
122
120
118
116
114
112
110
108
106
104
102
100
98
96
94
92
90
88
86
84
82
80

44
130
128
126
124
122
120
118
116
114
112
110
108
106
104
102
100
98
96
94
92
90
88
86
84
82
80

46
130
128
126
124
122
120
118
116
114
112
110
108
106
104
102
100
98
96
94
92
90
88
86
84
82
80

48
130
128
126
124
122
120
118
116
114
112
110
108
106
104
102
100
98
96
94
92
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50
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F

Screenshots

Figure F.1: Payoff table in normalized treatments.

Figure F.2: Payoff calculator in normalized treatments.

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Figure F.3: Feedback about individual choices and payoffs in normalized treatments.

Figure F.4: Feedback about the history of own choices and payoffs in previous rounds.

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