Potters, J.J.M.

Suetens, S.

English

Research Papers in Economics

Exp Econ (2007) 10:71–77
DOI 10.1007/s10683-006-9132-2
Bertrand colludes more than Cournot
Sigrid Suetens · Jan Potters
Received: 27 April 2006 / Revised: 14 September 2006 /
Accepted: 18 September 2006 / Published online: 31 January 2007
C   Economic Science Association 2007
Abstract On the basis of evidence of past oligopoly experiments, we argue that there
is often signiﬁcantly more tacit collusion in Bertrand price-choice than in Cournot
quantity-choice markets.
Keywords Collusion . Quantity-choice and price-choice experiments
JEL Classiﬁcation L00, L13, D21
1 Introduction
An important issue in industrial organisation is the incidence of tacit collusion.
Given that Bertrand price competition and Cournot quantity competition are the two
workhorsemodelsinIO,anaturalquestioniswhetheroneismorepronetotacitcollu-
sion than the other. Somewhat surprisingly perhaps, this issue has not been explicitly
addressed in any experiments.
Based on a survey of experimental IO research, Holt (1995) hypothesises that in
Cournot settings there is less collusion than in Bertrand settings. However, apart from
Fouraker and Siegel (1963), referred to as FS in what follows, the studies on which
this hypothesis is based have either Cournot or Bertrand treatments, but not both.
Recently, a series of papers have investigated the impact of information feedback
on behaviour in Cournot and Bertrand settings (see Huck et al., 2000; Davis, 2002;
Altavilla et al., 2006, further referred to as HNO, Davis and ALS, respectively). As a
S. Suetens ()
University of Antwerp, Department of Economics,
Prinsstraat 13, B-2000 Antwerp Belgium
e-mail: sigrid.suetens@ua.ac.be
J. Potters
Tilburg University, CentER, P.O.Box 90153, 5000 LE Tilburg, The Netherlands
e-mail: j.j.m.potters@uvt.nl
Springer72 S. Suetens, J. Potters
by-product,theunderlyingdataallowforadirectcomparisonofthetendencytotacitly
collude between Cournot and Bertrand environments with two and more sellers. In
the present paper we review the evidence and ﬁnd support for Holt’s hypothesis.
2 Overview of experimental designs
FS, HNO, Davis and ALS examine the impact of information on behaviour in ex-
periments in which subjects play repetitions of static Cournot and Bertrand games.
The reported experiments have similar designs and contain BASIC and EXTRA treat-
ments.1 In the BASIC treatments only aggregate information was provided on the
competitors’ quantities or prices and no information about their proﬁt. In the EXTRA
treatments information was provided on all individual quantities or prices as well as
on proﬁts.2
Althoughtherearedifferencesinexperimentalproceduresbetweenthefourstudies
with respect to type of matching, information about duration and number of players,
these procedures are the same across Cournot and Bertrand treatments within each
study.
The following inverse demand function for ﬁrm i producing substitute goods is
used in all Cournot treatments:
pi = a − b
 
qi + θ
 
j =i
qj
 
(1)
with a,b > 0 and 0 <θ≤ 1.
IntheBertrandtreatmentstheformofthedemandfunctiondependsonwhetherone
assumes homogeneous goods. FS assume homogeneous goods and use the following
demand function:
qi =
⎧
⎨
⎩
A − Bpi
L
: pi ≤ pj
0: pi > pj
where L is the number of producers tied for the lowest price. Only the producer
setting the lowest price will produce to meet the entire demand and in case of a tie
total production is evenly split up amongst the tied producers. The other Bertrand
treatments are based on a setting with differentiated goods. The demand function of
ﬁrm i, which is a transformation of inverse demand (Eq. (1)) provided that qi ≥ 0, is
qi = α − βpi + γ
 
j =i pj
n − 1
(2)
1 The Cournot and Bertrand experiments of FS are discussed in part three of their book. BASIC refers
to their incomplete information treatment and EXTRA to their complete information treatment. For ALS,
BASIC and EXTRA respectively refer to their ED 1 and ED 3 treatments.
2 In the BASIC treatments of FS subjects were not informed about their competitors’ proﬁt function. In
the EXTRA treatments of FS the competitors’ proﬁt could be calculated on the basis of proﬁt tables, apart
from the Cournot triopoly treatment, where only the sum of the competitors’ proﬁt could be calculated.
SpringerBertrand colludes more than Cournot 73
with
α =
a
b[1 + (n − 1)θ]
(3)
β =
1 + (n − 2)θ
b(1 − θ)[1 + (n − 1)θ]
(4)
γ =
(n − 1)θ
b(1 − θ)[1 + (n − 1)θ]
. (5)
Under Cournot competition, πi = (pi − c)qi − C is maximised with respect to qi
and under Bertrand competition, with respect to pi.F Su s ea = 2.4, b = 0.04, θ = 1
and c,C = 0 in the Cournot treatments, and A = 56, B = 8, c = 0, C = 13 for a pro-
ducer who actually produced and C = 25 for a producer who does not produce in the
Bertrand treatments. These cost parameters were chosen in order to facilitate compe-
tition.3 HNO use a = 300, b = 1, θ = 2/3, c = 2 and C = 0 in all treatments. Davis
uses a = 225, b = 1, θ = 2/3 and C = 0 in all treatments, c = 33 in all Bertrand
treatments and the EXTRA Cournot treatment and c = 1 in the BASIC Cournot treat-
ment.4 ALS use a = 24 and c,C = 0 in all treatments, b = 1/2 and θ = 1i nt h e
Cournot treatment with homogeneous products (C1), and b = 2/3 and θ = 1/2i nt h e
Cournot treatment with differentiated products (C2) and the Bertrand treatment.
Table 1 summarises the theoretical benchmarks that correspond to these parameter
choices and includes other features (such as number of ﬁrms, number of rounds,
type of matching and minimum and maximum feasible action) in the four sets of
experiments (C refers to Cournot and B to Bertrand). Note that in the FS, HNO and
ALS experiments, theoretical benchmarks are the same across BASIC and EXTRA.
In Table 1 the Nash equilibrium choice of the stage game is in bold. In ﬁnitely
repeated games with a known end, the unique subgame perfect Nash equilibrium cor-
respondstorepeatedplayoftheuniqueNashequilibriumofthestagegame.However,
from earlier studies we know that even in ﬁnitely repeated games, players manage to
cooperate to some degree and behave similarly as in inﬁnitely repeated games or as in
games with an unknown end (see Selten and Stoecker, 1986; Normann and Wallace,
2004). One measure for the scope for tacit collusion in a repeated game is the ratio
between the gains from cooperation and the gains from defection,
πJPM−πNash
πDefect−πJPM, which
in the table we refer to as the Friedman index after Friedman (1971). The higher the
index, the higher the possibility that tacit collusion can be sustained as an equilibrium
in an inﬁnitely repeated game context as part of a grim trigger strategy. Martin (1995),
van Wegberg (1995) and Potters et al. (2004), for instance, use this index to measure
the scope for tacit collusion. It is clear from Table 1 that on the basis of this index one
3 In the Bertrand triopolies, payoff of producers tied at the lowest price is set equal to 0. The payoff table
FS use in the Cournot treatments is a rough approximation of the payoffs based on the parameter values.
In fact, all payoffs in the table are multiples of 50. The calculations in Section 3 will be based on the exact
payoff function.
4 Davis expected Nash behaviour under BASIC and imitation under EXTRA. Therefore he implemented
different cost parameters to ensure equivalence between Nash equilibria in Bertrand and Cournot settings
in the former case and between imitation equilibria in Bertrand and Cournot settings in the latter case (see
Section 3 for more on the imitation equilibrium).
Springer74 S. Suetens, J. Potters
Table 1 Summary of features of Cournot/Bertrand experiments
FS duopoly FS triopoly HNO Davis ALS
C B C BCB CB C 1 C 2 B
n 2 2 3 344 44 2 2 2
# rounds T 22 15 22 15 40 40 40 40 20 20 20
duration not known not known known not known known
matching ﬁxed ﬁxed ﬁxed ﬁxed random in BASIC
ﬁxed in EXTRA
minimum 8 0.5 8 0.5 0 0 0 0 0 0 0
maximum 32 5 32 5 400 1000 −− 48 48 24
pBASIC
Nash 0.8 0.5 0.6 0.5 76.5 39.25 57 57 8 9.6 8
pBASIC
JPM 1.2 3.5 1.2 3.5 151 151 113.1 129 12 12 12
qBASIC
Nash 20 26 15 17.33 74.5 86.92 56 56 16 14.4 16
qBASIC
JPM 15 14 10 9.33 49.67 49.67 37.3 32 12 12 12
πBASIC
Nash 16 0 9 0 5550 3238 3136 1344 128 138.2 128
πBASIC
JPM 18 36 12 20 7400 7400 4181 3072 144 144 144
πBASIC
Defect 20 83 16 83 9867 16915 5575 7022 162 150 162
pEXTRA
Nash 81 57
pEXTRA
JPM 129 129
qEXTRA
Nash 48 56
qEXTRA
JPM 32 32
πEXTRA
Nash 2304 1344
πEXTRA
JPM 3072 3072
πEXTRA
Defect 4096 7022
Friedman 1 0.77 0.75 0.32 0.75 0.44 0.75 0.44 0.89 0.96 0.89
JPM = joint proﬁt maximising.
would a priori expect that the tendency to tacitly collude is higher in Cournot than in
Bertrand treatments. For ALS one would expect the same tendency to cooperate when
comparing C1 with B.
3 Degree of tacit collusion
In this section we examine to which extent behaviour in Bertrand and Cournot treat-
ments deviates from the Nash prediction toward full collusion. We use two alternative
measures of the degree of collusion. Both are indices in which the degree of collusion
of oligopoly k in round t is deﬁned as follows:
ρkt =
¯ xkt − xNash
xJPM − xNash
. (6)
SpringerBertrand colludes more than Cournot 75
For ¯ xkt we take the average price of oligopoly k in round t as a ﬁrst measure, and the
average proﬁt as a second measure.5 If ρkt = 0, oligopoly k behaves on average ac-
cordingtotheNashequilibriumofthestagegameinroundt,if0<ρ kt ≤ 1,behaviour
of oligopoly k in round t is collusive, and if ρkt < 0, behaviour is more competitive
than in the Nash equilibrium.
Table 2 provides averages and standard deviations of the degree of collusion based
on prices and proﬁts for the different experiments, where ¯ ρ = 1
NT
 N
k=1
 T
t=1 ρkt.
The standard deviations refer to cross-sectional variability.6 Also included in the table
foreachexperimentare p-valuesofMann-Whitney-U testsof H0 :¯ ρCournot = ¯ ρBertrand
based on independent data.7
Table 2 shows that average degrees of collusion are higher in Bertrand than in
Cournot treatments in all possible comparisons except under BASIC information con-
ditions in the experiment of Davis.8 Under BASIC information conditions in the
experiment of HNO, the difference is in the ‘right’ direction but is not signiﬁcant,
while the difference is highly signiﬁcant under BASIC in FS and ALS. Under EX-
TRA information conditions, the difference between Cournot and Bertrand settings
is for all experiments highly signiﬁcant. In fact, in ALS, when comparing C2 and
B, the effect is so strong that it overturns the standard theoretical results that price
competition leads to lower prices than quantity competition, under both BASIC and
EXTRA conditions.
The ﬁnding that Bertrand environments are more prone to tacit collusion than
Cournot environments is clearly in contrast to what the Friedman index predicts, thus
other explanations should be looked for.
One possibility is that the location of the Nash equilibrium in the strategy spaces
induces relatively competitive behaviour under Cournot and relatively cooperative
behaviour in the Bertrand treatments.9 After all, in most studies both the equilibrium
quantity and the equilibrium price are in the lower end of the strategy space. It should
be noted though that the Bertrand games are signiﬁcantly more cooperative than the
5 For the Cournot treatments the price of each oligopoly member is calculated on the basis of the inverse
demand function taking into account that pi ≥ 0. Another option would be to calculate the degree of
collusion on the basis of the quantity decisions and benchmarks. As long as prices calculated on the basis
of inverse demand curves and quantities calculated on the basis of demand curves are not below 0, both
alternativesgivethesameresult.IntheCournottreatmentsoftheFSexperimentsandintheALS,HNOand
Davis experiments zero prices and quantities had to be enforced because calculated prices and quantities
were sometimes below 0. This adjustment results in slight differences in degrees of collusion, depending on
whether calculated on the basis of prices or quantities. However, all analyses based on the degree of price
or quantity collusion yield exactly the same conclusions. With respect to the Bertrand treatments in the FS
experiments,largerdifferencesnaturallyoccurredwhetherthedegreeofcollusionwasmeasuredonthebasis
of the price choices or the calculated quantities, because the proﬁt function deﬁned in terms of quantities is
not a linear transformation of the proﬁt function deﬁned in terms of prices (due to the homogeneous good
environment). Yet, even with respect to FS, the conclusions of the tests are exactly the same, whether based
on the degree of price or quantity collusion. For ALS BASIC we only have subject-speciﬁc decisions, such
that the degree of collusion in their BASIC Cournot and Bertrand treatments is respectively calculated on
the basis of quantity and price choices.
6 Onlypre-mergerdatafromDavisareusedforourpurposessincepost-mergerdataareexanteasymmetric.
7 The ALS BASIC data are subject-speciﬁc (cf. random matching) and thus strictly speaking not indepen-
dent. We do not have the data on proﬁts in this treatment.
8 The same conclusions are obtained on the basis of other subsets of rounds.
9 We thank one of the referees for raising this issue.
Springer76 S. Suetens, J. Potters
Table 2 Average degrees of collusion in Cournot/Bertrand experiments and test statistics of H0 :
¯ ρCournot = ¯ ρBertrand
BASIC EXTRA
¯ ρ ¯ ρ
Price Proﬁt N Price Proﬁt N
FS duopoly
Cournot −0.12 (0.16) −0.53 (0.36) 16 −0.24 (0.70) −1.37 (1.70) 11
Bertrand 0.24 (0.15) 0.14 (0.20) 17 0.52 (0.26) 0.44 (0.34) 10
p-value 0.000 0.000 33 0.002 0.005 21
FS triopoly
Cournot −0.24 (0.18) −0.70 (0.49) 16 −0.25 (0.19) −0.78 (0.49) 11
Bertrand 0.09 (0.07) −0.25 (0.26) 17 0.18 (0.06) −0.31 (0.26) 10
p-value 0.000 0.010 33 0.000 0.004 21
HNO
Cournot 0.01 (0.03) −0.05 (0.15) 6 −0.23 (0.15) −0.63 (0.38) 6
Bertrand 0.04 (0.07) 0.05 (0.12) 6 0.04 (0.06) −0.01 (0.07) 6
p-value 0.394 0.394 12 0.004 0.002 12
Davis
Cournot 0.14 (0.54) −0.38 (1.01) 5 −0.50 (0.09) −1.48 (0.33) 5
Bertrand −0.10 (0.02) −0.24 (0.06) 5 −0.05 (0.03) −0.12 (0.07) 5
p-value 0.151 0.151 10 0.008 0.008 10
ALS
Cournot1 −0.70 (0.82) 18 −0.86 (0.69) −3.38 (2.57) 10
Cournot2 −1.30 (0.47) 18 −1.88 (0.31) −8.60 (1.96) 10
Bertrand 0.75 (0.43) 18 0.08 (0.36) −0.83 (0.89) 10
p-value1 0.000 36 0.006 0.011 20
p-value2 0.000 36 0.000 0.000 20
Standard deviations are in brackets and p-values refer to a Mann-Whitney-U test.
N = number of independent observations for FS, HNO, Davis and ALS EXTRA and number of players
for ALS BASIC.
Cournotgamesnotintheearlierrounds,butespeciallyinthelaterroundswhensubjects
havehadampleoppontunitytolearn.Nevertheless,thissurelyisanissuethatwarrants
further attention from experimentalists.10
Another explanation is that under EXTRA information conditions, where detailed
informationisprovidedonthesuccessofcompetitors,imitationofthemostsuccessful
competitormaydrivethedifferencebetweenCournotandBertrandsettings.Indeed,if
playersimitatethebestperformerofthepreviousroundandifwithasmallprobability
playerschoosearandomaction,itcanbeshownthattheoutcome—sometimesreferred
to as the imitation equilibrium—is more competitive and results in a lower degree of
cooperationunderCournotthanunderBertrandcompetition.Thismayespeciallyhave
played a role in the EXTRA treatments of the HNO and Davis experiments, where
groups consisted of four players and subjects only had a proﬁt calculator and no proﬁt
tables (see also Selten and Ostmann, 2001). However, the imitation argument is less
10 A recent study that still ﬁnds a difference in degree of cooperation while controlling for the location of
the Nash equilibrium is Potters and Suetens (2006).
SpringerBertrand colludes more than Cournot 77
compelling under BASIC information conditions because subjects are then unable to
identify the most successful competitor.
An alternative explanation is provided by the indirect evolutionary model of Bester
andG¨ uth(1998)whoﬁndthatthescopeforaltruistic(cooperative)preferencesislarger
in games of strategic complements (cf. Bertrand games) than in games of strategic
substitutes (cf. Cournot games). In other words, the type of strategic interdependence
(strategic substitutes versus complements) may be an important determinant of the
degree of tacit collusion (see also Rotemberg, 1994).
4 Conclusion
On the basis of experimental data from oligopoly experiments with Cournot and
Bertrand treatments, we ﬁnd statistical support for the suggestion of Holt (1995)
that there seems to be more tacit collusion in Bertrand price-choice than in Cournot
quantity-choice experiments.
Acknowledgments We gratefully acknowledge Luigi Luini, Hans Normann, J¨ org Oechssler and Patrizia
Sbriglia for providing their data, and Jan Bouckaert, Doug Davis, the editor (Tim Cason) and two referees
of this journal for useful suggestions.
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