The paper analyzes a Cournot model with two types of firms: Maximizers of profits and maximizers of relative payoffs. It is shown that the equilibrium is located somewhere between the regular Cournot-Nash equilibrium and the competitive Walrasian (or Bertrand-) equilibrium.Bertrand Equilibrium

Thomas Riechmann

English

Research Papers in Economics

Mixed motives in a Cournot game 
Thomas Riechmann
University of Magdeburg, FEMM
Abstract
The paper analyzes a Cournot model with two types of firms: Maximizers of profits and
maximizers of relative payoffs. It is shown that the equilibrium is located somewhere
between the regular Cournot-Nash equilibrium and the competitive Walrasian (or Bertrand-)
equilibrium.
I thank a referee for his helpful comments and suggestions.
Citation: Riechmann, Thomas, (2006) "Mixed motives in a Cournot game." Economics Bulletin, Vol. 4, No. 29 pp. 1-8
Submitted: February 16, 2006.  Accepted: September 20, 2006.
URL: http://economicsbulletin.vanderbilt.edu/2006/volume4/EB-06D40004A.pdf1 Introduction
The Cournot model of simultaneous oligopolistic quantity choice is one of the
classic workhorses of economic theory. Since its origin in 1838, countless varia-
tions of the original model have been brought up. More or less all aspects of the
model have been changed, varied and re–organized, analyzed and re–analyzed.1
What has very rarely been looked at is the aspect of the ﬁrms’ behavioral mo-
tives: What happens if ﬁrms have aims other than the mere maximization of their
proﬁts? Apart from ‘classical’ proﬁt maximization, there is another out-standing
way of behavior: maximization of relative payoff, meaning that a ﬁrm aims to
have higher proﬁts than the competitors. Besides the fact that individuals may
hold certain preferences about relative payoffs (Fehr and Schmidt, 1999; Bolton
and Ockenfels, 2000), there are many more reasons why a ﬁrm might concentrate
on being better than the others, instead of just trying to be as proﬁtable as possi-
ble. One frequently named reason is the ﬁrm’s wish to increase its market share,
which can serve as a means of pushing other ﬁrms out of the market or to pre-
vent market entry. A second possible reason for maximization of relative payoffs
is the fact that managers are paid due to relative performance of their ﬁrm: The
manager of the largest ﬁrm in the market gets the highest pay. A third reason is
a lack of information. Vriend (2000) shows that ﬁrms that cannot rely on private
information and are forced to determine their production quantity by mimicking
other ﬁrms’ decisions are de–facto maximizers of relativepayoff. Finally, it seems
worthwhile noting that, in a somewhat broader sense, it is the Bertrand model of
oligopoly that represents the most severe model of maximizing relative payoff by
maximizing the market share.
Schaffer (1989) was probably the ﬁrst author to analytically analyze ﬁrm be-
havior in a Cournot model in an evolutionary context. Referring to the concept
of spite from evolutionary biology (Hamilton, 1970), he shows that there is a way
of unilaterally deviating from a Cournot equilibrium that decreases the proﬁt of
the deviator, but at the same time decreases the other ﬁrms’ proﬁts even more.
Given a force that ‘selects for’ the ﬁrm with the highest proﬁt, the deviator will
be better off than the others. The Cournot equilibrium is not ‘stable’ in an evolu-
tionary sense. The evolutionarily stable solution results from a process of every
ﬁrm trying to be ahead of every other one (maximizing relative payoff), which
in the Cournot model results in the Walrasian (competitive market) equilibrium.
Schaffer shows this for a model with zero costs.
For the basic evolutionary concept of ‘being better than the others’, it does not
matter if a ﬁrm’s ‘relative payoff’ means the ratio of its own payoff to the total
payoff of all ﬁrms or if it means the difference between the ﬁrm’s payoff and the
1For an instructive survey on oligopoly theory, see Shapiro (1989).
1average payoff of all ﬁrms. The latter concept, which should more aptly be named
‘differential payoff’ is the one used by Schaffer (1988, 1989).
In an important subsequent paper, Vega-Redondo (1997), using differential
payoffs, replicates Schaffer’s results in a dynamic framework, while Riechmann
(2006) ﬁnds the same results for a more general class of Cournot games, again
using differential payoffs. Relative payoffs in form of ratios are in frequent use
in basic evolutionary dynamics, but turn into the differential formulation as soon
as these dynamics take place in continuous time (see, e.g. Weibull 1995; Vega-
Redondo 1996; Samuelson 1997; Fudenberg and Levine 1998).
Thus, the results of two extreme forms of Cournot models are quite clear: If
all ﬁrms follow the classical motive of proﬁt maximization, the result will be the
Cournot equilibrium. If all ﬁrms maximize relative payoff, they will all end up
in the Walrasian equilibrium of a competitive market (which, in turn, is identical
to the oligopolistic Bertrand equilibrium). What has not been analyzed yet is the
question of what happens if in the same market there are both types of ﬁrms,
maximizers of absolute payoff as well as maximizers of relative payoff.
In the ﬁeld of experimental economics, a ﬁnding common to most Cournot
experiments is the one that the experimental outcomeis ‘usually more competitive
than the Cournot prediction’ (Holt, 1995, p. 367). This paper will show that this
ﬁnding can be explained by a heterogeneity of individuals’ motives in the game:
As soon as both, maximizers of absolute and maximizers of relative payoff are
active in the same market, the resulting equilibrium must necessarily be located
somewhere in between the oligopolistic Cournot- and the competitive Walrasian
outcome.
The paper proceeds as follows. In the second section, the oligopolistic model
is introduced. Section three analyzes a duopolistic version of the model, before
the fourth section presents the general model of mixed motivesin a Cournot game.
The paper ends with a summary.
2 The Model
The basic model is the following: Market demand is given by
D = 1− p, (1)
with p giving the (market) equilibrium price. Let si denote the quantity of ﬁrm i.
Firms must supply non–negative quantities. Aggregate supply, S, is given as the
sum ofthesuppliedquantitiesofthen ﬁrms involved,S=å
n
i=1si. The equilibrium
price, p, results as
p = 1−
n
å
i=1
si. (2)
2Assume that the total of all ﬁrms’ joint capacities are too low to reach or even
exceed autonomous demand, S < 1, such that prices will be positive.
Let the n ﬁrms have identical quadratic cost functionsC( ),
C(si) =
1
2
s2
i . (3)
Fixed costs are assumed to be zero.
The (absolute) proﬁt of each ﬁrm i is given by
pi(si, S−i) = psi−C(si) = (1−S) si−
1
2
s2
i , (4)
where S−i gives the aggregate quantity of all ﬁrms except of i, S−i = å
n
j=1
j =i
sj.
Relative payoff to ﬁrm i will be deﬁned in the tradition of evolutionary
game theory (Samuelson 1997, p. 66; Weibull 1995, pp. 72–74) as the differ-
ence between i’s absolute payoff and the average absolute payoff off all ﬁrms,
p = 1
n å
n
j=1pj:
pr
i (si, S−i) = pi(si,S−i)−p(si, S−i) , (5)
=
n−1
n
￿
(1−S)si−
1
2
s2
i
￿
−
1
n
n
å
j=1
j =i
￿
(1−S)sj−
1
2
s2
j
￿
. (6)
3 Duopoly
For a start, let us take a look at a duopolistic version of the model. For only two
ﬁrms in the market, A and R, (4) becomes
pi(si, s−i) = (1−s−i) si−
3
2
s2
i , i, −i ∈ {A, R}. (7)
From this, the reaction function for ﬁrm A, aiming to maximize absolute pay-
offs, becomes
s⋆
A =
1
3
(1−s−i) . (8)
A maximizer of relative payoff, though, derives the reaction function from the
duopolistic version of (5), which is
pr
i (si, s−i) =
1
2
[pi(si, s−i)−p−i(si, s−i)] . (9)
Maximizing (9) with respect to si results in the reaction function for ﬁrm R, max-
imizing relative payoff:
s⋆
R =
1
3
. (10)
3Thisisinfact identicalto theWalrasian (competitivemarket)equilibriumquantity:
A ﬁrm neglecting its inﬂuence on the equilibrium price and consequently using
the rule ‘produce the quantity that equates the price to your marginal costs’ to
determine the output level will produce exactly s⋆
R. (A Bertrand model would of
course result in the same equilibrium quantity.)
It is remarkable to see that the reaction function for relative payoffs (10) is a
degenerate function. For a maximizer of relative payoffs in a duopoly, producing
thequantity s⋆
R is the optimumstrategy as long as the opponent is restricted to pro-
ducing quantities that keep prices positive(which is guaranteed by the assumption
that åi si <1): No matter what the opponent does, a maximizer of relative payoffs
should produce the Walrasian quantity. Note, thought, that this is a special trait of
the duopolistic case. As soon as there are more than two ﬁrms involved, no type
of ﬁrms has a constant best strategy any more. (See equation (15) below, which
shows the general best response function for an R–type ﬁrm.)
The equilibrium is easily derived as
s⋆
A =
2
9
, s⋆
R =
1
3
. (11)
Obviously, s⋆
A < s⋆
R. The R–ﬁrm produces a higher quantity than the A–ﬁrm.
Moreover, it can be shown that the relative–payoff–maximizerhas higher absolute
payoff. (Of course it has, because it maximized the difference.)
pR(s⋆
R, s⋆
A) > pA(s⋆
A, s⋆
R) . (12)
Thus, the maximizer of relative payoff does exactly this: She maximizes her rela-
tive payoff. All that is left to do for the maximizer of absolute payoff is to ﬁnd his
best response to the strategy of his opponent. The maximizer of absolute payoff
does indeed maximize his payoff given the relative-payoff-maximizers quantity.
Considering this outcome, it might be asked why ﬁrm A does not switch to us-
ing R’s strategy, too. The answer to this question is straightforward: By switching
from s⋆
A to s⋆
R, he reduces his (absolute) payoffto a level less than his previouspay-
off from playing sA (This should of course be obvious from the reaction function
(8).):
pA(s⋆
R, s⋆
R) < pA(s⋆
A, s⋆
R) . (13)
Moreover, following the usual deﬁnition of efﬁciency as a measure in absolute
payoffs, the resulting equilibrium is inefﬁcient for the ﬁrms.2 Still, it should be
2Of course, if we looked at a broader measure of efﬁciency like the sum of producers’and con-
sumers’ surpluses, efﬁciency would probably rise compared to the original Cournot situation. An
efﬁcient state for the producers would have both players use the standard Cournot-Nash quantity
or even collude on the monopolistic quantity.
4kept in mind that absolute payoff is not what ﬁrm R cares for, such that the regular
measures of welfare might be inadequate in this model.
This result implies a structure that holds true for the general case of the n–
player model. As will be shown further down in this paper, in the general case,
too, maximizers of relative payoff will at the same time achieve higher payoffs
than maximizers of absolute payoff.
All in all, the model subsumes at three different outcomes. It has been shown
before (Riechmann, 2006) that, given both players aim to maximize absolute pay-
offs, the result will be the usual Cournot equilibrium, but if both players care for
relative payoffs instead, the result will be Walrasian. If players hold different mo-
tives,the result will bethe onepresented in (11). A special case ofthis equilibrium
bears a nice interpretation. For the case of no variable costs (d = 0), the result is
equal to a Stackelberg equilibrium where the R–type ﬁrm is the Stackelberg leader
and the A–type ﬁrm is the follower.3 This outcome derives from the fact that in
the duopolistic case, the R–type ﬁrm has an optimum strategy it needs not condi-
tion on what the A–type ﬁrm will do. In a strategic sense, this implicitlymakes the
R–ﬁrm the Stackelberg–leader, who (trivially) decides ﬁrst. The A–type ‘follows’
by playing a best response.
4 The General Case
The derivation of respective results for the general n–player case is not compli-
cated, but involvessome rather tedious computation. In order to preserve readabil-
ity, this section only gives the most important results, while the technical details
are postponed to the appendix.
From (4), the reaction function for A–type ﬁrm number j derives as
s⋆
A,j =
1
3
￿
1−S−j
￿
, (14)
with S−j giving the total quantity minus (A–type) ﬁrm j’s quantity, S−j = S−sA,j.
The respective function for R–type ﬁrm k derives from (6):
s⋆
R,k =
1
3
￿
1−
n−2
n−1
S−k
￿
, (15)
with S−k giving the total quantity minus ﬁrm k’s quantity.
Note that the reaction function of R–types explicitely contains the number
of ﬁrms while the reaction function of A–types does not. The reason for this is
straightforward. A–types effectively play against the rest of the economy’s supply
side, while R–types effectively play against every single supplier.
3I am thankful to a referee to point out this fact.
5The equilibriumwill be a semi–symmetricone(an equilibriumwith all players
ofthesametypebehavingidentically). WithnR givingthenumberofR–typeﬁrms,
the equilibrium quantities result as
s⋆
A =
2n−nR−1
2(n2−1)+3(n−nR)
, (16)
s⋆
R =
3n−nR−2
2(n2−1)+3(n−nR)
. (17)
Notably, the equilibrium quantities are determined by both the size of the sup-
ply side of the market (i.e. the number of ﬁrms, n) and the composition of the
supply side (measured by the number of R–types, nR). There are no ‘dominant’
strategies any more.
All the other results remain true in the n–player game. Again, R–type ﬁrms
produce a higher quantity than A–type ﬁrms, such that R–type ﬁrms are better off
than A–type ﬁrms even in terms of absolute payoff. Again, switching from s⋆
A to
s⋆
R is not worthwhile.
Again, the result in (16) and (17) includes two special cases, namely the Wal-
rasian and the Cournot equilibrium. For a market with only A–type ﬁrms, the
equilibrium quantity becomes
s⋆
A(nR = 0) =
1
n+2
, (18)
the Cournot equilibrium quantity.
In a market with only R–type ﬁrms, the individual equilibrium quantity is
s⋆
R(nR = n) =
1
n+1
, (19)
which is the Bertrand equilibrium quantity and the Walrasian competitive market
equilibrium quantity.
These two special cases represent the limiting cases for the model. The
more R–types there are in the market, the more the market tends to the
Bertrand/Walrasian outcome. The more A–types there are, the closer the re-
sult will be to the Cournot outcome. Generally, the result will always fall into the
range between (including) the Cournot and the Walrasian equilibrium.
5 Summary
This paper makes one short point: In a Cournot model of oligopolistic quantity
choice with regular proﬁt maximizers and maximizers of relative payoffs active at
the same time, the resulting equilibrium quantity will generally be located in the
range between (including) the Cournot and the Walrasian/Bertrand quantity.
6Appendix: The General Case
In the general n–player case, there are n ﬁrms, nR of them R–types and n−nR
A–types. Let sA,j denote the quantity that A–type ﬁrm number j produces and use
the respective notation sR,k for R–types.
S, the total quantity, is the sum of the individual quantities of A– and R–types:
S =
n−nR
å
j=1
sA,j+
nR
å
k=1
sR,k. (20)
From (4), the reaction function for A–type ﬁrms number j derives as
s⋆
A,j =
1
3
−
1
3
S−j, (21)
with S−j giving the total quantity minus (A–type) ﬁrm j’s quantity, S−j = S−sA,j.
The respective function for R–type ﬁrms k derives from (6):
s⋆
R,k =
1
3
−
1
3
n−2
n−1
S−k, (22)
with S−k giving the total quantity minus ﬁrm k’s quantity.
As the equilibrium will be a semi–symmetric one (an equilibrium with all
players of the same type behaving identically, i.e. sA,j = s⋆
A,j = s⋆
A for all A–types
and sR,k = s⋆
R,k = s⋆
R for all R–types), the equilibrium quantities can be derived
from a simple system of two equations. We ﬁnd that
s⋆
A =
1
2+n−nR
−
nR
2+n−nR
s⋆
R, (23)
s⋆
R =
(n−1)
2n−2nR+nnR−1
−
(n−2)(n−nR)
2n−2nR+nnR−1
s⋆
A (24)
The equilibrium quantities result as
s⋆
A =
2n−nR−1
2(n2−1)+3(n−nR)
, (25)
s⋆
R =
3n−nR−2
2(n2−1)+3(n−nR)
. (26)
From
s⋆
A =
2n−nR−1
3n−nR−2
s⋆
R, (27)
it can be seen that
s⋆
R(nR = i) > s⋆
A(nR = i) ∀1 < i ≤ n, (28)
7and
pR(nR = i) > pA(nR = i) ∀1 < i ≤ n. (29)
Switching from s⋆
A to s⋆
R is not worthwhile:
pR(s⋆
A, nR = i) > pR(s⋆
R, nR = i+1) ∀1 < i ≤ n−1. (30)
For a market with only A–type ﬁrms, the equilibrium quantity becomes the
Cournot quantity:
s⋆
A(nR = 0) =
1
n+2
. (31)
In a market with only R–type ﬁrms, the individual equilibrium quantity is
Walrasian:
s⋆
R(nR = n) =
1
n+1
. (32)
References
Bolton, G. E. and Ockenfels, A. (2000). ERC: A theory of equity, reciprocity, and
competition. American Economic Review, 90:166–193.
Fehr, E. and Schmidt, K. M. (1999). A theory of fairness, competition, and coop-
eration. The Quarterly Journal of Economics, 114(3):817–868.
Fudenberg, D. and Levine, D. K. (1998). The Theory of Learning in Games. MIT
Press Series on Economic Learning and Social Evolution. MIT Press, Cam-
bridge, MA.
Hamilton, W. D. (1970). Selﬁsh and spiteful behavior in an evolutionary model.
Nature, 228:1218–1220.
Holt, C. A. (1995). Industrial organization: A survey of laboratory research. In
Kagel, J. H. and Roth, A. E., editors, The Handbook of Experimental Eco-
nomics, chapter 5, pages 349–443. Princeton University Press, Princeton, NJ.
Riechmann, T. (2006). Cournot or Walras? Long–run results in oligopoly games.
Journal of Institutional and Theoretical Economics (JITE), forthcoming.
Samuelson, L. (1997). EvolutionaryGamesandEquilibriumSelection. MITPress
Series on Economic Learning and Social Evolution. MIT Press, Cambridge,
MA, London.
Schaffer, M. E. (1988). Evolutionary stable strategies for a ﬁnite population and
a variable contest size. Journal of Theoretical Biology, 132:469–478.
8Schaffer, M. E. (1989). Are proﬁt maximisers the best survivors? Journal of
Economic Behavior and Organization, 12:29–45.
Shapiro, C. (1989). Theories of oligopoly behavior. In Schmalensee, R. and
Willig, R. D., editors, Handbook of Industrial Organization, volume I, chap-
ter 6, pages 329–414. North Holland, Amsterdam, London, New York, Tokio.
Vega-Redondo, F. (1996). Evolution, Games, and Economic Behavior. Oxford
University Press, Oxford, UK.
Vega-Redondo, F. (1997). The evolution of Walrasian behavior. Econometrica,
65:375–384.
Vriend, N. J. (2000). An illustration of the essential difference between individual
and social learning, and its consequences for computational analyses. Journal
of Economic Dynamics and Control, 24:1–19.
Weibull, J. W. (1995). Evolutionary Game Theory. MIT Press, Cambridge, MA,
London.
9

A theory of fairness, competition, and cooperation.

An illustration of the essential difference between individual and social learning, and its consequences for computational analyses.

Are proﬁt maximisers the best survivors?

Cournot or Walras? Long–run results in oligopoly games.

ERC: A theory of equity, reciprocity, and competition.

Evolution, Games, and Economic Behavior.

Evolutionary Game Theory.

Evolutionary stable strategies for a ﬁnite population and a variable contest size.

Mixed motives in a Cournot game

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