ORIGINAL ARTICLE
A dynamic Cournot duopoly model with different strategies
A.A. Elsadany *
Department of Basic Science, Faculty of Computers and Informatics, Suez Canal University, Ismailia, Egypt
Received 4 December 2013; accepted 2 January 2014
KEYWORDS
Different strategies;
Chaos;
Boundedly rational player;
Local approximation player;
Nash equilibrium point;
Parameter variation method
Abstract This paper analyzes the dynamics of a Cournot duopoly model with different strategies.
We offer results on existence, stability and local bifurcations of the equilibrium points. The bifurcation
diagrams and Lyapunov exponents of the model are presented to show that the model
behaves chaotically with the variation in the parameters. The state variables feedback and parameter
variation methods are used to control the chaos of the model.
2010 MATHEMATICS SUBJECT CLASSIFICATION: 39A10; 91A25; 37G35
ª 2014 Production and hosting by Elsevier B.V. on behalf of Egyptian Mathematical Society.
1. Introduction
A Cournot duopoly game is an oligopoly market with two
players. Oligopoly was first introduced by Cournot [1]. In a
typical Cournot oligopoly, there are two or more players, no
other players can enter the market, and collusive behavior is
prohibited. Each player in the oligopoly market aims to maximize
its expected profit, and profit is maximized when marginal
revenue equals marginal cost. Recently, it has also been
shown that even oligopolistic markets may become chaotic under
certain conditions [2–10]. In general, in order to adjust his
output, a player can choose his expectation rule among many
available strategies. Naive, adaptive, boundedly rational and
local approximation expectations are only a few examples. In
the literature on oligopoly games, most papers focus on games
with homogeneous strategies, that is, players who adopt the
same expectation rule. Another branch of the literature is
made up of studies in which games with different strategies
are taken into consideration. The assumption of players adopting
heterogeneous rules to decide their production is, in our
opinion, more realistic than the opposite case. This approach
characterizes the works by Leonard and Nishimura [11],
Den-Haan [12], Agiza et al. [13], Agiza and Elsadany [14,15].
Zhang et al. [16] used the technique of Agiza and Elsadany
to analyze a duopoly game with heterogeneous players and
nonlinear cost function. Angelini et al. [17] and Tramontana
[18] studied a duopoly game with heterogeneous firms assuming
a microfounded nonlinearity on the demand function.
The main purpose of this paper is to investigate the dynamic
behavior and control of duopoly game with different
strategies. We considered that each player forms a different
strategy in order to compute its expected output. We assume
that the first player represents a boundedly rational player
and the second player has local approximation expectations.
The main aim of this work is to investigate the dynamic behaviors
of the two players game using different expectation rules.
Moreover, from a mathematical point of view, it is shown that
the loss of the market equilibrium stability may occur through
* Tel.: +20 1140734666; fax: +20 643320267.
E-mail address: aelsadany1@yahoo.com
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Please cite this article in press as: A.A. Elsadany, A dynamic Cournot duopoly model with different strategies, Journal of the
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a flip bifurcation and that a cascade of flip bifurcations may
lead to periodic cycles and chaos.
The paper is organized as follows. In Section 2 we describe
a nonlinear duopoly game model. In Section 3 we study the
fixed points and the dynamics of the model, showing explicit
parametric conditions of the existence and local stability of
the market equilibrium. Section 4 the results of the previous
section are numerically illustrated and showing the occurrence
of complex behaviors. In Section 5, we exerted control on the
duopoly game model. Section 6 concludes.
2. The model
We consider a duopoly Cournot game where xt
i; i ¼ 1; 2 represent
the quantity supplied by ith player during period
t ¼ 0; 1; 2 . . .. Assume that the inverse demand function
[19,20] is
pðXÞ ¼ a À b
ffiffiffiffi
X
p
ð1Þ
where X ¼ x1 þ x2 is the total quantity in the market, a and b
are positive constants. This function is convex as the isoelastic
demand function but it does not tend to infinity as p ! 0. In
fact, ða=bÞ2
represents the maximum amount of output that
can be brought to the market. Those properties are important/relevant
from an economic point of view. Moreover this
form has also used in others oligopoly models and in laboratory
experiments economics dealing with learning and expectations
formation (see e.g. [21–23]). The cost function of the
players is as follows:
CiðxiÞ ¼ cixi; i ¼ 1; 2: ð2Þ
where the positive parameters ci are the marginal costs. The
profit function of player i is
Piðx1; x2Þ ¼ xi a À b
ffiffiffiffi
X
p 
À cixi; i ¼ 1; 2: ð3Þ
From the profit maximization by player i, the marginal
profits are obtained as:
Ui ¼
@Pi
@xi
¼ a À ci À b
ffiffiffiffi
X
p
À
bxi
2
ffiffiffiffi
X
p ; i ¼ 1; 2 ð4Þ
We assume different expectations: i.e., player 1 is boundedly
rational and player 2 is local approximation. The boundedly
rational player 1 has no complete knowledge of the market;
hence they try to use local information based on the marginal
profit @P1
@x1
. It decides to increase (decrease) its quantity if it has
a positive (negative) marginal profit. This adjustment mechanism
has been called myopic by Dixit [2]. Thus, the dynamic
adjustment mechanism can be modeled as follows:
xtþ1
1 ¼ xt
1 þ a1xt
1
@p1 xt
1; xt
2
À Á
@xt
1
; ð5Þ
where a1 is a positive parameter which represents the speed of
adjustment of first player.
The second duopolist is a local approximation player (see
[19,23]), i.e.
xtþ1
2 ¼
xtþ1
2
2
þ
c1 À fðXt
Þ
2f0ðXt
Þ
ð6Þ
Therefore, given these types of strategies formation, the
two-dimensional system that characterizes the dynamics of a
Cournot duopoly game is the following:
xtþ1
1 ¼ xt
1 þ a1xt
1 a À c1 À b
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
xt
1 þ xt
2
p
À
bxt
1
2
ffiffiffiffiffiffiffiffiffi
xt
1
þxt
2
p
 
xtþ1
2 ¼
2ðaÀc2Þ
ffiffiffiffiffiffiffiffiffi
xt
1
þxt
2
p
À2bxt
1Àbxt
2
2b
8
><
>:
ð7Þ
We are interested only in positive trajectories. Note also that
the game is not defined in the origin (0,0). In the next section,
we study the dynamical behaviors of the map (7).
3. Equilibrium points and local stability
In this section, we determine the equilibrium points of the map
(7) by solving the following nonlinear algebraic system:
xÃ
1 a À c1 À b
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
xÃ
1 þ xÃ
2
p
À
bxÃ
1
2
ffiffiffiffiffiffiffiffiffi
xÃ
1
þxÃ
2
p
 
¼ 0
2ða À c2Þ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
xÃ
1 þ xÃ
2
p
À 2bxÃ
1 À 3bxÃ
2 ¼ 0
8
<
:
ð8Þ
This map has two equilibrium points:
E1 ¼ 0;
4ða À c2Þ2
9b2
!
ð9Þ
E2 ¼ xÃ
1; xÃ
2
À Á
where
xÃ
1 ¼
4ð2a À c1 À c2Þða þ 2c2 À 3c1Þ
25b2
; xÃ
2 ¼
4ð2a À c1 À c2Þða þ 2c1 À 3c2Þ
25b2
ð10Þ
E2 is called Nash equilibrium point and has positive coordinates
provided that
2a > c1 þ c2
a > 3c1 À 2c2
a > 3c2 À 2c1
8
><
>:
ð11Þ
In order to investigate the local stability of the equilibrium
points, we must consider the Jacobian matrix of the map (7) is
the following:
J ¼
1 þ a1 a À c1 À b
ffiffiffiffi
X
p
À bx1
2
ffiffiffi
X
p À
3bx2
1þ4bx1x2
4
ffiffiffiffi
X3
p

À
a1 bx2
1þ2bx1x2ð Þ
4
ffiffiffiffi
X3
p
aÀc2
2b
ffiffiffiffiffiffiffiffiffix1þx2
p À 1 aÀc2
2b
ffiffiffiffiffiffiffiffiffix1þx2
p À 1
2
2
6
4
3
7
5
ð12Þ
The equilibrium points will be stable if the eigenvalues ki,
i ¼ 1; 2 of the above Jacobian matrix satisfy inequalities
kij j < 1; i ¼ 1; 2. By applying the stability condition to the
equilibrium E1 we have the following result:
Proposition 1. If the Nash equilibrium E2 is strictly positive,
then the equilibrium point E1 is a saddle point.
Proof. In fact at E1, the Jacobian matrix becomes a triangular
matrix:
JðE1Þ ¼
1 þ a1
ðaþ2c2À3c1Þ
3
0
À1
4
1
4
" #
whose eigenvalues are given by the diagonal entries. They are
k1 ¼ 1 þ a1
ðaþ2c2À3c1Þ
3
and k2 ¼ 1
4
. If the Nash equilibrium point
has positive coordinates, then a > 3c1 À 2c2, hence jk1j > 1
and jk2j < 1. Then E1 is a saddle point. h
2 A.A. Elsadany
Please cite this article in press as: A.A. Elsadany, A dynamic Cournot duopoly model with different strategies, Journal of the
Egyptian Mathematical Society (2014), http://dx.doi.org/10.1016/j.joems.2014.01.006
3.1. Stability of the Nash equilibrium point E2
In order to investigate the local stability properties of the Nash
equilibrium point (10) of the two-dimensional system (7), the
Jacobian matrix evaluated at E2, that is:
JðE2Þ ¼
1 À
a1 3bxÃ2
1 þ4bxÃ
1xÃ
2ð Þ
4
ffiffiffiffiffi
XÃ3
p À
a1 bxÃ2
1 þ2bxÃ
1xÃ
2ð Þ
4
ffiffiffiffiffi
XÃ3
p
aÀc2
2b
ffiffiffiffiffiffiffiffiffi
xÃ
1
þxÃ
2
p À 1 aÀc2
2b
ffiffiffiffiffiffiffiffiffi
xÃ
1
þxÃ
2
p À 1
2
2
6
4
3
7
5 ð13Þ
where XÃ
¼ xÃ
1 þ xÃ
2.
Whose trace and determinant are given by:
T :¼ TrðJðE2ÞÞ ¼
1
2
Àa1hþ/
D :¼ DetðJðE2ÞÞ ¼ /À
1
2
Àa1w/þa1r
where
h ¼
3bxÃ2
1 þ4bxÃ
1xÃ
2
À Á
4
ffiffiffiffiffiffiffi
XÃ3
p ;/ ¼
aÀc2
2b
ffiffiffiffiffiffi
XÃ
p ;w ¼
bxÃ2
1 þbxÃ
1xÃ
2
À Á
2
ffiffiffiffiffiffiffi
XÃ3
p ð14Þ
and r ¼
bxÃ2
1
8
ffiffiffiffiffiffiffi
XÃ3
p
so the characteristic polynomial of (13) is:
PðkÞ ¼ k2
À Tk þ D; ð15Þ
whose discriminant is D ¼ T2
À 4D.
From the stability theory we know that the fixed point E2 is
locally asymptotically stable as long as the eigenvalues of Jacobian
matrix JðE2Þare inside the unit circle of the complex
plane. This is true if and only if the following Jury’s [24] stability
criteria are hold:
ð1Þ : F :¼ 1 þ T þ D > 0;
ð2Þ : TC :¼ 1 À T þ D > 0;
ð3Þ : H :¼ 1 À D > 0:
8
><
>:
ð16Þ
The above inequalities (16) define a region in which the Nash
equilibrium point E2 is local stable. The violation of any single
inequality in (16), with other two being simultaneously fulfilled
leads to: (1) a flip bifurcation (real eigenvalue that passes
through À1) when F ¼ 0; (2) a fold or transcritical bifurcation
(a real eigenvalue that passes through +1) when TC ¼ 0; (3) a
Neimark-Sacker bifurcation (i.e., the modulus of a complex
eigenvalue pair that passes through 1) when H ¼ 0 and
Tj j < 2 (see [25,26]).
The discriminant of (13) is
D ¼ T2
À 4D ¼
3
2
À a1h À /
 2
þ 2a1ð2w/ À 2h/ þ h À 2rÞ;
From Eqs. (10), (11), (14) and since a; b; c1; c2 and a1 are positive
parameters. Then w > h and h > 2r. Hence D > 0, then
the eigenvalues of Nash equilibrium are real.
For the special case of the Jacobian matrix (13), the stability
conditions in (16) can be written as follows:
ð1Þ : F :¼ 1 À a1h þ 2/ À a1w/ þ a1r > 0;
ð2Þ : TC :¼ a1h þ a1r À a1w/ > 0;
ð3Þ : H :¼ 3
2
À / þ a1w/ À a1r > 0:
8
><
>:
ð17Þ
Since the discriminant is positive, the existence of complex
eigenvalues of JðE2Þ is prevented. Then the condition (3) is
always fulfilled. While from Eqs. (10), (11) and (14) it is clear
that the condition (2) is satisfied. Therefore, the Nash equilibrium
point E2 can loose stability only through a flip bifurcation.
So, we have the following proposition about local
stability of Nash Equilibrium point E2.
Proposition 2. The Nash equilibrium point E2 is asymptotically
stable if a < 1þ2/
wþ/hÀr. The system (7) undergoes a flip bifurcation
at E2 when aÃ
¼ 1þ2/
wþ/hÀr. Moreover, period-2 points bifurcate
from E2 when a > 1þ2/
wþ/hÀr.
Proof. From above results, the conditions (2) and (3) are
always satisfied. However, condition (1) can be violated, since
the flip bifurcation occurs when F ¼ 0. Then
1 þ T þ D ¼ 0;
1 À a1h þ 2/ À a1w/ þ a1r ¼ 0
Then from stability situation, the Nash equilibrium point can
loose stability when aÃ
¼ 1þ2/
wþ/hÀr
. Hence the Nash equilibrium
point is asymptotically stable if a < aÃ
. Moreover, period-2
points bifurcate from E2 when a > aÃ
. h
Next we present the results of the numerical simulations for
the system (7).
4. Analysis and numerical simulation
The main purpose of this section is to show that the qualitative
behavior of the solutions of the nonlinear duopoly game with
heterogeneous players (described by the dynamic system (7)).
To provide some numerical evidence for the existence of chaotic
motions, we use several standard tools, bifurcations diagrams,
basin of attraction, lyapunov exponents, strange
attractors, sensitive dependence on initial conditions and so
on.
Fig. 1 presents a bifurcation diagram of system (7) in
ða1 À x1x2Þ plane when a ¼ 10; b ¼ 1; c1 ¼ 1 and c2 ¼ 3. From
Fig. 1, we can see that the orbit with initial values ð0:25; 0:2Þ
approaches to the stable fixed point for a1 < 0:415. With a1
increasing, a flip bifurcation for system (7) takes place at a1
¼ 0:415 and period-2 points bifurcate as a1 ¼ :415 , which is
verifies Proposition 2. Furthermore, the period-2 points are
attracting when a1varies in the interval ð0:415; 0:565Þ. As long
Figure 1 Bifurcation diagram of the model (7) with respect to a1.
A dynamic Cournot duopoly model with different strategies 3
Please cite this article in press as: A.A. Elsadany, A dynamic Cournot duopoly model with different strategies, Journal of the
Egyptian Mathematical Society (2014), http://dx.doi.org/10.1016/j.joems.2014.01.006
as the parameter a1increases, the Nash equilibrium point E2
becomes unstable and the bifurcation scenario occurs and ultimately
leads to unpredictable (chaotic) motions that are observed.
This means that immediately after the NS bifurcation
the system displays complicated dynamics, differently from
what happens after the period doubling bifurcation in which
aperiodic attractor appears. While bifurcation and chaos occur,
the output of players acutely fluctuate making it difficult
for players to forecast their output and to make decisions in
the future.
In order to classify the dynamic behavior of system (7) for
many different parameters set, we compute the largest lyapunov
exponent. Fig. 2 shows the largest lyapunov exponent corresponding
to the bifurcation diagram Fig. 1. As it is positive,
there is evidence of chaos and beyond that it is even possible
to differentiate between cycles of very high order and aperiodic
(chaotic) behavior of the system. Moreover, by comparing the
standard bifurcation diagram in Fig. 1 with the diagram of largest
lyapunov exponent, one can obtain a better understanding of
the particular properties of the dynamic behavior of the system.
Fig. 3 shows the bifurcation diagram with respect to the
parameter a with b ¼ 1; c1 ¼ 1; c2 ¼ 3 and a1 ¼ 0:6. The Nash
equilibrium E2 is locally stable for small values of a. As a
increases, the Nash equilibrium point becomes unstable and
complex dynamic behavior occurs, including higher-order
cycles and chaos. The bifurcation diagram with respect to c1
when the other parameters take the values ða; b; c2; a1Þ ¼
ð10; 1; 3; 0:6Þ is given Fig. 4. We can see that the system dynamics
is chaotic if the return rate c1 is small. As c1 increases, there
exist period-halving bifurcations. The game experiences chaos
and period-halving bifurcation.
A strange attractor is another characteristic of chaos of the
system, and it reflects the inherent regularity of the complex
phenomena in a chaotic state. Thus, players can forecast the
market output in a short term according to inherent regularity
while the system is in a chaotic state. Fig. 5 shows the graph of
strange attractors of system (7) when a ¼ 10; b ¼ 1; c1 ¼ 1;
c2 ¼ 3 and a1 ¼ 0:6. Strange attractors show the complexity
of players dynamic output competition in chaos.
5. Chaos control
The appearance of chaos in the economic system is not expected
and even is harmful. Thus, people hope to find some
methods to control the chaos of economic system. A wide variety
of methods have been proposed for controlling chaos in
oligopoly models, for example, chaos control with OGY method
in the Kopel duopoly game model was applied in [27],
chaos control with modified straight-line stabilization method
Figure 2 Largest Lyapunov exponents corresponding to Fig.1.
Figure 3 Bifurcation diagram of the model (7) with respect to a.
Figure 4 Bifurcation diagram of the model (7) with respect to c1.
Figure 5 Strange attractor for the model (7).
4 A.A. Elsadany
Please cite this article in press as: A.A. Elsadany, A dynamic Cournot duopoly model with different strategies, Journal of the
Egyptian Mathematical Society (2014), http://dx.doi.org/10.1016/j.joems.2014.01.006
in an output duopoly competing evolution model have been
studied by Du et al. [28], Holyst and Urbanowicz [29] had
studied chaos control with time-delayed feedback method in
an economical model. The dynamics and adaptive control of
a duopoly advertising model based on heterogeneous expectations
is presented in Ding et al. [30], and so on. Elabbasy et al.
[31] have considered such a feedback control in their triopoly
with heterogeneous players. Also Ding et al. [32] have applied
feedback control on multi-team Bertrand model.
The above section results show that the duopoly market
will become unstable and fall into chaos when output adjustment
speed parameter out of the stable region. Therefore, it
is necessary to take a control strategy to delay or eliminate
the occurrence of bifurcation and chaos. Feedback and parameter
variation are two methods for the chaos control. Recently,
Luo et al. [33] proposed a new control method which is called
as control strategy of the state variables feedback and parameter
variation. Pu and Ma [34] have considered the state variables
feedback and parameter variation method in their four
oligopolist model. In this article, the same method will be used
to control the chaos of system (7). We change two-dimensional
discrete dynamic system (7) into the following format:
xtþ1
1 ¼ ð1ÀlÞ xt
1 þa1xt
1 aÀc1 Àb
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
xt
1 þxt
2
p
À
bxt
1
2
ffiffiffiffiffiffiffiffiffi
xt
1
þxt
2
p
 & '
þlxt
1
xtþ1
2 ¼ ð1ÀlÞ
2ðaÀc2Þ
ffiffiffiffiffiffiffiffiffi
xt
1
þxt
2
p
À2bxt
1Àbxt
2
2b
& '
þlxt
2
8
>>><
>>>:
ð18Þ
System (7) will fall into instability region and chaos with the
change of output modification speed for player 1. The chaotic
state of system (7) with the change of speed adjustment a1 for
player 1 is controlled. Fig. 6 is the bifurcation diagram of controlled
system (18) with the change of control parameter l
after adding control to the chaotic state (a ¼ 10; b ¼ 1; c1 ¼ 1;
c2 ¼ 3 and a1 ¼ 0:6). It can be seen from Fig. 6 that the system
(18) is in a chaotic state when l < 0:0465; the first players
output and the second players output are controlled in the
four-fold period bifurcation state when l ¼ 0:0475. The first
player output and the second player output are controlled in
the period-doubling bifurcation when l ¼ 0:0875. Also, the
period-doubling bifurcation disappears, and the system stabilizes
at the Nash equilibrium point when l > 0:27. Fig. 7 shows
that the chaotic system is controlled at fixed point when
l ¼ 0:3. Thus chaos control is successful. The method presented
here can be applied to many chaotic dynamical systems.
6. Conclusion
A Cournot duopoly game model with different strategies is
analyzed. The stability of the equilibrium points has been analyzed.
From bifurcation diagrams and phase portraits, the basic
properties of the game are presents. The results show that
the quantity adjustment speed of boundedly rational player
has an obvious impact on the stability of the players’ dynamic
quantity competition model. When it continues to increase, a
series of chaotic phenomena occur: period doubling bifurcation,
a positive Lyapunov exponent and strange attractors
have been obtained. Finally, the model is quickly arrived at
the Nash equilibrium point when a suitable controlling factor
is chosen.
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Egyptian Mathematical Society (2014), http://dx.doi.org/10.1016/j.joems.2014.01.006
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6 A.A. Elsadany
Please cite this article in press as: A.A. Elsadany, A dynamic Cournot duopoly model with different strategies, Journal of the
Egyptian Mathematical Society (2014), http://dx.doi.org/10.1016/j.joems.2014.01.006