arXiv:nlin/0210035v1[nlin.CD]16Oct2002
Nonlinear dynamics in the Cournot duopoly
game with heterogeneous players
H. N. Agiza and A. A. Elsadany
Department of Mathematics, Faculty of Science
Mansoura University, Mansoura 35516, P. O. Box 64, Egypt
E-mail: agizah@mans.edu.eg (H.N. Agiza)
Abstract
We analyze a nonlinear discrete-time Cournot duopoly game, where
players have heterogeneous expectations. Two types of players are
considered: boundedly rational and naive expectations. In this study
we show that the dynamics of the duopoly game with players whose
beliefs are heterogeneous, may become complicated. The model gives
more complex chaotic and unpredictable trajectories as a consequence
of increasing the speed of adjustment of boundedly rational player.
The equilibrium points and local stability of the duopoly game are
investigated. As some parameters of the model are varied, the stability
of the Nash equilibrium point is lost and the complex (periodic
or chaotic) behavior occurs. Numerical simulations is presented to
show that players with heterogeneous beliefs make the duopoly game
behaves chaotically. Also we get the fractal dimension of the chaotic
attractor of our map which is equivalent to the dimension of Henon
map.
Keywords: Discrete dynamical systems; Cournot duopoly games;
Complex dynamics; Heterogeneous expectations
1 Introduction
An oligopoly is a market system which is controlled by a few number of
ﬁrms producing homogeneous products. The dynamic of oligopoly game is
1
more complex because oligopolist must consider not only the behaviors of
the consumers, but also the reactions of the other competitors. Cournot, in
1838 [1] introduced the ﬁrst formal theory of oligopoly, who treated the case
with naive expectations, so that in every step each player assumes the last
values taken by the competitors without estimation of their future reactions.
Recently several works have shown that the Cournot model may lead to
complex behaviors such as cyclic and chaotic, see for example [2, 3, 4, 5, 6,
7]. Among the ﬁrst to do this was Puu [3, 4] who found a variety complex
dynamics arising in the Cournot duopoly case including the appearance of
attractors with fractal dimension. Others study of the dynamics of oligopoly
models with more ﬁrms and other modiﬁcations include Ahmed and Agiza [8],
Agiza [5] and Agiza et al. [9] such eﬀorts have been extended by Bischi and
Kopel [10] in a duopoly game with adaptive expectations. The development
of complex oligopoly dynamics theory have been reviewed in [11].
Expectations play a key role in modelling economics phenomena. A producer
can choose his expectations rules of many available techniques to adjust
his production outputs. May be in the market of duopoly model each ﬁrm
behaves with diﬀerent expectations strategies, so we are going to apply this
kind of expectations in our model which is common in reality.
In this paper we consider a duopoly model which is introduced in [7] but
each player form a diﬀerent strategy in order to compute his expected output.
We take ﬁrm 1 represent a boundedly rational player while ﬁrm 2 has
naive expectations. Each player adjusts his outputs towards the proﬁt maximizing
amount as target and use his expectations rule. Recently examples
of oligopoly games with homogenous expectations are studied by Puu [4],
Kopel [6], Agiza [12], Agiza et al [13, 14]. It was shown that the dynamics of
Cournot oligopoly game may never settle to a steady state, and in the long
run they exhibit bounded dynamic which may be periodic or chaotic. Economic
model with heterogeneous players is introduced see [15,16]. Also the
dynamics of heterogeneous two-dimensional cobweb model have been studied
by Onozaki et al, see [17].
The main aim of this work is to investigate the dynamic behaviors of a
heterogenous model representing two ﬁrms using heterogeneous expectations
rules. This mechanism was applied in cobweb model [17] and gave us a guide
to apply it in our study.
The paper is organized as follows. In section 2 we describe the evolution of
dynamical systems of players with heterogeneous expectations rules. In section
3, the dynamics of a duopoly game with boundedly rational player and
2
naive player is modelled by a two dimensional map .The existence and local
stability of the equilibrium points of the nonlinear map are analyzed. Complex
dynamics of behavior occur under some changes of control parameters
of the model which are shown by numerical experiments. Fractal dimension
of the strange attractor of the map is measured numerically.
2 Heterogeneous expectations
In oligopoly game players choose simple expectations such as naive or complex
as rational expectations. The players can use the same strategy (homogenous
expectations) or use diﬀerent strategies (heterogeneous expectations).
Several economic models represent the dynamics of heterogeneous
ﬁrms, have been proposed in recent years see for example [17, 18]. In this
study we consider duopoly game where each player has diﬀerent strategy to
maximize his proﬁt.
Firms can use rational expectations if they assume perfect knowledge of
underlying market and this may not be available in real economic market.
Also it is well known that in a duopoly model with a heterogeneous ﬁrms
their outputs depend upon expectations of all competitors. Hence rational
expectations can only be achieved under unrealistic assumptions. For this
reason ﬁrms try to use another and my more realistic method which is called
bounded rationality. Firms usually do not have a complete knowledge of
the market, hence they try to use partial information based on the local
estimates of the marginal proﬁt. At each time period t each ﬁrm increases
(decreases) its production qi at the period (t + 1) if the marginal proﬁt is
positive (negative). If the players use this kind of adjustments then they are
boundedly rational players.
Let us consider a Cournot duopoly game where qi denotes the quantity
supplied by ﬁrm i, i = 1, 2. In addition let P(qi + qj), i = j, denote a
twice diﬀerentiable and nonincreasing inverse demand function and let Ci(qi)
denote the twice diﬀerentiable increasing cost function. Hence the proﬁt of
ﬁrm i is given by
Πi = P(qi + qj)qi − Ci(qi) (1)
At each time period every player must form an expectation of the rival’s
output in the next time period in order to determine the corresponding proﬁtmaximizing
quantities for period t + 1. If we denote by qi(t) the output of
ﬁrm i at time period t, then its production qi(t + 1), i = 1, 2 for the period
3
t + 1 is decided by solving the two optimization problems.
q1(t + 1) = arg maxq1 Π1(q1(t), qe
2(t + 1))
q2(t + 1) = arg maxq2 Π2(qe
1(t + 1), q2(t)),
(2)
where the function Πi(., .) denotes the proﬁt of the i the ﬁrm and qe
j (t + 1)
represents the expectation of ﬁrm i about the production decision of ﬁrm
j, (j = 1, 2, j = i). Cournot [1] assumed that qe
i (t + 1) = qi(t) , ﬁrm i
expects that the production of ﬁrm j will remain the same as in current
period ( naive expectations). The solution of the optimization problem of
producer i can be expressed as q1(t + 1) = f (q2(t)),and q2(t + 1) = g(q1(t)),
so that the time evolution of the duopoly system is obtained by the iteration
of the two-dimensional map T : R2
→ R2
given by
T :
q′
1 = f(q2)
q′
2 = g(q1)
(3)
where′
represents the one-period advancement operator.
The map (3) describes the duopoly game in the case of homogenous expectations
(naive). The ﬁxed points of the map (3) are located at the intersections
of the two reaction functions q1 = f(q2) and q2 = f(q1) and are
called Cournot-Nash equilibria of the two-players game.
Firms try to use more complex expectations such as bounded rationality,
hence they try to use local information based on the marginal proﬁt ∂Πi
∂qi
.
At each time period t each ﬁrm increases (decreases) its production qi at the
period (t + 1) if the marginal proﬁt is positive (negative). If the players use
this kind of adjustments then they are boundedly rational players and the
dynamical equation of this game has the form
qi(t + 1) = qi(t) + αiqi(t)
∂Πi
∂qi(t)
, t = 0, 1, 2, .. (4)
where αi is a positive parameter which represents the speed of adjustment.
The dynamics of the game (4) was studied by Bischi and Naimzada [7 ].
In duopoly game with players use a diﬀerent expectations for example
the ﬁrst player is boundedly rational player and the other is naive. Hence
the duopoly game in this case are composed from the ﬁrst equation of (4)
and the second equation of (3). Thus the discrete dynamical system in this
case is described by:
4
q1(t + 1) = q1(t) + α1q1(t) ∂Π1
∂q1(t)
q2(t + 1) = g(q1(t))
(5)
Therefore Eq. (5) describes the dynamics of a duopoly game with two
players using heterogenous expectations. In the next section we are going
to apply this technique to a duopoly model with linear demand and cost
functions.
3 The model
Let qi(t), i = 1, 2 represents the output of ith supplier during period t, with a
production cost function Ci(qi). The price prevailing in period t is determined
by the total supply Q(t) = q1(t) + q2(t) through a linear demand function
P = f(Q) = a − bQ (6)
where a and b positive constants of demand function. The cost function
is taken in the linear form
Ci(qi) = ciqi, i = 1, 2 (7)
where ci is the marginal cost of ith ﬁrm. With these assumptions the
single proﬁt of ith ﬁrm is given by
Πi = qi(a − bQ) − ciqi, i = 1, 2 (8)
Then the marginal proﬁt of ith ﬁrm at the point (q1, q2) of the strategy
space is given by
∂Πi
∂qi
= a − ci − 2bqi − bqj, i, j = 1, 2, j = i (9)
This optimization problem has unique solution in the form
qi = ri(qj) =
1
2b
(a − ci − bqj) (10)
If the two ﬁrms are naive players, then the duopoly game is describe from
Eq.(3) by using Eq. (10) which has a linear form and the Nash equilibrium is
asymptotically stable [6]. In this study we consider two players with diﬀerent
5
expectation, which the ﬁrst is boundedly rationality player and the other
naive player. The dynamic equation of the ﬁrst player (boundedly rational
player) is obtained from inserting (9) in (3) which has the form:
q1(t + 1) = q1(t) + αq1(t)(a − c1 − 2bq1(t) − bq2(t)) (11)
Using second equation of Eq. (3) the second player (naive) updates his
output according to the dynamic equation:
q2(t + 1) =
1
2b
(a − c2 − bq1(t)) (12)
Then the duopoly game with heterogeneous players is described by a
two-dimensional nonlinear map
T(q1, q2) → (q′
1, q′
2)
which is deﬁned from coupling the dynamic equations (11) and (12) as
follows:
T :
q′
1 = q1 + αq1(a − c1 − 2bq1 − bq2)
q′
2 = 1
2b
(a − c2 − bq1)
(13)
The map (13) is a noninvertable map of the plane. The study of the
dynamical properties of the map (13) allows us to have information on the
long-run behavior of heterogenous players. Starting from given initial condition
(q10,q20 ), the iteration of equation (13) uniquely determines a trajectory
of the states of ﬁrms output.
(q1(t), q2(t)) = Tt
(q10,q20 ), t = 0, 1, 2, ...
3.1 Equilibrium points and local stability
The ﬁxed points of the map (13) are obtained as nonnegative solutions of
the algebraic system
q1(a − c1 − 2bq1 − bq2) = 0
(a − c2 − 2bq2 − bq1) = 0
which is obtained by setting q′
i = qi, i = 1, 2 in Eq. (13). We can have at
most two ﬁxed points E0 = (0, a−c2
2b
) and E∗ = (q∗
1, q∗
2). The ﬁxed point E0 is
6
called a boundary equilibrium [7] and have economic meaning when c2 < a.
The second equilibrium E∗ is called Nash equilibrium where
q∗
1 =
a + c2 − 2c1
3b
and q∗
2 =
a + c1 − 2c2
3b
(14)
provided that
2c1 − c2 < a
2c2 − c1 < a
(15)
It is easy to verify that the equilibrium point E∗ is located at the intersection
of the two reaction curves which represent the locus of points of
vanishing marginal proﬁt in Eq. (9). In the following, we assume that Eq.
(15) is satisﬁed, so the Nash equilibrium E∗ exists.
The study of the local stability of equilibrium solutions is based on the
localization, on the complex plane of the eigenvalues of the Jacobian matrix
of the two dimensional map (Eq. (13)).
The study of the local stability of equilibrium solutions is based on the
localization, on the complex plane of the eigenvalues of the Jacobian matrix
of the two dimensional map (Eq. (13)).
The Jacobian matrix of the map (13) at the state (q1, q2) has the from:
J(q1, q2) =
1 + α(a − 4bq1 − bq2 − c1) −αbq1
−1
2
0
(16)
The determinant of the matrix J is
Det = −
1
2
αbq1
Hence the map (13) is dissipative dynamical system when |αbq1| < 2.
Lemma 1: The ﬁxed point E0 of the map (Eq. (13)) is unstable .
Proof: In order to prove this results, we estimate the eigenvalues of
Jacobian matrix J at E0. The Jacobian matrix has the form
J(E0) =
1 + α
2
(a − 2c1 + c2) 0
−1
2
0
The matrix J(E0) has two eigenvalues λ1 = 1+ α
2
(a−2c1 +c2) and λ2 = 0.
From condition (15), it follows that |λ1| > 1. Then E0 is unstable ﬁxed point
(saddle point) for the map (13) and this completes the proof
7
3.1.1 Local stability of Nash equilibrium
We study the local stability of Nash equilibrium of two-dimensional map
(13). The Jacobian matrix (16) at E∗, which take the form
J(E∗) =
1 − 2αbq∗
1 −αbq∗
1
−1
2
0
(17)
The characteristic equations of J(E∗) is P(λ) = λ2
−Trλ+Det = 0 where
Tr is the trace and Det is the determinant of the Jacobian matrix deﬁnes in
(17), Tr = 1 − 2αbq∗
1 and Det = −1
2
αbq∗
1,
Since (Tr)2
−4Det = (1−2αbq∗
1)2
+2αbq∗
1. It is clear that (Tr)2
−4Det > 0
( has positive discriminant), then we deduce that the eigenvalues of Nash
equilibrium are real. The local stability of Nash equilibrium is given by
using Jury’s conditions [21] which are:
1. |Det| < 1
2. 1 − Tr + Det > 0, and
3. 1 + Tr + Det > 0
The ﬁrst condition is |αbq∗
1| < 2, which implies that
α <
6
(a − 2c1 + c2)
(18)
The second condition 1−Tr+Det = 3
2
αbq∗
1 > 0, then the second condition
is satisﬁed. Then the third condition becomes:
5
2
αbq∗
1 − 2 < 0
This inequality is equivalent to
α <
12
5(a − 2c1 + c2)
(19)
Form (18) and (19) it follows that the Nash equilibrium is stable if α <
12
5(a−2c1+c2)
and hence the following lemma is proved.
Lemma 2: The Nash equilibrium E∗ of the map (Eq. (13)) is stable
provided that α < 12
5(a−2c1+c2)
.
8
From the previous lemma, we obtain information of the eﬀects of the
model parameters on the local stability of Nash equilibrium point E∗. For
example, an increase of the speed of adjustment of boundedly rational player
with the other parameters held ﬁxed, has a destabilizing eﬀect. In fact, an
increase of α, starting from a set of parameters which ensures the local stability
of the Nash equilibrium, can bring out the region of the stability of
Nash equilibrium point, crossing the ﬂip bifurcation surface α = 12
5(a−2c1+c2)
.
Similar argument apply if the parameters α, b, c1 and c2 are ﬁxed parameters
and the parameter a, which represents the maximum price of the good
produced, is increased. In this case the region of stability of E∗ becomes
small, and this implies that E∗ losses its stability. Complex behaviors such
as period doubling and chaotic attractors are generated where the maximum
Lyapunov exponents of the map (16) become positives.
3.2 Numerical investigations
The main purpose of this section is to show that the qualitative behavior. of
the solutions of the duopoly game (13 ) with heterogeneous player generates
a complex behavior that the case of duopoly game with homogenous (naive)
player.
To provide some numerical evidence for the chaotic behavior of system
(13), we present various numerical results here to show the chaoticity, including
its bifurcations diagrams, strange attractors, Laypunov exponents,
Sensitive dependence on initial conditions and fractal structure. In order to
study the local stability properties of the equilibrium points, it is convenient
to take the parameters values as follows: a = 10, b = .5, c1 = 3 and c2 = 5.
The Figure 1 shows the bifurcation diagram with respect to the parameter
α (speed of adjustment of boundedly rational player), while the other
parameters are ﬁxed ( a = 10, b = .5, c1 = 3, and c2 = 5). In fact a bifurcation
diagram of a two-dimensional map (13) shows attractor of the model
(13) as a multi-valued of two-dimensional map of one parameter. In Figure
1 the bifurcation scenario is occurred, if α is small then three exists a stable
equilibrium point (Nash). As one can see the Nash equilibrium point (6, 2) is
locally stable for small values of α. As α increases, the Nash equilibrium becomes
unstable, inﬁnitely many period doubling bifurcations of the quantity
behavior becomes chaotic, as α increased. It means for a large values of speed
of adjustment of bounded rational player α, the system converge always to
complex dynamics. Also one can see that the period doubling bifurcation
9
occur at α = 4
15
. If the case of α > 4
15
, one observes ﬂip bifurcation occurs
and complex dynamic behavior begin to appear for 4
15
< α < 1.
A bifurcation diagram with respect to the marginal cost of the ﬁrst player
c1, while other parameters are ﬁxed as follow a = 10, b = .5, c2 = 5 and
α = .335, is shown in Figure 2.
We show the graph of a strange attractor for the parameter constellation
(a, b, c1, c2, α) = (10, .5, 3, 5, .42) in Figure 3, which exhibits a fractal
structure similar to Henon attractor [22].
In order to analyze the parameter sets for which aperiodic behavior occurs,
one can compute the maximal Lyapunov exponent depend on α. For
example, if the maximal Lyapunov exponent is positive, one has evidence
for chaos. Moreover, by comparing the standard bifurcation diagram in α,
one obtains a better understanding of the particular properties of the system.
In order to study the relations between the local stability of the Nash
equilibrium point and the speed of adjustment of boundedly rational player
α, one can compute the maximal Lyapunov exponents for adjustment factor
in the environment of 1. Figure 4 displays the related maximal Lyapunov
exponents as a function of α. From Figure 4, one can easily determine the
degree of the local stability for diﬀerent values of α ∈ ( 4
15
, 1). At value of
α > 4
15
the maximal Lyapunov exponents is positive. A positive value of
maximal Lyapunov exponents implies sensitive dependence on initial condition
for chaotic behavior. From the maximal Lyapunov exponents diagram
it is easy to determine the parameter-sets for which the system converges to
cycles, aperiodic, chaotic behavior. Beyond that its even possible to diﬀerentiate
between cycles of very higher order and aperiodic behavior of the map
(13) see Figure 4.
3.2.1 Sensitive dependence on initial conditions
To demonstrate the sensitivity to initial conditions of system (13), we compute
two orbits with initial points (q10 , q20 ) and (q10 +.0001, q20 ), respectively.
The results are show in Fig. 5 and Fig. 6. At the beginning the time series
are indistinguishable; but after a number of iterations, the diﬀerence between
them builds up rapidly.
Figure 5 and ﬁgure 6 show sensitive dependence on initial conditions,
q1-coordinates of the two orbits, for system (13), plotted against the time
with the parameter constellation (a, b, c1, c2, α) = (10, .5, 3, 5, .4); the q1coordinates
of initial conditions diﬀer by 0.0001, the other coordinate kept
10
equal.
3.2.2 Fractal dimension of the map (13)
Strange attractors are typically characterized by fractal dimensions. We
examine the important characteristic of neighboring chaotic orbits to see
how rapidly they separate each other. The Lyapunov dimension see [23] is
deﬁned as follows:
dL = j +
i=j
i=1 λi
|λj|
with λ1, λ2, ...., λn, where j is the largest integer such that i=j
i=1 λi ≥ 0
and i=j+1
i=1 λi < 0.
In our case of the two dimensional map has the Lyapunov dimension
which is given by
dL = 1 +
λ1
|λ2|
, λ1 > 0 > λ2
By the deﬁnition of the Lyapunov dimension and with help of the computer
simulation one can show that the Lyapunov dimension of the strange attractor
of system (13). At the parameters values (a, b, c1, c2, α) = (10, .5, 3, 5, .41)
two Lyapunov exponents exists and are λ1 = 0.28 and λ2 = −1.06. Therefore
the map (13) has a fractal dimension dL ≈ 1 + 0.28
1.06
≈ 1.26, which is the same
fractal dimension of Henon map [24].
4 Conclusions
We have investigated the dynamics of a nonlinear, two-dimensional duopoly
game, which contains two-types of heterogeneous players: boundedly rational
player and naive player. This game is described by a two-dimensional
noninvertible map. The stability of equilibria, bifurcation and chaotic behavior
are analyzed. The inﬂuence of the main parameters (such as the speed
of adjustment of boundedly rational player, the maximum price of demand
function and the marginal costs of players) on the local stability is studied.
We deduced that introducing heterogeneous expectations for players in the
duopoly game cause a market structure to behave chaotically.
Acknowledgment : We thank T. Onazaki for his comments.
11
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13
Figure Captions
Fig.1. Shows the bifurcation diagram with respect to the parameter
α speed of adjustment of bounded rational player, with
other ﬁxed parameters a = 10, b = .5, c1 = 3, and c2 = 5.
Fig.2. A bifurcation diagram with respect to the marginal
cost of the ﬁrst player c1, with other parameters ﬁxed at a =
10, b = .5, c2 = 5 and α = .335
Fig.3. We show the graph of a strange attractor for the parameter
constellation (a, b, c1, c2, α) = (10, .5, 3, 5, .41).
Fig.4. displays the related maximal Lyapunov exponents as
a function of α.
Fig.5. and Fig 6 show sensitive dependence on initial conditions,
q1-coordinates of the two orbits, for system (12), at the
parameter values (a, b, c1, c2, α) = (10, .5, 3, 5, .4);
14