# Complex Dynamics of an Adnascent-Type Game Model

Authors: Baogui Xin; Junhai Ma; Qin Gao
Journal: Discrete Dynamics in Nature and Society
 (2008)

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## Abstract

The paper presents a nonlinear discrete game model for two oligopolistic firms whose products are adnascent. (In
biology, the term adnascent has only one sense, “growing to or on
something else,” e.g.,  “moss is an adnascent plant.” See Webster's Revised Unabridged Dictionary published in 1913 by C. & G. Merriam
Co., edited by Noah Porter.) The bifurcation of its Nash equilibrium is analyzed with Schwarzian derivative and normal form theory. Its complex dynamics is demonstrated by means of the largest Lyapunov exponents, fractal dimensions, bifurcation diagrams, and phase portraits. At last, bifurcation and chaos anticontrol of this system are studied.


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## Body

## 1. Introduction

Economic  thought has  had some significant influence on the development of ecological
theory [1]. (Worster
claimed that Darwin was influenced in his development of the theory of
evolution of species by the views of Malthus.) In
the opposite direction, many scientists such as Marshall [2] and Lotka [3], have stated that biology
can be a source of inspiration for economics. (Marshall [2] suggested that “The
Mecca of the economist lies in economic biology rather than in economic
dynamics;" Lotka [3] said that “Man's industrial activities are merely
a highly specialized and greatly form of the general biological struggle for
existence,…, the analysis of the biophysical foundations of economics, is one
of the problems coming within the program of physical biology.") Thus further analogies between
biology and economics can be discovered as both disciplines adopt concepts such
as competition, mutualism and adnascent relation. Such ideas have greatly
influenced a good many researchers in economics, for example, Barnett and Glenn [4] investigated competition and
mutualism among early telephone companies;  Hens and Schenk-Hoppé  [5] studied evolutionary
stability  of portfolio rules in incomplete markets; Levine [6] Compared products and
production in ecological and industrial systems.In addition, there are a lot of phenomena with
adnascent relation in economics, for example, a car key ring is adnascent to a
car. In this paper, the definition of adnascent will be applied into economics
to investigate a novel game model with two oligopolistic firmsX and Y, where
product B of the firm Y is adnascent to product A of the firm X, and the output
of product B is determined by the output of product A, but not vice versa.In 1838, Cournot proposed the classical oligopoly game
model. In 1883, Bertrand reworked Cournot's duopoly game model using prices
rather than quantities as the strategic variables. In 1991, Puu [7] introduced chaos and
bifurcation theory into duopoly game models. Over the past decade, many
researchers, such as Tramontana et al.   [8], Ahmed and Agiza [9] and Ahmed et al. [10],  Agiza and Elsadany  [11],  Bischi et al.  [12], Kopel [13] and Den Haan [14], have paid a great attention to the dynamics of
games.As mentioned above, if one draws an analogy between
species in biology and products in economics, it is easy to find that some of
relationships among different products are substitutable or parasitic, and
others are supportive or adnascent. But all the models cited above are based on
the assumption that all players(firms) produce goods which are perfect
substitutes in an oligopoly market. In this paper, we assume that the
relationship of two players' products are not substitutable but adnascent.This paper is organized as follows. In Section2, a
nonlinear discrete adnascent-type game model is presented. In Section 3, local
stability of the Nash equilibrium of this system is studied. In Section 4, the
bifurcation is studied with Schwarzian derivative and normal form theory. In
Section 5, bifurcation and chaos anticontrol of the model is considered with
nonlinear feedback anticontrol technology. In Section 6, the model's complex
dynamics is numerically simulated by the largest Lyapunov exponents, fractal
dimensions, bifurcation diagrams and phase portraits.


## 2. An adnascent-type dynamical game
model

### 2.1. Assumptions

This model is
based on these following assumptions.Assumption 2.1.
There are
two heterogeneous firmsX and Y producing adnascent products. The production
decision of firm Y must depend on firm X, but not vice versa.Assumption 2.2.
Each firm is
a monopoly of its products market.Assumption 2.3.
Firms have
respective nonlinear variable cost functions [15] and nonlinear inverse
demand functions [16].  (The
linear cost function C(x)=x or C(x)=a+bx is usually adopted in the classical economics.
Indeed, quadratic cost functions are often met in many applications (see [17–19]).)Assumption 2.4.
FirmX can
compete solely on price and then make its output decision, which can have
effect on firm Y.Assumption 2.5.
Firms always make the optimal output
decision for the maximal margin profit in every period.


### 2.2. Nomenclature

The following
is a list of notations that will be used throughout the paper.(i)
xt,yt are outputs of
firms X and Y in period t,
respectively, and they must be positive for any t>0.(ii)
Pxt=a1−b1xt2,Pyt=a2−b2yt2 are nonlinear
inverse demand functions [16] for firms X and Y in period t,
respectively, where a1,b1,a2,b2>0.(iii)
Cxt=c1xt2,Cyt=c2yt2 are nonlinear
variable cost functions [15]
for firms X and Y in period t,
respectively, where c1,c2>0. (The
nonlinear variable cost function C(x)=cx2 can be derived from a Cobb-Douglas-type production
function (see [19–21]).)(iv)
Πxt=Pxtxt−Cxt=xt(a1−b1xt2)−c1xt2,Πyt=Pytyt−Cyt=yt(a2−b2yt2)−c2yt2 are single
profits of firms X and Y in period t,
respectively.(v)
α1,α2>0 are respective
output adjustment parameters of firms X and Y, which represent the fluctuation
of two firms' output decisions. Generally speaking, the two parameters should
be very small.


### 2.3. Model

With Assumptions (2.5), the margin profits of firms X
and Y in period t are give, respectively, by(2.1)∂Πxt∂xt=a1−3b1xt2−2c1xt,∂Πyt∂yt=a2−3b2yt2−2c2yt.One of the methods to find out the Nash equilibrium is
to let (2.1) be equal to 0. Thus one can get firms' reaction functions, that is,
the optimal outputs xt* and yt*.
Under Assumptions (2.1) and (2.4), the dynamic adjustment of the adnascent-type
game can be written as follows:(2.2)xt+1=xt+α1xt∂Πxt∂xt,yt+1=yt+α2xt∂Πyt∂yt.The game model with bounded
rational players has the following nonlinear form:(2.3)xt+1=xt+α1xt(a1−3b1xt2−2c1xt),yt+1=yt+α2xt(a2−3b2yt2−2c2yt).Note that the model has a
particular form, it is a so-called triangular map which is the class of maps in
which one dynamic variable is independent on the other, that is of the type x′=f(x), y′=g(x,y),
while the other, y,
strongly depends on the first. A peculiarity of this class of maps is that the
eigenvalues in any point of the phase plane are always real, and that many
bifurcations are explained via the one-dimensional map x′=f(x).


## 2.1. Assumptions

This model is
based on these following assumptions.Assumption 2.1.
There are
two heterogeneous firmsX and Y producing adnascent products. The production
decision of firm Y must depend on firm X, but not vice versa.Assumption 2.2.
Each firm is
a monopoly of its products market.Assumption 2.3.
Firms have
respective nonlinear variable cost functions [15] and nonlinear inverse
demand functions [16].  (The
linear cost function C(x)=x or C(x)=a+bx is usually adopted in the classical economics.
Indeed, quadratic cost functions are often met in many applications (see [17–19]).)Assumption 2.4.
FirmX can
compete solely on price and then make its output decision, which can have
effect on firm Y.Assumption 2.5.
Firms always make the optimal output
decision for the maximal margin profit in every period.


## 2.2. Nomenclature

The following
is a list of notations that will be used throughout the paper.(i)
xt,yt are outputs of
firms X and Y in period t,
respectively, and they must be positive for any t>0.(ii)
Pxt=a1−b1xt2,Pyt=a2−b2yt2 are nonlinear
inverse demand functions [16] for firms X and Y in period t,
respectively, where a1,b1,a2,b2>0.(iii)
Cxt=c1xt2,Cyt=c2yt2 are nonlinear
variable cost functions [15]
for firms X and Y in period t,
respectively, where c1,c2>0. (The
nonlinear variable cost function C(x)=cx2 can be derived from a Cobb-Douglas-type production
function (see [19–21]).)(iv)
Πxt=Pxtxt−Cxt=xt(a1−b1xt2)−c1xt2,Πyt=Pytyt−Cyt=yt(a2−b2yt2)−c2yt2 are single
profits of firms X and Y in period t,
respectively.(v)
α1,α2>0 are respective
output adjustment parameters of firms X and Y, which represent the fluctuation
of two firms' output decisions. Generally speaking, the two parameters should
be very small.


## 2.3. Model

With Assumptions (2.5), the margin profits of firms X
and Y in period t are give, respectively, by(2.1)∂Πxt∂xt=a1−3b1xt2−2c1xt,∂Πyt∂yt=a2−3b2yt2−2c2yt.One of the methods to find out the Nash equilibrium is
to let (2.1) be equal to 0. Thus one can get firms' reaction functions, that is,
the optimal outputs xt* and yt*.
Under Assumptions (2.1) and (2.4), the dynamic adjustment of the adnascent-type
game can be written as follows:(2.2)xt+1=xt+α1xt∂Πxt∂xt,yt+1=yt+α2xt∂Πyt∂yt.The game model with bounded
rational players has the following nonlinear form:(2.3)xt+1=xt+α1xt(a1−3b1xt2−2c1xt),yt+1=yt+α2xt(a2−3b2yt2−2c2yt).Note that the model has a
particular form, it is a so-called triangular map which is the class of maps in
which one dynamic variable is independent on the other, that is of the type x′=f(x), y′=g(x,y),
while the other, y,
strongly depends on the first. A peculiarity of this class of maps is that the
eigenvalues in any point of the phase plane are always real, and that many
bifurcations are explained via the one-dimensional map x′=f(x).


## 3. Nash equilibrium and its local
stability of system (2.3)

A Nash
equilibrium, named after John Nash, is a solution concept of a game involving
two or more players, such that no player has incentive to unilaterally change
his or her action. In other words, players are in equilibrium if a change in
strategies by any one of them would lead that he (she) to earn less than if
he (she) remained with his (her) current strategy.System (2.3) is a two-dimensional non-invertible that
depends on eight parameters. The Nash equilibrium point of system (2.3) is
the solution of the following algebraic system:(3.1)α1x(a1−3b1x2−2c1x)=0,α2x(a2−3b2y2−2c2y)=0.Note that system (3.1) does not
depend on the parameters α1 and α2.
By simple computation of the above algebraic system it was found that there
exists one interesting positive Nash equilibrium as follows:  (3.2)E*(x*,y*)=(A−c13b1,B−c23b2),where A=c12+3a1b1,B=c22+3a2b2.The Jacobian matrix of system (2.3) at the Nash
equilibrium E*(x*,y*) has the following form:(3.3)J(E*)=[1−2α1(a1−c1x*)001−2α2Bx*].Thus its eigenvalues can be expressed asλ1=1−2α1Ax* and λ2=1−2α2Bx*.
Then the condition λ1<1 is always satisfied while λ1>−1 holds if(3.4)α1<1Ax*=3b1A(A−c1)=C,and the condition λ2<1 is always satisfied while λ2>−1 holds if(3.5)α2<1Bx*=3b1B(A−c1)=D.As a result, the following
proposition holds.Proposition 3.1.
The Nash equilibriumE*(x*,y*) is called
(i)
a sink ifα1<C and α2<D,
so the sink is locally asymptotically stable;(ii)
a source
ifα1>C and α2>D,
so the sink is locally unstable;(iii)
a saddle
ifα1<C and α2>D or α1>C and α2<D;(iv)
non-hyperbolic if eitherα1=C or α2=D.


## 4. Bifurcation analysis

Due to the fact
that the map is triangular, the stability of the variablex is independent on the other, thus the
bifurcation analysis for this variable can be easily performed with the
one-dimensional map x′=f(x),
which is a cubic, and the interest is only in the positive part.The best known and most popular projective
differential invariant is the Schwarzian derivative. The map's Schwarzian
derivative [22] is(4.1)Sf(x)=f′′′(x)f′(x)−32(f′′(x)f′(x))2=−6α1(3b1+3α1a1b1+54α1b12x2+24α1b1c1x+4α1c12)(1+α1a1−9α1b1x2−4α1c1x)2.Obviously Sf(x)<0 for x>0,
so that all the flip bifurcations are supercritical [23].An example of supercritical flip bifurcation will be
presented with normal form theory as follows.Generally speaking, for given firmsX and Y, their
parameters a1,b1,a2,b2,c1, and c2 are invariable, and their output adjustment
parameters α1 and α2 are changeable. In what follows, for
convenience of studying the bifurcation parameter α1 and α2,
we let a1=10, b1=0.5, a2=9.75, b2=0.182, c1=5, and c2=4.
Then we can get the following system:(4.2)xt+1=xt+α1xt(10−1.5xt2−10xt),yt+1=yt+α2xt(9.75−0.546yt2−8yt).Howevere, (4.2) exists a Nash equilibrium point E*(0.883,1.132) which is independent of the parameters α1 and α2.
The Jacobian matrix at E*(0.883,1.132) is(4.3)A=(1−11.17α1001−8.1557α2).Obviously, its eigenvalues
satisfy (i) λ1=−1 if α1=0.179;
(ii) λ2=−1 if α2=0.245.
Thus system (4.2) may undergo flip bifurcation at α1=0.179 or α2=0.245.Lemma 4.1 (Topological norm form for the flip bifurcation [24]).
Any generic, scalar,
one-parameter systemx↦f(x,α), having at α=0 the fixed point x0=0 with μ=fx(0,0)=−1,
is locally topologically equivalent near the origin to one of the following
normal forms: η↦−(1+β)η±η3.The following
system can be obtained withα2=0.2,(4.4)xt+1=xt+α1xt(10−1.5xt2−10xt),yt+1=yt+0.2xt(9.75−0.546yt2−8yt).Proposition 4.2 (Critical norm form for flip bifurcation).
System (4.4) can be written as following critical normal form for flip
bifurcation:  (4.5)ξt+1=−ξt+cξt3,where c=12.23.Proof.
To compute coefficients of normal form, we translate
the origin of the coordinates to this Nash equilibriumE*=(0.883,1.132) by the change of variables as by the change of
variables(4.6)x=0.883+u,y=1.132+v.This transforms system (4.2) with
parameters α1=0.179 into(4.7)ut+1=0.883−ut−2.5ut2−0.269ut3,vt+1=1.131−0.001ut−0.63vt−1.85utvt−0.1vt2−0.11utvt2.This system can be written
as(4.8)Xn+1=AXn+12B(Xn,Xn)+16C(Xn,Xn,Xn)+O(Xn4),where(4.9)A0=A(E*)=(−100−0.63).and the multilinear functions B:ℝ2×ℝ2→ℝ2 and C:ℝ2×ℝ2×ℝ2→ℝ2 are also defined, respectively,
by(4.10)Bi(x,y)=∑j,k=12∂2Xi(ξ,0)∂ξj∂ξk|ξ=0xjyk,Ci(x,y,z)=∑j,k,l=12∂3Xi(ξ,0)∂ξj∂ξk∂ξl|ξ=0xjykzl.
For system (4.7),(4.11)B(ξ,η)=(−5ξ1η1−1.85ξ1η2−0.2ξ2η2),C(ξ,η,ζ)=(−1.61ξ1η1ζ1−0.22ξ1η2ζ2).The eigenvalues of the matrix J
are λ1=−1 and λ2=−0.63.
Letq,p∈ℝ2 be eigenvectors corresponding to λ1,λ1,
respectively: (4.12)q=(10),p=(10).satisfy A0q=−q,A0Tp=−p and 〈p,q〉=1.
So the coefficient of the normal form of system (4.7)
can be computed by the following invariant formula:(4.13)c=16〈p,C(q,q,q)〉−12〈p,B(q,(A−In)−1B(q,q))〉=12.23.
The proposition is proved.The bifurcation type is determined by the stability of
the Nash equilibrium as at the critical parameter value. According to the above
Proposition4.2, for system (4.7), the critical parameter c=12.23>0,
so the flip bifurcation at the Nash equilibrium E*(0.883,1.132) is supercritical.


## 5. Bifurcation and chaos anticontrol

A government may pay attention to chaos anticontrol
on the game system. Its motivations are as follows. Chaos exhibits high
sensitivity to initial conditions, which manifests itself as an exponential
growth of perturbations in the initial conditions. As a result, two firms'
decision behaviors of the anticontrolled chaotic game systems appear to be
random. So it can weaken the negative effect of excessive monopoly at least. In
addition, Huang [25]
has proved that, in some sense, chaos is beneficial not only to all
oligopolistic firms but also to the economy as a whole.There are various methods can be used to control or
anticontrol bifurcations and chaos, for example, impulsive control [26], adaptive feedback
control [27], linear
and nonlinear feedback control [28–30]. In this section, the nonlinear feedback technique
will be employed to anticontrol system (4.4). As mentioned above, system (4.4) is a adnascent-type game model, that is, firm Y must depend on firm X,
but not vice versa. In other words, the production decision of firm X is
independent. Since firm X of system (4.4) undergoes bifurcation and chaos, one
may merely anticontrol firm Y. Considering the principle of simplification and
maneuverability, one may choose a generalized nonlinear feedback
anticontroller (e.g., production tax rebate) on firm Y as follows:(5.1)u=∑i=1nkiyi,where the linear terms in the
anticontroller are used to shift the location of the equilibrium and
bifurcation because only the linear part affects the Jacobian matrix of the
linearized system, the nonlinear terms are used to change the property of the
bifurcation and chaos. But it is not necessary to take too much components
unless one wants to preserve all equilibria of the original system. In this
paper, since it is unnecessary to preserve all equilibria of system, the
anticontroller can be greatly simplified as(5.2)u=ky2.Then the anticontrolled system
can be represented as(5.3)xt+1=xt+α1xt(10−1.5xt2−10xt),yt+1=yt+0.2xt(9.75−0.546yt2−8yt)+kyt2,for system (5.3), it is easy to
get its Nash equilibria(5.4)E1(0.88,106−14964−38743k150k−14.47),E2(0.88,106+14964−38743k150k−14.47)and Jacobian
matrix(5.5)J(E*)=[1+α1(10−20x−4.5x2)01.95−0.11y2−1.6y1−0.22xy−0.6x+2ky].As mentioned above, system (4.4) undergoes a flip bifurcation at α1=0.179 and x=0.88.
Like system (5.3), after a anticontroller u=ky2 is put on firm Y of system (4.4), firm X is
uninfluenced. As a result, in system (5.3), when x=0.88 and α1=0.179,
the two conditions of flip bifurcation at Nash equilibria can be expressed as
follows: (5.6)1+α1(10−20x−4.5x2)=−1,|1−0.22xy−0.6x+2ky|<1holdwith0.1<k<0.39.


## 6. Numerical simulations

In this section,
some numerical simulations are presented to confirm the above analytic results
and to demonstrate added complex dynamical behaviors. To do this, one will use
the largest Lyapunov exponents, fractal dimensions, bifurcation diagrams and
phase portraits to show interesting complex dynamical behaviors.In system (4.2), the largest Lyapunov exponents,
fractal dimensions and bifurcation diagrams with two parameters α1 and α2 are shown in Figure 1.For system (4.2) with α1∈[0,0.27] and α2∈[0,0.2], (a) bifurcation diagram of firms X; (b) bifurcation diagram of firms Y; (c)
largest Lyapunov exponents (LLEs); (d) fractal dimensions (FDs).
(a)(b)(c)(d)Figure1(a) is the outputs   bifurcation diagram of
firm X with the parameters α1∈[0,0.27] and α2∈[0,0.2].
When the output adjustment parameter α1 increases, the outputs of firm X present
complex dynamics as follows. Its outputs change from Nash equilibrium to
bifurcation till chaos. Obviously the output adjustment parameter α2 of firm Y has no effect on firm X, which just
verifies the adnascent relationship between firms X and Y.Figure1(b) is the outputs bifurcation diagram of
firm Y with the parameters α1∈[0,0.27] and α2∈[0,0.2].
It is obviously that there is no bifurcation and chaos in Figure 1(b).Figure1(c) is the largest Lyapunov exponents
diagram of system (4.2) with the parameters α1∈[0,0.27] and α2∈[0,0.2].
The Lyapunov exponent of a dynamical system is a quantity that characterizes
the rate of separation of infinitesimally close trajectories. A positive
Lyapunov exponent is usually taken as an indication that the system is
chaotic [31].Figure1(d) is a fractal dimensions diagram of
system (4.2) with the parameters α1∈[0,0.27] and α2∈[0,0.2].
A fractal dimension is taken as a criterion to judge whether the system is
chaotic. There are many specific definitions of fractal dimension and none of
them should be treated as the universal one. This paper adopts the following
definition of fractal dimension [32].(6.1)dL=k−1λk+1∑i=1kλiwhere λ1≥λ2≥,…,≥λn are the Lyapunov exponents and k is the largest integer for which ∑i=1kλi≥0and∑i=1k+1λi<0.
If λi≥0 for all i=1,2,…,n then dL=n.
If λi<0 for all i=1,2,…,n then dL=0.In system (4.4), firm X has supercritical flip
bifurcation at α1=0.179 shown in Figure 2(a), while firm Y undergoes
neither bifurcation nor chaos.(a) The largest Lyapunov exponents and bifurcation
diagram of system (4.4) versus α1∈[0,0.27]; (b) the largest Lyapunov exponents and
bifurcation diagram of system (5.3) versus k=0.2 and α1∈[0,0.27].
(a)(b)In system (5.3), when one fixes k=0.2,
he can get the largest Lyapunov exponents and bifurcations diagram shown in
Figure 2(b) and chaotic attractor portrait shown in Figure 3. Obviously firms
X and Y undergo synchronously bifurcations and chaos with k=0.2.
The goverment can anticontrol the synchronization of bifurcation and chaos by
varying the anticontrol parameter k.For system (5.3) with k=0.2. (a) Chaotic attractor versus α1=0.24; (b) chaotic attractor versus α1=0.27.
(a)(b)


## 7. Conclusion

In this paper,
we have presented a nonlinear adnascent-type game dynamical model with two
oligopolistic firms, and emphatically reported its some complex dynamics, such
as Nash equilibrium, bifurcations, chaos and their anticontrol. By means of
the largest Lyapunov exponents, fractal dimensions, bifurcation diagrams and
phase portraits, we have demonstrated numerically its complex dynamics. For the
system, other complexity anticontrol theory and methodology will be considered
in future work.



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*Source: 467972-2009-03-10.xml*