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.
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 firms and , where product of the firm is adnascent to product of the firm , and the output of product is determined by the output of product , 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 Section 2, 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 firms and producing adnascent products. The production decision of firm Y must depend on firm , 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 or is usually adopted in the classical economics. Indeed, quadratic cost functions are often met in many applications (see [17–19]).)
Assumption 2.4. Firm X 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) are outputs of firms and in period , respectively, and they must be positive for any .(ii) are nonlinear inverse demand functions [16] for firms and in period , respectively, where .(iii) are nonlinear variable cost functions [15] for firms and in period , respectively, where . (The nonlinear variable cost function can be derived from a Cobb-Douglas-type production function (see [19–21]).)(iv) are single profits of firms and in period , respectively.(v) are respective output adjustment parameters of firms and , 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 and in period are give, respectively, by
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 and . Under Assumptions (2.1) and (2.4), the dynamic adjustment of the adnascent-type game can be written as follows:The game model with bounded rational players has the following nonlinear form: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 , , while the other, , 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 .
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:Note that system (3.1) does not depend on the parameters and . By simple computation of the above algebraic system it was found that there exists one interesting positive Nash equilibrium as follows: where
The Jacobian matrix of system (2.3) at the Nash equilibrium has the following form:
Thus its eigenvalues can be expressed as and . Then the condition is always satisfied while holds ifand the condition is always satisfied while holds ifAs a result, the following proposition holds.
Proposition 3.1. The Nash equilibrium is called
(i)a sink if and ,
so the sink is locally asymptotically stable;(ii)a source
if and ,
so the sink is locally unstable;(iii)a saddle
if and or and ;(iv)non-hyperbolic if either or .
4. Bifurcation analysis
Due to the fact that the map is triangular, the stability of the variable is independent on the other, thus the bifurcation analysis for this variable can be easily performed with the one-dimensional map , 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] isObviously for , 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 firms and , their parameters , and are invariable, and their output adjustment parameters and are changeable. In what follows, for convenience of studying the bifurcation parameter and , we let , , , , , and . Then we can get the following system:
Howevere, (4.2) exists a Nash equilibrium point which is independent of the parameters and . The Jacobian matrix at isObviously, its eigenvalues satisfy (i) if ; (ii) if . Thus system (4.2) may undergo flip bifurcation at or .
Lemma 4.1 (Topological norm form for the flip bifurcation [24]). Any generic, scalar, one-parameter system having at the fixed point with , is locally topologically equivalent near the origin to one of the following normal forms:
The following system can be obtained with ,
Proposition 4.2 (Critical norm form for flip bifurcation). System (4.4) can be written as following critical normal form for flip bifurcation: where .
Proof. To compute coefficients of normal form, we translate
the origin of the coordinates to this Nash equilibrium by the change of variables as by the change of
variablesThis transforms system (4.2) with
parameters intoThis system can be written
aswhereand the multilinear functions and are also defined, respectively,
by
For system (4.7),The eigenvalues of the matrix
are and .
Let be eigenvectors corresponding to ,
respectively: satisfy and .
So the coefficient of the normal form of system (4.7)
can be computed by the following invariant formula:
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 Proposition 4.2, for system (4.7), the critical parameter , so the flip bifurcation at the Nash equilibrium 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 must depend on firm , but not vice versa. In other words, the production decision of firm is independent. Since firm of system (4.4) undergoes bifurcation and chaos, one may merely anticontrol firm . Considering the principle of simplification and maneuverability, one may choose a generalized nonlinear feedback anticontroller (e.g., production tax rebate) on firm as follows: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 asThen the anticontrolled system can be represented asfor system (5.3), it is easy to get its Nash equilibriaand Jacobian matrixAs mentioned above, system (4.4) undergoes a flip bifurcation at and . Like system (5.3), after a anticontroller is put on firm of system (4.4), firm is uninfluenced. As a result, in system (5.3), when and , the two conditions of flip bifurcation at Nash equilibria can be expressed as follows:
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 and are shown in Figure 1.
(a)
(b)
(c)
(d)
Figure 1(a) is the outputs bifurcation diagram of firm with the parameters and . When the output adjustment parameter increases, the outputs of firm present complex dynamics as follows. Its outputs change from Nash equilibrium to bifurcation till chaos. Obviously the output adjustment parameter of firm has no effect on firm , which just verifies the adnascent relationship between firms and .
Figure 1(b) is the outputs bifurcation diagram of firm with the parameters and . It is obviously that there is no bifurcation and chaos in Figure 1(b).
Figure 1(c) is the largest Lyapunov exponents diagram of system (4.2) with the parameters and . 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].
Figure 1(d) is a fractal dimensions diagram of system (4.2) with the parameters and . 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].where are the Lyapunov exponents and k is the largest integer for which . If for all then . If for all then .
In system (4.4), firm has supercritical flip bifurcation at shown in Figure 2(a), while firm undergoes neither bifurcation nor chaos.
(a)
(b)
In system (5.3), when one fixes , 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 and undergo synchronously bifurcations and chaos with . The goverment can anticontrol the synchronization of bifurcation and chaos by varying the anticontrol parameter .
(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.
Acknowledgments
The authors acknowledge partial financial support by the National Natural Science Foundation of China (Grant no. 60641006). They also would like to express sincere gratitude to anonymous referees for their valuable suggestions and comments.