## Abstract

In this chapter, we assume that two bounded rational firms not only pursue profit maximization but also take consumer surplus into account, so the objections of all the firms are combination of their profits and the consumer surplus. And then a dynamical duopoly Cournot model with bounded rationality is established. The existence and stability of the boundary equilibrium points and the Nash equilibrium of the model are discussed, respectively. And then the stability condition of the Nash equilibrium is given. The complex dynamical behavior of the system varies with parameters in the parameter space is studied by using the so-called 2D bifurcation diagram. The coexistence of multiple attractors is discussed through analyzing basins of attraction. It is found that not only two attractors can coexist, but also three or even four attractors may coexist in the established model. Then, the topological structure of basins of attraction and the global dynamics of the system are discussed through invertible map, critical curve and transverse Lyapunov exponent. At last, the synchronization phenomenon of the built model is studied.

### Keywords

- bifurcation
- chaos
- duopoly
- consumer surplus
- synchronization

## 1. Introduction

Oligopoly is a market between perfect monopoly and perfect competition [1]. With the application of chaos theory and nonlinear dynamic system into oligopoly models, the static game evolves into a dynamic game. Especially in recent years, with the rapid development of computer technology, a powerful tool has been provided for dealing with the complex nonlinear problems. And hence, the economists and the mathematicians can simulate the complex dynamical behavior of oligopoly market by using computer technology. Recently, a large number of scholars have improved the oligopoly models, and introduced bounded rationality (see [2, 3]), incomplete information [4], time delay [5], market entering and entering barriers [6], differentiated products [7] and other factors [8, 9] into the classical oligopoly models, and the bifurcation and chaos phenomenon were founded in the process of repeated game.

However, all of the above discussions are mainly based on private enterprises, which pursuit the maximization of their own profits. In fact, the public ownership enterprises, which always aim at maximizing social welfare, and mixed ownership enterprises, which always aim at maximizing the weighted average of the social welfare and their own profits, are also widespread in the real economic environment. De Fraja and Delbono [10] found that the social welfare might be higher when a public ownership enterprise is a profit-maximizer rather than a social-welfare-maximizer. Matsumura [11] proposed that the social welfare could be improved through partial privatization of public enterprises. The research of Fujiwara [12] suggested that partial privatized public enterprises are more efficient than private enterprises. Elsadany and Awad [13] explored the complex dynamical behavior of competition between two partial public enterprises under the assumption of bounded rationality. However, the global dynamical behavior and synchronization behavior of semi-public enterprises, which corporate social responsibility into their objectives, are rarely studied. In this chapter, the occurrence of synchronization, the coexistence of attractors and the global dynamic of a duopoly game corporate consumer surplus are mainly discussed.

## 2. The model

Considering a duopolistic market where two firms produce homogeneous goods. In order to study the long-term behaviors of the duopoly market with quantity competition, we briefly present the economic setup leading to the final model in this chapter. The price and quantity of product of firm

Following Dixit [14] and Singh and Vives [15], we suppose that the utility function used in this chapter is quadratic and can be given by,

where

Suppose that the budget constraint of consumer is,

where

This chapter discusses homogenous products, so here it is assumed that all these two players have the identical marginal cost. Therefore, the cost function of firm 1 and firm 2 are same and can be given by,

In the real market, there are a lot of firms, who not only pursue their own profits but also take corporate social responsibility into account. A large number of empirical studies have shown how the introduced corporate social responsibility affects firm’s performance, where we interpret corporate social responsibility as either consumer surplus (for short CS) or social welfare (for short SW). In this chapter we take CS into account to analyze which firms have an incentive to exhibit corporate social responsibility as a means of maximizing their profits in a Cournot competition. Based on the above assumptions and the definition of consumer surplus, CS can be written as,

where

According to the above assumptions, the objective function of the firm

where

And the first-order condition of the objection function (7) is given as,

It is now significant to specify the information set of both players regarding the objection functions, to determine the behaviors of the players with the change of time. We assume a discrete time

where

Since the output of a firm cannot be negative, the initial conditions of map

By setting

and the only Nash equilibrium is

where

## 3. Stability properties

The local stability analyses of system (10) near the fixed points are too difficult to carry on. For the sake of analyzing the local stability of the system, we firstly let

Then all the equilibrium points are substituted into the Jacobian matrix (12). According to the characteristic values of the Jacobian matrix evaluated at each equilibrium, the type and stability of the equilibrium can be analyzed and the following results can be obtained.

**Proposition 1**. The equilibrium point

**Proof**. It is clear that the Jacobian matrix of map

The eigenvalues of

**Proposition 2.**

**Proof**. By substituting the equilibrium

The eigenvalues of

Similar to the case of the equilibrium

For the purpose of research of the local stability near the Nash equilibrium, we should compute the Jacobian matrix evaluated at the Nash equilibrium

It can be seen that the form of the Jacobian matrix is so complex. In order to simplify the calculation, let

Then the trace and the determinant of the Jacobian matrix evaluated at the Nash equilibrium

According to Jury condition, if we substitute the specific mathematical expressions of

Since all the equilibrium points are non-negative when the parameters meet

Then the stability region of the Nash equilibrium can be obtained by substituting

The stability condition of the Nash equilibrium gives a parameters region, in which the Nash equilibrium is always stable. For the sake of better analysis of the stability of the Nash equilibrium under different set of parameters, a useful tool called “two-dimensional bifurcation diagram” (also called 2-D bifurcation diagram) is employed. From (16), we can find that the stability region of the Nash equilibrium is related to the difference of parameters

Figure 1 is a two-dimensional bifurcation diagram of system (10) with a set of fixed parameter

Figure 2a shows the coexistence of attractors with the parameters chosen as

Figure 3 shows a series of two-dimensional bifurcation diagram under different parameters. It shows a very beautiful gallery, from which we can enjoy the system (10) with full complex dynamics phenomenon. We can observe from Figure 3 that the difference between the maximum price of the unit product

## 4. Global dynamics and synchronization

The type and stability of the equilibrium points have been analyzed as above. And the boundary equilibrium

Subsequently, we assume that both firms have the same speed of adjustment. It means that the latter discussion is based on

It can be proved that the map

### 4.1 Critical curve and noninvertible map

We divide the discrete dynamical system into invertible and noninvertible. The invertible discrete dynamical system refers that an image

### 4.2 Invariant sets

The dynamics of the system on the diagonal is studied by analyzing the invariant sets. Firstly, we can prove that the coordinates are invariant sets of map

It is easy to verify that the dynamics on the axis

and the parameter

It can also be proved that the diagonal

Similarly, through the following linear transformation

we can also prove that the map (21) is topologically conjugate to the standard logistic map

Through the standard logistic map, we can easily analyze the dynamical behavior of the two-dimensional map

As shown in Figure 4b, which the parameters is the same as Figure 4a, a two-dimensional bifurcation diagram of the system with

**Proposition 3**. If we let

In order to analyze the effect of any slight perturbation of one parameter on the system, we study the transverse stability of an attractor

Then, the characteristic values of the Jacobian matrix

where the corresponding eigenvectors are

It is assumed that a period-*k* cycle

Since the stability conditions of the period-*k* cycle on the diagonal

Through Eq. (27), we can draw the following conclusions directly. That is, when all the parameters satisfy

As we know that an attractor

where *-k* cycle, then *k* cycle is transversely stable. When the initial condition

If all cycles embedded in

Figure 5 gives the natural transverse Lyapunov exponent and the bifurcation diagram with the fixed parameter

### 4.3 Global bifurcation and basins of attraction

A closed invariant set

For the sake of analyzing the topological structure of the basin of attraction

The result is given as,

Since *k* preimages *1* preimages of a point

We can easily obtain the rank-1 preimages of the origin, which are

Through the discussion above, we can get the following propositions,

**Proposition 4**. Let

Basins of attraction may be connected or not. The connected basins of attraction are divided into simple connected and complex one, and the complex connected basins of attraction means the existence of holes. If

Figure 6 shows the coexistence of attractors and their basins of attraction for given parameters *A* located on the diagonal and the other consisting of 4-piece chaos attractor is in symmetrical positions with respect to the diagonal, i.e.,

We have analyzed the global bifurcations that occur when the attractor’s boundary contact to the critical curve, and we discuss the global bifurcation when the attractor contacts to the boundary of its basin of attraction. We also denote global bifurcation as “boundary crisis,” the attractor is destroyed when it contacts to its basin of attraction. Figure 7 shows the coexistence of attractors and their basins of attraction corresponding to the parameter

Figures 6 and 7 give two different global bifurcations, such bifurcations which can be restored clearly by numerical simulation method only. With the set of parameters in Figure 9 being identical to Figure 4, we select different speed of adjustment to analyze the change of attractors and their basins of attraction. We can observe that as the speed of adjustment changes from 0.79 to 0.9, the period-4 cycle being in symmetrical positions with respect to the diagonal of the system generates smooth limit cycle via a Neimark-Sacker bifurcation, as shown in Figure 9b, and the limit cycle becomes non-smooth gradually, and finally forms four-piece chaotic attractor, as shown in Figure 9d. The basin of attraction shrinks as the speed of adjustment

### 4.4 Synchronization

In this section we study the formation mechanism of the synchronization trajectories. The trajectories starting from different initial conditions return to the diagonal eventually, i.e.,

## 5. Conclusion

In this chapter, the nonlinear dynamics of a Cournot duopoly game with bounded rationality is investigated. Unlike the existing literature, we suppose that the two firms not only pursue profit maximization but also take consumer surplus into account. Meanwhile, the objection of firms is supposed as the weighted sum of profit and consumer surplus. Based on the theory of gradient adjustment, all the firms adjust the output of next period according to the estimation of “marginal goal.” The existence and stability of fixed points are analyzed. It is found that the boundary equilibrium point is always unstable, no matter what the parameters of the system are satisfied. At the same time, with the two-dimensional bifurcation diagram as the tool, the stability of the Nash equilibrium is analyzed. We found that the Nash equilibrium will lose its stability when the speed of adjustment of firms is too large, which maybe lead the market into chaos. The stability region of the Nash equilibrium will be only affected by the weight of consumer surplus. And the parameters

Moreover, with the theory of invertible mapping and the critical curves of the system, the topological structure of basin of attraction is analyzed. By calculating the transverse Lyapunov exponent, the weak chaotic attractor of the system in the sense of Milnor is found, and the synchronization of the system is further studied. If we fix the other parameters of the system, and only change the weight of the firm to the consumer surplus, we can find on-off intermittency phenomenon and synchronization phenomenon. With the increasing of