Discrete Dynamics in Nature and Society

Volume 2017 (2017), Article ID 2363804, 12 pages

https://doi.org/10.1155/2017/2363804

## Dynamics of a Duopoly Game with Two Different Delay Structures

Faculty of Science, Jiangsu University, Zhenjiang 212013, China

Correspondence should be addressed to Shumin Jiang

Received 23 March 2017; Revised 8 May 2017; Accepted 14 May 2017; Published 13 June 2017

Academic Editor: Douglas R. Anderson

Copyright © 2017 Shumin Jiang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Two different time delay structures for the dynamical Cournot game with two heterogeneous players are considered in this paper, in which a player is assumed to make decision via his marginal profit with time delay and another is assumed to adjust strategy according to the delayed price. The dynamics of both players output adjustments are analyzed and simulated. The time delay for the marginal profit has more influence on the dynamical behaviors of the system while the market price delay has less effect, and an intermediate level of the delay weight for the marginal profit can expand the stability region and thus promote the system stability. It is also shown that the system may lose stability due to either a period-doubling bifurcation or a Neimark-Sacker bifurcation. Numerical simulations show that the chaotic behaviors can be stabilized by the time-delayed feedback control, and the two different delays play different roles on the system controllability: the delay of the marginal profit has more influence on the system control than the delay of the market price.

#### 1. Introduction

A monopoly market is the case where a trade is completely controlled by a small number of firms. The few firms produce the same or homogeneous products and they must take into account all the information in the market and the actions of the competitors. Cournot [1] first introduced the game model which gives a mathematical description of the dynamic competition in a duopoly market. In a classic Cournot model, each firm is assumed to have naïve expectation and thus guesses that the opponent’s output remains at the same level as in the previous period and then chooses a optimal production strategy in the current period. The classic research has paid much attention to the stability and the complex phenomena in the dynamics of a Cournot game played by players with naïve expectation (e.g., [2–8]).

Rather than the naïve expectation, a so-called boundedly rationality based on players’ marginal profits has received great attention in recent years. Bischi and Naimzada [9] first gave the general formula of bounded rationality in a duopoly game, where each firm uses the local knowledge of its marginal profit to adjust strategy in a new period: increasing its output if it perceives a positive marginal profit and decreasing its production if the perceived marginal profit is negative. A great deal of research work has been done for the dynamical Cournot game played by homogeneous players with such a kind of rationality (e.g., [9–12]).

Besides the work on the models with homogeneous players, there is another branch of literature that is interested in the games with heterogeneous players. The dynamic duopoly model with one boundedly rational player and one naïve player has been studied by Agiza and Elsadany [13]. By modifying the cost function in [13], Zhang et al. [14] studied the case with nonlinear cost function. A duopoly game with heterogeneous players and isoelastic demand function has been studied by Tramontana [15] and Angelini et al. [16]. Ma and Xie [17] considered the changing demand to develop the dynamic game models for the two scenarios and analyzed the model’s dynamic behavior. In the model with two heterogeneous players, Fan et al. [18] supposed that the naïve player does not know the rival’s output and adjusts his production according to the previous market price. Ding et al. [19] studied a two-team game played by one team consisting of two boundedly rational players and one team consisting of a naïve player.

In the work on the dynamic Cournot model, time delay has also attracted the attention of many scholars. Many researches examine the effect of delay on dynamics. One-time delay, two-time delay, continuously distributed time delay, and geometric delay are systematically reviewed and studied in [20–22], which offer results on existence, stability, and local bifurcations of the equilibrium points. Besides, the work in [23–28] studied the dynamical Cournot game with delayed bounded rationality and found that time delay can increase the system stability and delay the occurrence of complex behaviors. Ma and Si [29] also confirmed that the stability of the system depends on the delay and weight. The result showed that the stability of price was closely related to the parameters and the reasonable region of price benefited the firm profit. Further, Ma and Wu [30] studied the influence of parameter change on the market stability and chaos on the sensitivity level. By comparing parameter basin plots, the influences of decision parameters on the dynamics behaviors of the two models are further analyzed [31]. In [23–25, 27, 28] time delay is done for the output variable so that decision makers’ expectations are based on the delayed data of their outputs, which implicitly means that any player must know each opponent’s output in the previous periods. These models may not be suitable for the market with incomplete information where a decision maker has too limited knowledge to get the opponents’ output data [26]. It is also pointed out [26] that any producer is able to know his own marginal profit, which is the profit of the last unit production and can be observed in its accounting. Then Ding et al. [26] considered a case that each player makes adjustment strategy according to the delayed marginal profit, the one that averages his own previous marginal profits with different weights. Besides the information of marginal profit, in this work we will consider another fact that the market price history is usually a common knowledge to every player. In the market, there may be players who simply make decision according to an expected price, for example, an average from several previous periods.

In this work, we reconsider the duopoly model in Ding et al. [26]. But different from [26] that only considers homogeneous players and only discusses the time delay structure built for the marginal profit, this paper revises the model and assumes that in the duopoly market the two players are heterogeneous and they consider different time delay structures. One player is still assumed to set delay on the marginal profit and thus adjusts strategy by weighting his marginal profits in the previous periods, as done in [26]. But another player is assumed to make decision according to an expected price that averages the price history with proper weights. For this kind of duopoly game with different delay structures, this paper establishes the dynamical system and discusses its dynamic properties form theoretical and numerical approaches.

#### 2. The Model

##### 2.1. Two Different Delay Structures

We assume that there are two firms producing homogeneous goods for sale and the total supply determines the market price through an inverse demand function . If denotes the cost function of firm , then the profit of firm will be given by

Suppose that firm 1 adjusts its output with the same time delay structure as discussed in [26]. That is, firm 1 sets delay on the marginal profit , and by weighting the marginal profit history firm 1 updates its strategy in period according to the following equation (see [26]):where is a weight coefficient to the delayed period , , and is a positive constant representing the relative adjustment rate. As done in [26], we also consider a delay structure with one step: , then (3) can be written aswhere is the weight coefficient assigned to the nondelayed period and hence is the one to the delayed period .

Next, we consider firm 2 that is supposed to have a time delay structure built for the price history. In a real market, the market price is usually a common knowledge to all players so that the information of the price history may be perceived by all of them. In our model, we suppose that in period firm 2 makes an expectation for the market price, which averages the above price history with different weight coefficients and hence takes its form aswhere is the weight coefficient and . For player 2, we also consider the case of one step delay; that is,where is the weight to the nondelayed period and is associated with the delayed period . From such a time delay structure, firm 2 may expect a profit for its output choice :

It is supposed that in period firm 2 will make its decision in accordance with a best reply to the expected profit . That is, to maximize its expected profit firm 2 will choose its output strategy according to the following formula:

##### 2.2. The Dynamical System

For a duopoly game with two homogeneous players, Section 2.1 has built two different time delay structures and obtained (4) and (9), which describe two-dimensional and discrete dynamics:

To give a specific form for dynamics (10), in our model we also assume linear inverse demand function [5, 9, 10, 13, 14, 18, 23, 24, 26, 32–34]:and a quadratic form of cost function [10, 14, 24, 26]:where , , and are all positive. Then the profit of firm is given byfrom which we obtain the marginal profit of firm 1 in period :And from firm ’s expected price in (7), the market price expression in (11), and the cost function in (12), we obtain firm ’s expected profit as follows:which gives a maximization solution

Taking (14) and (16) into (10), we obtain a dynamical system with one step delay:

To study dynamics (17) conveniently, we write for and for ; then we rewrite (17) as a four-dimensional system as follows:

#### 3. Stability of the Equilibriums

Letting () in system (18), we get two nonnegative equilibrium points: where , and . is a boundary point and is an interior equilibrium.

At an equilibrium point , the Jacobian matrix of (18) takes its form aswhere

We know that the local stability of an equilibrium point is determined by the eigenvalues of the above matrix. That is, an equilibrium will be stable if the inequality holds for every eigenvalue of the Jacobian matrix ; it will be unstable if there exists an eigenvalue such that . And in a discrete system, there maybe exist a critical case that holds for each eigenvalue and holds for at least one eigenvalue . In a critical case, the system may be stable or unstable, and it may be needed to check the high order items to discuss the system stability.

Taking the expression of the boundary point into (20), we have that the Jacobian matrix at is simply given by which has four eigenvalues: Since holds evidently, we conclude that the boundary equilibrium is unstable.

Next we study the local stability of the interior equilibrium . Notice that, at the interior point , the item must be zero. Then from (20), we have of which the characteristic polynomial , where

The local stability of is determined by the roots of : if all the roots lie inside the unit disk (i.e., holds for ever root ), then will be asymptotically locally stable. About a polynomial , the following three statements give the equivalent conditions for all the roots lie inside the unit disk (Schur-Cohn Criterion, see, e.g., [35]):(i)(ii)(iii)The determinants of the matrices and the matrices are all positive, where

In our model, and ; then is obviously positive and by calculating we get which tells that holds also. So we conclude that in our model the equilibrium will be asymptotically stable if and . These conditions can be reduced to the following inequality systems:(S1)(S2)

If the inequalities in (S1) and (S2) are all satisfied, then the equilibrium will be locally asymptotically stable in the evolution of (18). If these conditions fail to hold, then the system may lose stability and even exhibit much complicated dynamical behaviors, which will be shown in the following section of numerical simulation.

#### 4. Numerical Simulation

In this section, the constants , , , and are fixed and the different influence of the two delay weights and on the system stability are simulated. The complicated dynamical behaviors are shown when the system loses its stability. Simply, in the next numerical simulations, the constants , , , and are fixed as , , , and .

##### 4.1. The Influence of the Delay Weight

In Figure 1, for different and the same with respect to the adjustment speed , the delay weight (by player 1 for the marginal profit) is fixed as and the bifurcation diagrams are plotted for different (the delay weight by player 2 for the market price): . We find that the system will be stable when the value of the adjustment speed stays at a low level, while when increases, the system will lose stability through bifurcations. Comparing the three diagrams in Figure 1, there is little difference in the values of for the system to lose stability.