Original articles
Nonlinear analysis and chaos control of the complex dynamics of multi-market Cournot game with bounded rationality

https://doi.org/10.1016/j.matcom.2019.01.004Get rights and content

Abstract

In this work, a dynamic multi-market Cournot model is introduced based on a multi-markets’ specific inverse demand function. Puu’s incomplete information approach, as a realistic method, is used to contract the corresponding dynamical model under this function. Therefore, some stability analysis is used by the model to detect the stability and instability conditions of the system’s Nash equilibrium. Based on the analysis, some dynamic phenomena such as bifurcation and chaos are found. Numerical simulations and the Maximum Lyapunov exponent are used to provide experimental evidence for the complicated behaviors of the system evolution. It is observed that the equilibrium points of the system can loose stability via flip bifurcation or Neimark–Sacher bifurcation and time-delayed feedback control is used to stabilize the chaotic behaviors of the system.

Introduction

The dynamical behaviors of oligopoly games are complex because every oligopolistic producer in each period must consider not only its own decision but also the reactions of all other competitors. Cournot competition is an economic model used to describe the competition between some companies based on the amount of output they will produce [12]. Thus, a generalization of this game to the case of two markets is done. It is shown that the resulting dynamics is quite rich. In the classic model, each participant uses a naïve expectation to suppose that the opponents’ output keeps the same level as the previous period’s and adopts an output strategy to maximize the expected profit. Many researchers have analyzed the system stability and the complex phenomena in Cournot oligopoly games with this kind of expectation [1], [6], [7], [18], [20], [21], [22], [25]. According to an early work by Bischi and Naimzada [10], players behave adaptively in the dynamical Cournot game, following a bounded rationality adjustment process where each producer updates its production by the marginal profit maximization method based on complete knowledge of the market. In recent years, a great amount of works has been done on the dynamical Cournot games with homogeneous or heterogeneous expectations. Bounded rationality in the marginal profit method is assumed to that all producers in the models consider homogeneous expectation [4], [5], [8], [10]. The models with heterogeneous expectations (naïve, boundedly rational or adaptive) have been discussed in many other works [2], [3], [15], [16], [17], [27], [29].

In the model of dynamical Cournot game, output is a key variable and each player can adjust output according to marginal profit maximization; thus, it is based on an incomplete assumption that all players could provide sufficient quantity of products in the market. Investment plays the most important role in multi-market. As a strategic behavior in these economic activities, investment accumulation plays a significant role in terms of achieving a good production level. Only when the investment comes up to a certain level can a firm provide as much goods as the market demands. Therefore, during the competitive period of the multi-market, the competition among producers lies mainly in their investment strategy distribution in the multi-market. To obtain a competitive market share and get superiority over opponents, producers must consider their investment strategies distribution in successive periods.

In this study, the paper starts with the monopoly in two markets using Puu’s incomplete information dynamics. It is more realistic than the standard bounded rationality since profits may not be known as functions but only as quantities. A dynamic multi-market Cournot model is introduced based on a specific inverse demand function. The main purpose of our work is to formulate a novel model, which puts decision as a substitute for output adjustment into the dynamical Cournot game. In the model, all producers are also assumed to have bounded rationality and make their decisions according to the marginal profit in the previous period. That is to say, each firm will increase its investment if it perceives a positive marginal profit and decrease its investment if the perceived marginal profit is negative. During a local adjustment process, this novel dynamical Cournot game aims to develop the equilibrium or demonstrate complex dynamic behaviors of the multi-market.

In this paper, time-delayed feedback control is used to stabilize the chaotic behaviors of the system. It is good to consider other chaos control methods in the future. For example, BP Neural Network Control of Chaotic System, RBF Neural Network Control of Chaotic System, Fuzzy sliding mode control for hyperchaotic systems, Fuzzy adaptive control for uncertain chaotic systems, Hybrid genetic neural network control of chaotic systems, Fuzzy Neural Network Adaptive Control for Uncertain Chaotic Systems, Identification and Control of Uncertain Chaotic Systems Based on Dynamic Neural Networks, fuzzy control of uncertain chaotic systems based on linear matrix inequalities and so on [24].

This article is organized as follows. In Section 2, we model the dynamical game played by players with bounded rationality. In Section 3, we discuss the existence and local stability of the equilibrium points for the system. In Section 4, we show the dynamic features of this system with numerical simulations, including bifurcation diagram, phase portrait and sensitive dependence on initial conditions. In Section 5, time-delayed feedback control is used to stabilize the chaotic behaviors of the system.

Section snippets

The multi-market Cournot model

In this model, the products are near substitutes but different in their quality levels. So, firms can charge different prices for different markets. Suppose there are i competing firms (i2) in two markets A and B. For the price in the market, we consider Bowley [11] has introduced the demand function pi=ψiqiξjiqiwhere 0ξ1. We also suppose that the production cost function of each firm takes a specific inverse demand function [9], [13], [23] used this form in their work. Billand et al. [9]

Analysis of the equilibrium points and stability

Each firm chooses the quantity according to the principle of their benefit maximization in the game of firm’s investment in the markets A and B. Eligible equilibrium points can be obtained by the Eq. (11) [14] as follows: E0=(0,0,0,0), E1=(0,0,G2a2,0),E2=(0,0,0,L2b2),E3=(0,K2b1,0,0),E4=(H2a1,0,0,0),E5=(0,0,G2a2,L2b2),E6=(0,K2b1,G2a2,0),E7=(H2a1,0,0,L2b2),E8=(H2a1,K2b1,0,0),E9=(0,q2,0,q4),E10=(q1,0,q3,0),E11=(0,q2,G2a2,q4),E12=(H2a1,q2,0,q4),E13=(q1,0,q3,L2b2),E14=(q1,K

Numerical simulation

In this section, the numerical simulations show how the system evolves under different levels of parameters, especially with the adjustment speed θ1 and θ2. In all the numerical simulations, the other parameters are fixed: ψ1=4, ψ2=5, β1=3, β2=5, a1=0.4, a2=0.5, b1=0.4, b2=0.6, c1=0.3, c2=0.4, d1=0.4 and d2=0.6.

Fig. 1(a)–(d) show the bifurcation diagram and the Maximum Lyapunov exponents with the adjustment speed θ1, while the other parameters are constant and have taken the value θ2=0.6. Fig. 1

Chaos control

From the numerical simulations, the adjustment rate and the weight coefficient have great influence in the stability of system (11). If the model parameters fail to locate into the stable region required, the behaviors of the dynamics will be much complicated. In a real economic system, chaos is not desirable and will not be expected, and it is needed to be avoided or controlled so that the dynamic system would work better. In this section, we use the time-delayed feedback control [28] to

Conclusion

In this study, we have taken into consideration a dynamic multi-market Cournot game based on line inverse demand cost function which is built with the same assumption of bounded rationality. We discuss the stability of each equilibrium solution by using the nonlinear system. Numerical simulations and the Maximum Lyapunov exponent are used to provide experimental evidence for the complicated evolution behaviors of the system. It proves that the chaotic behavior of the system can be controlled.

Acknowledgments

This work is supported by the National Nature Science Foundation of China (No. 71471076 and 71001028), and Jiangsu University of Technology talent introduction project (KYY18539).

References (29)

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