Complexity

Volume 2017 (2017), Article ID 5434680, 12 pages

https://doi.org/10.1155/2017/5434680

## Study on Triopoly Dynamic Game Model Based on Different Demand Forecast Methods in the Market

^{1}College of Management and Economics, Tianjin University, Tianjin 300072, China^{2}College of Science, Tianjin University of Science and Technology, Tianjin 300457, China^{3}School of Economics, Beijing Wuzi University, Beijing 101149, China

Correspondence should be addressed to Junhai Ma and Xueli Zhan

Received 11 February 2017; Revised 22 March 2017; Accepted 26 March 2017; Published 6 July 2017

Academic Editor: Christos Volos

Copyright © 2017 Junhai Ma 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

The impact of inaccurate demand beliefs on dynamics of a Triopoly game is studied. We suppose that all the players make their own estimations on possible demand with errors. A dynamic Triopoly game with such demand belief is set up. Based on this model, existence and local stable region of the equilibriums are investigated by 3D stable regions of Nash equilibrium point. The complex dynamics, such as bifurcation scenarios and route to chaos, are displayed in 2D bifurcation diagrams, in which and are negatively related to each other. Basins of attraction are investigated and we found that the attraction domain becomes smaller with the increase in price modification speed, which indicates that all the players’ output must be kept within a certain range so as to keep the system stable. Feedback control method is used to keep the system at an equilibrium state.

#### 1. Introduction

A Triopoly is a market structure dominated by three firms in the market. The market is known as Cournot game if firms choose quantities as their strategic variables to maximize their profits in an uncertain demand environment.

In conventional market games, players are supposed to have common accurate demand functions of market. The dynamics of such system with this assumption have been intensively investigated in literature [1–8].

Assuming cost function to be twice differentiable increasing, Elabbasy et al. [1] analyzed the dynamics of oligopoly games with three types of players: bounded rational, naive, and adaptive.

Ma and Liu [2] studied a generalized nonlinear Fokker-Planck diffusion equation with external force and absorption. They obtained the corresponding exact solution expressed by -exponential function and the solutions can have a compact behavior or a long tailed behavior.

Yassen and Agiza [3] studied a repeated Cournot game model with delayed bounded rationality in the duopoly market and demonstrated that the lagged structure is helpful to expand the stable region of the system via numerical simulations.

Ma and Ren [4] focused on the influence of parameters on the macroeconomics IS-LM model and improved the analysis capabilities of the traditional economic models to suit the actual macroeconomic environment. They found that the system order has an important influence on the running state of the system.

Tramontana and Elsadany [5] discussed a triopolistic market with heterogeneous firms when the demand function is isoelastic. He found that double routes lead to chaos, via period-doubling and Neimark–Sacker bifurcations.

Ma and Ji [6] built a Triopoly outputs game model in electric power market. They obtained that the Triopoly model is a chaotic system and it is better than the duopoly model in applications.

Ma and Wu [7] studied the complexity of a Triopoly price game model and influence of delayed decisions on the stability. All those approaches assume that there is one uniform and accurate market demand function available and shared by all player.

Ma and Pu [8] researched the Cournot-Bertrand duopoly model with the application of nonlinear dynamics theory. They analyzed the stability of the fixed points and gave the bifurcation diagram and Lyapunov exponent spectrum along with the corresponding chaotic attractor. The research results show that either the change of output modification speed or the change of price modification speed will lead the market to the chaotic state which is disadvantageous for both of the firms.

All those literatures assume that there is one accurate market demand function, which is shared by all players. In practice, demand functions may be influenced by lots of different factors and every player has to estimate his own market demand function on the basis of past experience.

Because it is impossible for the firms to get the whole information, they cannot know the accurate demand function, so all the players have to make an estimate for demand function. Compared to real market demand function, it is inevitable for all the players to make demand evaluation biases. Thus, it is important to study how dynamics of the Triopoly game will be influenced by those evaluation biases, in terms of equilibrium points, local stability, and system performance.

A few works have been done to investigate the system dynamics, equilibrium offset with inaccurate demand beliefs. Bischi et al. [9] studied a model of a quantity-setting duopoly market with misspecified demand, the global dynamics of this game was investigated, and the number of steady states and their welfare properties were characterized. The impact of misspecified demand on the steady state was also studied. However, the asymmetric system with heterogeneous players’ behavior has not been considered.

Wang and Ma [10] considered a Cournot-Bertrand mixed duopoly game model with limited information about the market and opponent. They studied the local stability of the game model at the Nash equilibrium point and discussed the influences of the parameters on the system’s performance.

Bischi et al. [11] considered a repeated oligopoly game in single product Cournot oligopolies and proposed a method to learn demand function in a repeated oligopoly game via a closed loop feedback of real market price, which adjusts the evaluated demand function.

Guo and Ma [12] built a collecting price game model for a close-loop supply chain system with a manufacturer and a retailer who have different rationalities. They analyzed the influences of parameters on complex dynamic phenomena, such as the bifurcation, chaos, and continuous power spectrum.

Qiu et al. [13] studied the impact of uncertain demand on dynamic output-setting market games. A dynamic game with uncertain demand for two heterogeneous players was built. Based on this model, the impact of uncertain demand on the game’s complex dynamics was investigated. Sun and Ma [14] constructed the three-oligopoly game model and investigated the existence of the fixed points. The 3D stable regions were given. The complex dynamic behavior of the game model is studied and the chaos was successfully controlled.

In this paper, we analyze the complex dynamics of a Triopoly model with heterogeneous players and demand evaluation bias, focusing on the following perspectives:(1)Impact of demand estimate bias on equilibrium, stable region, and profits.(2)Impact of adjustment strategy on basins of attraction.The paper is organized as follows. In Section 2, a Triopoly game model with inaccurate demand belief is established. In Section 3, the existence and local stability of equilibrium points are discussed. The effects of inaccurate demand on stable region, profit, and equilibriums are shown in Section 4. Dynamical behaviors of the game are investigated by numerical simulations using 2D bifurcation diagrams [15] in Section 5. Basins of attraction [16] of the model are given in Section 7. In Section 8, conclusions are drawn from our analysis.

#### 2. The Cournot Triopoly Game Model

We consider a Cournot Triopoly game in which the price and the demand of firm ’s product are denoted by and , , and the demand functions for the three firms are as follows:in which and are both positive constants. Assume that all the three firms have nonlinear cost function considering that if exceeds a certain level, the cost will increase quickly and the cost function of the th firm has a quadratic form [3]:

While in practice, not all the firms can get the whole information, they may do not know demand function (1), so all the players have to make an estimate for demand function.

For each player, we assume that the actual demand function held by player can be denoted by multiplying the demand function (1) with an error coefficient . And it has the following form:which is called its subjective demand function.

The error coefficient which is between means the imperfection degree of player about the market.

If , it indicates that the evaluated demand function is just the true demand function. If , it indicates that the demand is underestimated by player, while if , it indicates the case where the demand is overestimated.

So the firms can get their maximum profits according to the following profit functions:in which is the profit of firm . Hence, the marginal profit functions of firms in period are given by

while in practice, firms usually cannot get the whole information. For example, they cannot know other firm’s price in the next period in advance, for which they cannot compute the price by the marginal profit functions above. In this paper, we consider all the firms as bounded rational players and their next-period price decisions are made on the basis of the local estimate to their marginal profit in current period. So the players make their strategies as the following form:

The equation means that if the marginal profit of the current period is positive, the firm will raise its price the next period; otherwise, it will reduce it. So the dynamical Triopoly system can be described as where denote the players’ adjustment speeds, respectively.

#### 3. Equilibrium Points and Local Stability

##### 3.1. Equilibrium Points

According to system (7), let ; then eight equilibrium points can be obtained:and the Nash equilibrium point can be obtained, whereIt can be seen from above that is independent of the adjustment factors. From an economic point of view, that means the value of the system local stability point in this dynamic game is independent of the players’ adjustment speed, but just determined by the characteristics of the system. We can find from (11) that if of player increases and of other players remains the same, will increase.

##### 3.2. Nash Points in Error-Free System-Benchmark

If the players have perfect knowledge, their subjective demand functions totally coincide with the real ones. Setting in (11), we can get Note that the denominator of , , and is the same, the output depends on the cost of the players, and the greater the cost, the lower the yield. These results match the results in [17].

##### 3.3. Local Stability of System Equilibriums

In order to analyze the stability of the preceding equilibrium points, the Jacobian matrix for discrete dynamic system (7) is found as follows:in whichAccording to Routh-Hurwitz condition, the necessary and sufficient conditions for equilibrium points to be asymptotically stable are that all roots of the characteristic equations have magnitudes of eigenvalues less than 1.

*Remark 1. * are unstable equilibrium points.

As for , is one eigenvalue which corresponds to , so is an unstable equilibrium point.

As for , is one eigenvalue which corresponds to , so is an unstable equilibrium point. In the same way we can prove that and are unstable equilibrium points.

*Remark 2. *, , and are unstable equilibrium points.

As for , is one eigenvalue which corresponds to ; set , , and then , so .

So is an unstable equilibrium point. In the same way we can prove that and are unstable equilibrium points.

From an economic point of view, in the stable state of this dynamic game, no player is forced to withdraw from the market.

As for , the necessary and sufficient condition of asymptotic stability is that all the eigenvalues are inside the unit circle in complex plane. So a stable system must satisfy the following conditions: where is the characteristic polynomial at .

For convenience, we set the parameters as follows: and the initial values are chosen as .

According to the parameters above, Its Jacobian matrix isThe characteristic equation of Jacobian matrix (18) isin which As what can be shown in Figure 1, a stable region in the space of is determined by the above inequalities. In the stable region, the final prices of the three oligarchs will stay stable at after a number of games. From Figure 1, we can see that the market is stable when , but the market may be unstable when increases. The economic meaning of the stable region is that if is in the stable region, prices of three firms will achieve the Nash equilibrium at last.