Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 960380, 12 pages

http://dx.doi.org/10.1155/2015/960380

## Study and Simulation on Discrete Dynamics of Bertrand Triopoly Team-Game

^{1}Group of Nonlinear Dynamics and Chaos, College of Management and Economics, Tianjin University, Tianjin 300072, China^{2}College of Science, Tianjin University of Science and Technology, Tianjin 300457, China

Received 4 December 2014; Accepted 9 February 2015

Academic Editor: Domenico Mundo

Copyright © 2015 Lijian Sun and Junhai Ma. 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

A Bertrand Triopoly team-game model is considered in which two firms with bounded rational expectations make up a cooperative team and allocate common profits proportionate to their marketing strength. The existence and three-dimensional stable regions of the fixed points are investigated. Complex effects of on bifurcation scenarios and profits are displayed by parameter basin plots and average profits charts. Impact of assigning weight on stable regions, 2D-bifurcation phase portraits, and the average profits is investigated. We find and can cause chaos; chaos resulting from adjustment speed is harmful to all the players as for profits, while chaos resulting from is conducive to firm 3. Basins of attraction are investigated and we find that the attraction domain will become smaller with increase of price modification speed.

#### 1. Introduction

An oligopoly is a market structure dominated by a few firms. The market is known as Bertrand oligopolists if firms choose price as their strategic variable to maximize their profits in an uncertain market demand. It is called a Triopoly if there are three firms in the oligopoly.

Many markets in the world have always been with Triopoly market structure. Take a telecommunication market and a petroleum market in China as examples. All the oligopolists may choose price as their competitive strategy. Price decisions by the firms in the Triopoly need to take into account the likely responses of the other players. We assume that two firms with bounded rational expectations make up a cooperative team; the third firm has adaptive expectations.

Since Rand first proposed that chaos may occur in a system of the duopoly market with the oligopolists’ reaction functions, the literatures on dynamics of Bertrand game model are very rich in economics. Bester [1] studied the stability of price competition in a horizontally differentiated duopoly. Zhang et al. [2] investigated the complexity of price competition in a Bertrand model with bounded rationality and found that if the speed of adjustment of bounded rational player increased, the stability of the Nash equilibrium point may change and bifurcation and chaos would occur. Peng et al. [3] considered the competition of a Bertrand model with delayed bounded rationality and pointed out that lagged structure may expand the stable region or change the system’s chaotic state into stable state. Ma and Wu [4] analyzed the influence of delayed decision on the stability of a Triopoly price game model and found that the number of time delay decision makers cannot improve the system stability. Fanti et al. [5] investigated the dynamics of a Bertrand duopoly with differentiated products and found that if the product differentiation between the firms increased, the interior fixed point would be unstable and attractors would have a complex structure. Zhao et al. [6] considered pricing decisions for two substitutable products in a supply chain and analyzed the effects of manufacturers’ different competitive strategies on the optimal pricing decisions. Giri and Sharma [7] studied a two-echelon supply chain with advertising cost dependent demand. Results showed that it was always beneficial for the manufacturer to adopt different wholesale pricing strategy for the retailers.

In a real market, the coexistence of competition and cooperation makes the team composed of similar companies in order to get the maximum profits. The Renault-Nissan Alliance is one of the most successful cases. Both sides share the product design and the production platform and thereby reduce their production costs and improve asset utilization. In China, in order to deal with the fierce competition, Chery Automobile and GAC build the first domestic strategic alliance in auto industry. They both hope to reduce costs by cooperation and gain more profits.

In recent years, some scholars have studied the team competition model which has the features of bounded rationality. Ahmed and Hegazi [8] studied the vendors in the same team to make production strategy in accordance with the outputs of the team as a whole. Elettreby and Hassan [9] proposed two different versions of the multiteam model where a team of two firms competes with another team and studied the equilibrium solutions, the conditions of their local asymptotic stability. Ding et al. [10] proposed a Cournot game model and showed that chaos may occur with changing of the profit weighting coefficients and the adjustment rate of boundedly rational players in the cooperative team.

The literatures [8–10] showed the complex dynamics of the team competition model, but they did not show influence of chaos on profits of all the players. What is the impact of chaos on the profits of the players?

This paper mainly discusses the dynamic and repeated games among different competitors by considering the complex influence of parameters ( and ) on the game process and the influences which assigning weight and have on the profits. Theoretical analysis and numerical simulations of the system are made in detail.

The paper is organized as follows. In Section 2, a Bertrand Triopoly model with team-game is established. In Section 3, the existence and local stability of equilibrium points about parameters are discussed. Complex influence of on bifurcation scenarios and profits are investigated in Section 4. The effects of assigning weight on stable regions, profits, and prices are shown in Section 5 using parameter basin plots [11, 12]. Basins of attraction [13] are given in Section 6. At last, three conclusions are made.(1)If the strength difference of the team firms is too large, the market will be easy to fall into chaos. Chaos resulting from is conducive to firm 3.(2)With increase of price modification speed, the system will lose stability. The firm in the cooperative team who has a more assigning weight should pay more attention to control the speed of price adjustment. Chaos resulting from adjustment speed is harmful to all the players.(3)Team firms’ price must be kept within a certain range to keep the market stable.

#### 2. The Bertrand Triopoly Game Model

We consider a Bertrand Triopoly game in which the price and the demand of firm ’s product are denoted by and , respectively, . Firm 1 and firm 2 in the Triopoly make up a cooperative team and have the same cost of production , as they share the same production technology. They allocate common interests proportionate to the assigning weights and which reflect the strength difference between the two firms. Based on the classic Bertrand model, the demand functions for the three firms are as follows:in which are all positive constants and denote the mutual product substitution rates among three different firms. Assume that all the three firms have linear cost functions Hence, the profits functions of firms in period are given byin which is the profits of firm and is the profits of the cooperative team.

So the firms can get their maximum profit by the following marginal profits functions:

While, in practice, they may do not know other firm’s price in the next-period in advance, they cannot calculate their optimal prices by the marginal profits functions above. We consider the two firms of the cooperative team bounded rational players and their next-period price decision is on the basis of the local estimate to their marginal profits in current period. This means that if the marginal profits of the current period are positive, the firm will raise their prices in the next period; otherwise, they will reduce their prices. So firm 1 and firm 2 adopt their strategies in the following form:where and denote the first two players’ price adjustment speed, respectively. We assume that the third firm has adaptive expectations; that is, they take current period’s price and the naïve expectations into consideration when they decide the price . This means that the third firm considers the expectation about the price before and foresees other firm’s price decision. The proportion is given by and that regulates how reluctant the third player is to change its price in the th period.

Let ; the naïve expectations of the third firm isSo the dynamic equation of third player is given byHence, the dynamic Bertrand Triopoly game in this case is formed from combining (4), (5), (7), and (9). Then, the dynamic system with team-game is described by

#### 3. Equilibrium Points and Local Stability about Price Adjustment Speed

According to system (10), let ; then three boundary equilibriaand unique Nash equilibrium point can be obtained.

Boundary equilibria mean that at least one firm will be out of the market, while the Nash equilibrium point means that all the firms can survive ( must be positive).

Values of parameters in (10) (such as ) may affect the stability of the equilibrium points. So it is meaningful to study stability of the preceding equilibrium points about .

In order to analyze the stability of the preceding equilibrium points, the Jacobian matrix for discrete dynamic system (10) is found as follows:in which

*Remark 1. * is an unstable equilibrium point when .

*Proof. *At point ,is an eigenvalue corresponding to ; then if , then , and is an unstable equilibrium point.

From an economic point of view, if there is a great strength difference between firm 1 and firm 2, the market will be unstable.

*Remark 2. * and are unstable equilibrium points.

*Proof. *As for , is the eigenvalue corresponding to . At point , the price of the product provided by firm 1 is 0, so firm 1 is out of the market; if is a stable equilibrium point, then firm 2 must survive in the market, then , and then , so is an unstable equilibrium point. In the same way we can prove that is an unstable equilibrium point.

From an economic point of view, if one firm in the cooperative team withdraws from the market, the market will be unstable.

While stability of the Nash equilibrium point is affected by parameters , according to Routh-Hurwitz condition, the necessary and sufficient condition of asymptotic stability at is that all the eigenvalues are inside the unit circle in complex plane. So it must satisfy the following conditions:where is the characteristic polynomial at .

In order to show the three-dimensional stable regions, we set the parametersAccording to the parameters above,Its Jacobian matrix isThe characteristic equation of Jacobian matrix (20) isin which As can be shown in Figure 1, a stable region in the space of is determined by the above inequalities. It means that if the value of is in the stable region, is stable, and if the value of is out of the stable region, is unstable.