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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

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 [

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 [

The literatures [

This paper mainly discusses the dynamic and repeated games among different competitors by considering the complex influence of parameters (

The paper is organized as follows. In Section

If the strength difference of the team firms is too large, the market will be easy to fall into chaos. Chaos resulting from

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.

Team firms’ price must be kept within a certain range to keep the market stable.

We consider a Bertrand Triopoly game in which the price and the demand of firm

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:

Let

According to system (

Boundary equilibria

Values of parameters in (

In order to analyze the stability of the preceding equilibrium points, the Jacobian matrix for discrete dynamic system (

At 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.

As for

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

In order to show the three-dimensional stable regions, we set the parameters

The stable region of the Nash equilibrium point,

In the stable region, the final prices of the three oligopolists will stay stable at

The economic meaning of the stable region is that if the value of

In order to have a further investigation about the impact of the parameters on the bifurcation scenarios, parameter basin plots are displayed. Parameter basin plots are powerful tools in the numerical analysis [

In the 2D-bifurcation phase portrait, bifurcation scenarios and route to chaos can be displayed more clearly. We choose the same values above in (

In Figure

In the 2D-bifurcation phase portrait, the system exhibits a sequence of flip bifurcations to chaos (which means the market will fall into chaos) then to divergence at last (which means the players will be out of the market).

In Figure

As seen from Figure

The corresponding double largest Lyapunov exponent (DLLE) with

Double largest Lyapunov exponent with

If

Bifurcation diagram and the largest Lyapunov exponent with

According to Figure

The chaotic attractor of system with

If

Bifurcation diagram and the largest Lyapunov exponent with

Figure

The phase space diagrams of system (

Through the above analysis in Section

Figures

Profits bifurcation diagram with

Average profits with

(a) Average profits of firm 1. (b) Average profits of firm 2. (c) Average profits of firm 3.

In Figure

In Figure

We can conclude that higher adjustment speed is not good for players as for average profits. The team members must control their price adjustment speed in the stable region to ensure the stability of the system to maximize their profits.

In order to analyze the effects of parameter

The stable region of the Nash equilibrium point,

The stable region of the Nash equilibrium point,

From an economic point of view, the firm in the cooperative team who has a more assigning weight should pay more attention to control the speed of price adjustment, because their adjustment interval is smaller than the other’s. That means if the firm in the team is stronger than the other firm in the cooperative team, their range of price adjustment speed must be smaller in order to maintain the market stable. As for the third firm who has adaptive expectations, their reluctant parameter

In Figures

Next, we will discuss the effects of assigning weight

The effects of assigning weights

Effects of assigning weight

Effects of assigning weight

In Figure

In Figure

We can get the following results from Figures

If the strengths of the two firms in the cooperative team have a big gap, the market will experience cyclical shock and fall into chaos in the competition.

In the stable interval of

Chaos resulting from

The prices of the three oligopolists with different

If

When

Effects of assigning weight

Effects of assigning weight

In order to investigate the impact of price adjustment speed on the attraction domain, we introduce basins of attraction which are the sets of initial prices.

By fixing the system parameters as mentioned above, let

Basin of attraction when

Basin of attraction when

Basin of attraction when

The attraction domain is the set of initial prices where the same attractor will emerge after iteration if the initial prices are taken from the attraction domain. If the attractor is one equilibrium point, from an economic point of view, the corresponding attraction domain is a safe region. That means if the initial prices of two sides is in the safe region, the system will remain stable after iteration. If the initial price is in the escape area, the system will fall into divergence at last.

In Figure

Figures

(a) Difference (

In Figure

In Figure

In Figure

In Figure

From the comparison of Figures

In this paper, we have studied a Bertrand Triopoly model with team-game. The existence and local stability of the Nash equilibrium are analyzed. The stable regions show that players should control their price adjustment speed to maintain the market stable. If one firm is stronger than the other in the cooperative team, the range of their price adjustment speed is smaller.

With the increase of price modification speed, the system will fall into chaos via period-doubling bifurcations and Neimark-Sacker bifurcations. Chaos resulting from adjustment speed is harmful to all the players. The effects of assigning weight

Using basins of attraction, we have found that the attraction domains become smaller with increase of price modification speed, and, in order to maintain market stable, two team firms’ prices must be kept within a certain range.

The authors declare that they have no competing financial interests.

The authors would like to thank the reviewers for their careful reading and for providing some pertinent suggestions. The research was supported by the National Natural Science Foundation of China (no. 61273231) and Doctoral Fund of Ministry of Education of China (no. 20130032110073).