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The classical Stackelberg game is extended to boundedly rational price Stackelberg game, and the dynamic duopoly game model is described in detail. By using the theory of bifurcation of dynamical systems, the existence and stability of the equilibrium points of this model are studied. And some comparisons with Bertrand game with bounded rationality are also performed. Stable region, bifurcation diagram, The Largest Lyapunov exponent, strange attractor, and sensitive dependence on initial conditions are used to show complex dynamic behavior. The results of theoretical and numerical analysis show that the stability of the price Stackelberg duopoly game with boundedly rational players is only relevant to the speed of price adjustment of the leader and not relevant to the follower’s. This is different from the classical Cournot and Bertrand duopoly game with bounded rationality. And the speed of price adjustment of the boundedly rational leader has a destabilizing effect on this model.

In the oligopolistic market, oligopoly firms compete in quantity or price. We all know that Cournot model [

Compared with quantity competition, maybe price competition is more general in a real market. One of the most classical price game models is the Bertrand model [

In the above mentioned quantity and price game models, the one-short games are static; that is, each of the players in the game determines their quantity or price at the same time. However, dynamic game or sequential game is very common in the real market. We all know that the most classical sequential game is Stackelberg game [

This paper is organized as follows. In Section

We consider an oligopoly market served by two firms producing differentiated products. The two firms compete in price. Firm 1, as the leader, chooses a price

At the second stage of the period

In this work, the two firms (leader and follower) are boundedly rational players. They have no complete information of market, and they determine the price of production with the information of local profit maximization.

The leader determines its price of production of period

The follower is also a boundedly rational player and changes its price according to its marginal profit

Substituting (

One can see that the form of this dynamical system is so different from the previous classical Cournot and Bertrand duopoly game with bounded rationality. And the product price of the leader at period

We are interested only in nonnegative trajectories; hence, the system is not defined in the origin

Easily, we can get the following:

In addition, we remember that in Zhang’s model [

In order to investigate the local stability of the equilibrium points of

The boundary equilibrium point

The Jacobian matrix at

Next, we will investigate the local stability of the Nash equilibrium point

The Jacobian matrix at

So, we can state the following summarizing result.

The Nash equilibrium

These inferences will be demonstrated in Section

In order to be more clear and intuitive in understanding the complex dynamic behavior of the discrete dynamical system (

Figure

Bifurcation diagram with respect to

The largest Lyapunov exponent versus

Figure

Time sequence diagrams under different states.

We draw the two-dimension flip bifurcation diagram in the parameters

Flip bifurcation curves in the parameters

Figure

Evolution diagram with respect to

Strange attractor is one of the main features of chaotic motion. It exhibits fractal structure. Figure

Strange attractor when

Sensitive dependence on initial conditions is another main characteristic of chaotic system. In order to demonstrate sensitive dependence on initial conditions of system (

Shows sensitive dependence on initial conditions; the two orbits of

In this paper we have proposed a price Stackelberg duopoly game model with boundedly rational players. The complex dynamical behaviors have been studied. In Nash equilibrium, the price of the leader is less than the follower’s and also less than equilibrium price of Bertrand game under the same assumption. The theoretical and numerical analysis shows the result that the stability of the dynamical system (

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors would like to thank the anonymous referees very much for their valuable comments and suggestions. This work was supported by Major Program of National Natural Science Foundation of China (71390521 and 71390523) and by National Natural Science Foundation of China (71101067, 71271103, 71301062, and 71301070).