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In the real business world, player sometimes would offer a limiter to their output due to capacity constraints, financial constraints, or cautious response to uncertainty in the world. In this paper, we modify a duopoly game with bounded rationality by imposing lower limiters on output. Within our model, we analyze how lower limiters have an effect on dynamics of output and give proof in theory why adding lower limiters can suppress chaos. We also explore the numbers of the equilibrium points and the distribution of conditioned equilibrium points. Stable region of the conditioned equilibrium is discussed. Numerical experiments show that the output evolution system having lower limiters becomes more robust than without them, and chaos disappears if the lower limiters are big enough. The local or global stability of the conditional equilibrium points provides a theoretical basis for the limiter control method of chaos in economic systems.

Since the French economist Cournot [

In the real business world, it is commonly observed that competitive firms would limit their production for steadiness or economies of scale. Huang [

This paper aims at the new output duopoly game by imposing lower limiters on output and focuses on the impact of limiters on dynamics and unraveling stabilizing mechanism of limiter method to reduce fluctuation.

As He and Westerhoff [

The remainder of this paper is organized as follows. Section

The output game we introduce here is based on the assumption that the two firms (players) do not have a complete knowledge of the market. Then one firm is labeled by

Hence the after-tax profit

The duopoly model with bounded rational players can be written in the form:

In order to make the solution of the output duopoly model have the economical significance, we study the nonnegative stable state solution of the model in this paper. The equilibrium solution of the dynamics system (

From (

Since

The Nash equilibrium

The study of the local stability of equilibrium points is based on the eigenvalues of the Jacobian matrix of the (

As regards the conditions for the fixed point to be stable (see [

The boundary equilibria

At the boundary fixed point

The eigenvalues of

At the boundary fixed point

The bifurcation occurring at

From the similarity between

In order to study the local stability of Nash equilibrium

The characteristic equation is

Since

The first condition is satisfied and the second condition becomes

This equation defines a region of stability in the plane of the speeds of adjustment

For the values of

Therefore, from the previously mentioned derivation, we have following theorem illustrating the local stability of equilibrium

The stable region of equilibrium

Theorem

It is noticeable that the game is based on the bounded rationality. The two firms cannot reach the Nash equilibrium at once. They may reach the equilibrium point after rounds of games. But once one player or both players adjust the production too fast and push

Similar argument applies if the parameters

We can show the stability of the Nash equilibrium point

Partial numerical simulation of the system (

Next, we assume that the

Limiting the output is economically justified in the real world. It can be explained by capacity constraints, financial constraints, breakeven, and steadiness.

In order to study the qualitative behavior of the solutions of the nonlinear map (

where

where

where

From the previously mentioned analysis, we can see that the existence of equilibriums

The distributions of conditional equilibrium points of the system (

It is very interesting that conditional equilibrium

Using the method similar to Section

In the following, we will explore the stability of conditional equilibrium points

As for conditional equilibrium

The conditional equilibrium

Because

As Section

Note that

Now, when one of the previously mentioned two inequalities (a) and (b) becomes an equation, we prove that

Using the method of reduction to absurdity can prove this conclusion. We now assume that there is one strict inequality at least for

If

Then

Hence,

The global stability of

Considering the symmetry of the system (

The stable region of conditional equilibrium

In Figures

In the plane of the speeds of adjustment

Because

Note that the trajectories of output converge to

Note that

According to the system (

By the same way, we can obtain the following.

In the plane of the speeds of adjustment

Numerical experiments are simulated to show the influence of lower limiters on the stability of Nash equilibrium, which are based on the same parameter setting as Figures

Figure

Bifurcation diagrams of the system (

Figure

The strange attractor of the system (

This paper is concerned with complex dynamics of duopoly game without and with output lower limiters. We discussed that if the behavior of producer is characterized by relatively low speeds of adjustment, the local production adjustment process without limiters converges to the unique Nash equilibrium. Complex behaviors such as cycles and chaos occur for higher values of speeds of adjustment.

Furthermore, we investigate how output lower limiters, which function identically to a recently explored chaos control method: phase space compression, and the limiter method, affect the output dynamics. The existence of Nash equilibrium becomes conditional. The distribution of the conditional equilibrium points is displayed in this paper, and their relation with lower limiter is studied. It is funny that the feasible region of conditional equilibrium points is convex and Nash equilibrium is one of the vertices of the region. We find that simple output lower limiters may (a) reduce the fluctuation of production and profit; (b) make chaos of the original duopoly game disappear; (c) help the firms to avoid the explosion of the economic system. The size of lower limiters and initial output also has relation with the stable region of conditional equilibrium points

This work was supported in part by the National Natural Science Foundation of China under Grants 71171099, 71073070, 71001028, and 70773051, by the National Social Science Foundation of China for major invitation-for-bid project under Grant 11&ZD169, and by China Postdoctoral Science Foundation under Grant 20090461080. This work was also sponsored by