Analysis of a duopoly game with delayed bounded rationality

doi:10.1016/S0096-3003(02)00143-1

Applied Mathematics and Computation

Volume 138, Issues 2–3, 20 June 2003, Pages 387-402

https://doi.org/10.1016/S0096-3003(02)00143-1 Get rights and content

Abstract

A dynamic of Cournot duopoly game is analyzed, where players use different production methods and choose their quantities with bounded rationality. A dynamic of nonlinear Cournot duopoly game is analyzed, where players choose quantities with delayed bounded rationality and similar methods of production. The equilibria of the corresponding discrete dynamical systems are investigated. The stability conditions of Nash equilibrium under a local adjustment process are studied. The stability of Nash equilibrium, as some parameters of the model are varied, gives rise to complex dynamics such as cycles of higher order and chaos. We show that firms using delayed bounded rationality have a higher chance of reaching Nash equilibrium. Numerical simulations are used to show bifurcations diagrams, stability regions and chaos.

Introduction

Recently the complex dynamics of bounded rationality duopoly game of Bowley’s model have been studied [1]. In the Bowley’s model [2] the demand function has the linear form $p=f(Q)=a−bQ$ where a and b are the demand parameters. The constant a is the maximum price in the market and Q is the total quantity in the market. The dynamics of the duopoly model when firms use similar production methods was investigated in [1]. In this work we study the complex dynamics of duopoly Bowley’s model when firms use different production method and the cost function is proposed in the nonlinear form $C_{i} (q_{i})=c_{i} q^{2}_{i}$ where c_i, i=1,2 is a positive shift parameter of the cost function. Generally the complex dynamics of duoploy models was studied with different assumptions to the demand function (see for examples [3], [4], [5], [6]). In this paper, we study duopoly game which describes a market with two-players producing homogeneous goods, updates their production strategies in order to maximize their profits. Each player think with bounded rationality, adjust his output according to the expected marginal profit, therefore the decision of each player depends on local information about his output. In this Cournot game each player tries to maximize his profit according to local informations of his input. The plan of the paper is as follows. In Section 2, the description of the duopoly Bowley’s model with bounded rationality. The existence and local stability of the equilibrium points are studied. The stability region of Nash equilibrium is determined in the plane of speeds adjustment (v₁v₂-plane) and also complex behavior such as cycles of higher order and chaos is studied. Numerical simulation is presented to show such behaviour. In Section 3, duopoly Bowley’s model with delayed bounded rationality is studied when firms use similar production methods. The existence and local stability of the boundary equilibria and Nash equilibrium are investigated. The stability region of Nash equilibrium is determined in the plane of speeds adjustment (v₁v₂-plane). Numerical simulations are presented. Section 4 is the conclusion.

Section snippets

Duopoly model

We consider two firms, labelled by i=1,2 producing the same good for sale in the market. Production decisions of both firms occur at discrete time periods t=0,1,2… Let q_i(t) represent the output of ith firm during period t, with a production cost function C_i(q_i). The price prevailing in period t is determined by the total supply Q(t)=q₁(t)+q₂(t) through a demand function $p=f(Q)$ In this model the demand function is assumed linear [7], which has the form $f(Q)=a−bQ$ where a and b are positive

Duopoly Bowley’s model with delayed bounded rationality

Discrete time dynamical systems defined by so-called delay equations arise in economic models [11]. The primary reason for the occurrence of such a lagged structure in economic models is that

1.
Decisions made by economic agents at time t depend on past observed variables by means of a prediction feedback, and
2.
the functional relationships describing the dynamics of the model may not only depend on the current state of the economy but also in a nontrivial manner on past states [12].

It is customary

Conclusion

The dynamic of duopoly Bowley’s model with bounded rationality is analyzed, where players use different production methods. Also the dynamic of nonlinear duopoly Bowley’s model with delayed bounded rationality is analyzed, where players use similar methods of production. This study shows that the stability region of Nash equilibrium of duopoly Bowley’s model with bounded rationality is decreased when players use different production methods. The stability region of Nash equilibrium of duopoly

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Analysis of a duopoly game with delayed bounded rationality

Abstract

Introduction

Section snippets

Duopoly model

Duopoly Bowley’s model with delayed bounded rationality

Conclusion

Chaos, Solitons & Fractals

Chaos, Soliton & Fractals

Chaos, Soliton & Fractals

Mathematical and Computer Simulation

Chaos, Solitons & Fractals

Oligopoly theory

Simple and complex adjustment dynamics in Cournot duopoly model

Chaos, Soliton & Fractals