On the Dynamics of a Discrete Fractional-Order Cournot–Bertrand Competition Duopoly Game

Ali Ahmadian,Abdulrahman Al-Khedhairi,Amr Elsonbaty,Abdelalim A Elsadany

doi:10.1155/2022/8249215

Abstract

A discreet fractional-order Cournot–Bertrand competition duopoly game is introduced based on the fractional-order difference calculus of the Caputo operator. The model is designed when players can make long memory decisions. The local stability of equilibrium points is discussed for the proposed model. Some numerical simulations explore the model’s bifurcation and chaos by employing bifurcation diagrams, phase portraits, maximal Lyapunov exponents, and time series. According to our findings, the fractional-order parameter has an effect on the game’s stability and dynamics.

Highlights

Game theory is one of the most interesting and complex topics that many researchers are interested in understanding
Game theory is concerned with predicting results for strategic games in which participants, for example, two or more firms competing on the market, have incomplete information on the intentions of others
It is known that the game theory is relevant to the study of corporate behavior in oligopolistic markets, for example, the decisions that companies must make in terms of production and pricing levels, as well as the amount of money invested in research and development. e decision-making mechanism has an important role to play in the process of adjusting the production and profits of firms

Summary

Introduction

Game theory is one of the most interesting and complex topics that many researchers are interested in understanding. Fractional calculus, discrete fractional calculus, has attracted substantial interest in recent decades due to its extensive significance in a wide range of scientific disciplines, including complex systems with memory and heredity. Researchers turned their attention to a discreet fractional calculus and tried to develop a complete theoretical framework for this subject. Is reflects the long-term memory effects of Cournot–Bertrand dynamic games in the fractional-order form for the game. We will investigate the dynamics of the discreet fractional-order Cournot–Bertrand duopoly game such as the stability, bifurcation, and chaos of the proposed game.

Preliminaries

Discrete Fractional-Order Cournot–Bertrand Duopoly Game

The Equilibrium Points and Their Local Stability

Numerical Simulations