Analysis of Nonlinear Duopoly Games with Product Differentiation: Stability, Global Dynamics, and Control
Many researchers have used quadratic utility function to study its influences on economic games with product differentiation. Such games include Cournot, Bertrand, and a mixed-type game called Cournot-Bertrand. Within this paper, a cubic utility function that is derived from a constant elasticity of...
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| Format: | Article |
| Language: | English |
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Wiley
2017-01-01
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| Series: | Discrete Dynamics in Nature and Society |
| Online Access: | http://dx.doi.org/10.1155/2017/2585708 |
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| author | S. S. Askar A. Al-khedhairi |
| author_facet | S. S. Askar A. Al-khedhairi |
| author_sort | S. S. Askar |
| collection | DOAJ |
| description | Many researchers have used quadratic utility function to study its influences on economic games with product differentiation. Such games include Cournot, Bertrand, and a mixed-type game called Cournot-Bertrand. Within this paper, a cubic utility function that is derived from a constant elasticity of substitution production function (CES) is introduced. This cubic function is more desirable than the quadratic one besides its amenability to efficiency analysis. Based on that utility a two-dimensional Cournot duopoly game with horizontal product differentiation is modeled using a discrete time scale. Two different types of games are studied in this paper. In the first game, firms are updating their output production using the traditional bounded rationality approach. In the second game, firms adopt Puu’s mechanism to update their productions. Puu’s mechanism does not require any information about the profit function; instead it needs both firms to know their production and their profits in the past time periods. In both scenarios, an explicit form for the Nash equilibrium point is obtained under certain conditions. The stability analysis of Nash point is considered. Furthermore, some numerical simulations are carried out to confirm the chaotic behavior of Nash equilibrium point. This analysis includes bifurcation, attractor, maximum Lyapunov exponent, and sensitivity to initial conditions. |
| format | Article |
| id | doaj-art-a7b39d56fdc040a4af7df673e9d17037 |
| institution | Kabale University |
| issn | 1026-0226 1607-887X |
| language | English |
| publishDate | 2017-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Discrete Dynamics in Nature and Society |
| spelling | doaj-art-a7b39d56fdc040a4af7df673e9d170372025-02-03T01:22:28ZengWileyDiscrete Dynamics in Nature and Society1026-02261607-887X2017-01-01201710.1155/2017/25857082585708Analysis of Nonlinear Duopoly Games with Product Differentiation: Stability, Global Dynamics, and ControlS. S. Askar0A. Al-khedhairi1Department of Statistics and Operations Researches, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi ArabiaDepartment of Statistics and Operations Researches, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi ArabiaMany researchers have used quadratic utility function to study its influences on economic games with product differentiation. Such games include Cournot, Bertrand, and a mixed-type game called Cournot-Bertrand. Within this paper, a cubic utility function that is derived from a constant elasticity of substitution production function (CES) is introduced. This cubic function is more desirable than the quadratic one besides its amenability to efficiency analysis. Based on that utility a two-dimensional Cournot duopoly game with horizontal product differentiation is modeled using a discrete time scale. Two different types of games are studied in this paper. In the first game, firms are updating their output production using the traditional bounded rationality approach. In the second game, firms adopt Puu’s mechanism to update their productions. Puu’s mechanism does not require any information about the profit function; instead it needs both firms to know their production and their profits in the past time periods. In both scenarios, an explicit form for the Nash equilibrium point is obtained under certain conditions. The stability analysis of Nash point is considered. Furthermore, some numerical simulations are carried out to confirm the chaotic behavior of Nash equilibrium point. This analysis includes bifurcation, attractor, maximum Lyapunov exponent, and sensitivity to initial conditions.http://dx.doi.org/10.1155/2017/2585708 |
| spellingShingle | S. S. Askar A. Al-khedhairi Analysis of Nonlinear Duopoly Games with Product Differentiation: Stability, Global Dynamics, and Control Discrete Dynamics in Nature and Society |
| title | Analysis of Nonlinear Duopoly Games with Product Differentiation: Stability, Global Dynamics, and Control |
| title_full | Analysis of Nonlinear Duopoly Games with Product Differentiation: Stability, Global Dynamics, and Control |
| title_fullStr | Analysis of Nonlinear Duopoly Games with Product Differentiation: Stability, Global Dynamics, and Control |
| title_full_unstemmed | Analysis of Nonlinear Duopoly Games with Product Differentiation: Stability, Global Dynamics, and Control |
| title_short | Analysis of Nonlinear Duopoly Games with Product Differentiation: Stability, Global Dynamics, and Control |
| title_sort | analysis of nonlinear duopoly games with product differentiation stability global dynamics and control |
| url | http://dx.doi.org/10.1155/2017/2585708 |
| work_keys_str_mv | AT ssaskar analysisofnonlinearduopolygameswithproductdifferentiationstabilityglobaldynamicsandcontrol AT aalkhedhairi analysisofnonlinearduopolygameswithproductdifferentiationstabilityglobaldynamicsandcontrol |