Global and Local Analysis for a Cournot Duopoly Game with Two Different Objective Functions
In this paper, a Cournot game with two competing firms is studied. The two competing firms seek the optimality of their quantities by maximizing two different objective functions. The first firm wants to maximize an average of social welfare and profit, while the second firm wants to maximize their...
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| Format: | Article |
| Language: | English |
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MDPI AG
2021-12-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/9/23/3119 |
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| author | Sameh Askar Abdulaziz Foul Tarek Mahrous Saleh Djemele Emad Ibrahim |
| author_facet | Sameh Askar Abdulaziz Foul Tarek Mahrous Saleh Djemele Emad Ibrahim |
| author_sort | Sameh Askar |
| collection | DOAJ |
| description | In this paper, a Cournot game with two competing firms is studied. The two competing firms seek the optimality of their quantities by maximizing two different objective functions. The first firm wants to maximize an average of social welfare and profit, while the second firm wants to maximize their relative profit only. We assume that both firms are rational, adopting a bounded rationality mechanism for updating their production outputs. A two-dimensional discrete time map is introduced to analyze the evolution of the game. The map has four equilibrium points and their stability conditions are investigated. We prove the Nash equilibrium point can be destabilized through flip bifurcation only. The obtained results show that the manifold of the game’s map can be analyzed through a one-dimensional map whose analytical form is similar to the well-known logistic map. The critical curves investigations show that the phase plane of game’s map is divided into three zones and, therefore, the map is not invertible. Finally, the contact bifurcation phenomena are discussed using simulation. |
| first_indexed | 2024-03-10T04:47:59Z |
| format | Article |
| id | doaj.art-068351a542ce4b38b276f3cdd8c26099 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T04:47:59Z |
| publishDate | 2021-12-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-068351a542ce4b38b276f3cdd8c260992023-11-23T02:46:19ZengMDPI AGMathematics2227-73902021-12-01923311910.3390/math9233119Global and Local Analysis for a Cournot Duopoly Game with Two Different Objective FunctionsSameh Askar0Abdulaziz Foul1Tarek Mahrous2Saleh Djemele3Emad Ibrahim4Department of Statistics and Operations Research, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi ArabiaDepartment of Statistics and Operations Research, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi ArabiaDepartment of Statistics and Operations Research, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi ArabiaDepartment of Statistics and Operations Research, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi ArabiaDepartment of Statistics and Operations Research, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi ArabiaIn this paper, a Cournot game with two competing firms is studied. The two competing firms seek the optimality of their quantities by maximizing two different objective functions. The first firm wants to maximize an average of social welfare and profit, while the second firm wants to maximize their relative profit only. We assume that both firms are rational, adopting a bounded rationality mechanism for updating their production outputs. A two-dimensional discrete time map is introduced to analyze the evolution of the game. The map has four equilibrium points and their stability conditions are investigated. We prove the Nash equilibrium point can be destabilized through flip bifurcation only. The obtained results show that the manifold of the game’s map can be analyzed through a one-dimensional map whose analytical form is similar to the well-known logistic map. The critical curves investigations show that the phase plane of game’s map is divided into three zones and, therefore, the map is not invertible. Finally, the contact bifurcation phenomena are discussed using simulation.https://www.mdpi.com/2227-7390/9/23/3119cobweb modeldynamicsgradient mechanismstabilitybifurcation |
| spellingShingle | Sameh Askar Abdulaziz Foul Tarek Mahrous Saleh Djemele Emad Ibrahim Global and Local Analysis for a Cournot Duopoly Game with Two Different Objective Functions Mathematics cobweb model dynamics gradient mechanism stability bifurcation |
| title | Global and Local Analysis for a Cournot Duopoly Game with Two Different Objective Functions |
| title_full | Global and Local Analysis for a Cournot Duopoly Game with Two Different Objective Functions |
| title_fullStr | Global and Local Analysis for a Cournot Duopoly Game with Two Different Objective Functions |
| title_full_unstemmed | Global and Local Analysis for a Cournot Duopoly Game with Two Different Objective Functions |
| title_short | Global and Local Analysis for a Cournot Duopoly Game with Two Different Objective Functions |
| title_sort | global and local analysis for a cournot duopoly game with two different objective functions |
| topic | cobweb model dynamics gradient mechanism stability bifurcation |
| url | https://www.mdpi.com/2227-7390/9/23/3119 |
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