Complex Investigations of a Piecewise-Smooth Remanufacturing Bertrand Duopoly Game
This paper considers a Bertrand competition between two firms whose decision variables are derived from a quadratic utility function. The first firm produces new products with their own prices while the second firm re-manufactures returned products and sells them in the market at prices that may be...
| Main Author: | |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2021-10-01
|
| Series: | Mathematics |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2227-7390/9/20/2558 |
| _version_ | 1827679232297271296 |
|---|---|
| author | Sameh Askar |
| author_facet | Sameh Askar |
| author_sort | Sameh Askar |
| collection | DOAJ |
| description | This paper considers a Bertrand competition between two firms whose decision variables are derived from a quadratic utility function. The first firm produces new products with their own prices while the second firm re-manufactures returned products and sells them in the market at prices that may be less than or equal to the price of the first firm. Dynamically, this competition is constructed on which boundedly rational firms apply a gradient adjustment mechanism to update their prices in each period. According to this mechanism and the nature of the competition, a two-dimensional piecewise smooth discrete dynamic map was constructed in order to study the complex dynamic characteristics of the game. The phase plane of the map was divided into two different regions, separated by border curve. The equilibrium points of the map, in each region on where they are defined, were calculated, and their stability conditions were investigated. Furthermore, we conducted a global analysis to investigate the complex structure of the map, such as closed invariant curves, periodic cycles, and chaotic attractors and their basins, which cause qualitative changes as some parameters are allowed to vary. |
| first_indexed | 2024-03-10T06:25:09Z |
| format | Article |
| id | doaj.art-a5a45b840fde4ed19595f4e47381a09f |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T06:25:09Z |
| publishDate | 2021-10-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-a5a45b840fde4ed19595f4e47381a09f2023-11-22T19:01:42ZengMDPI AGMathematics2227-73902021-10-01920255810.3390/math9202558Complex Investigations of a Piecewise-Smooth Remanufacturing Bertrand Duopoly GameSameh Askar0Department of Statistics and Operations Research, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi ArabiaThis paper considers a Bertrand competition between two firms whose decision variables are derived from a quadratic utility function. The first firm produces new products with their own prices while the second firm re-manufactures returned products and sells them in the market at prices that may be less than or equal to the price of the first firm. Dynamically, this competition is constructed on which boundedly rational firms apply a gradient adjustment mechanism to update their prices in each period. According to this mechanism and the nature of the competition, a two-dimensional piecewise smooth discrete dynamic map was constructed in order to study the complex dynamic characteristics of the game. The phase plane of the map was divided into two different regions, separated by border curve. The equilibrium points of the map, in each region on where they are defined, were calculated, and their stability conditions were investigated. Furthermore, we conducted a global analysis to investigate the complex structure of the map, such as closed invariant curves, periodic cycles, and chaotic attractors and their basins, which cause qualitative changes as some parameters are allowed to vary.https://www.mdpi.com/2227-7390/9/20/2558small piecewise smoothBertrand gamestabilitybifurcation |
| spellingShingle | Sameh Askar Complex Investigations of a Piecewise-Smooth Remanufacturing Bertrand Duopoly Game Mathematics small piecewise smooth Bertrand game stability bifurcation |
| title | Complex Investigations of a Piecewise-Smooth Remanufacturing Bertrand Duopoly Game |
| title_full | Complex Investigations of a Piecewise-Smooth Remanufacturing Bertrand Duopoly Game |
| title_fullStr | Complex Investigations of a Piecewise-Smooth Remanufacturing Bertrand Duopoly Game |
| title_full_unstemmed | Complex Investigations of a Piecewise-Smooth Remanufacturing Bertrand Duopoly Game |
| title_short | Complex Investigations of a Piecewise-Smooth Remanufacturing Bertrand Duopoly Game |
| title_sort | complex investigations of a piecewise smooth remanufacturing bertrand duopoly game |
| topic | small piecewise smooth Bertrand game stability bifurcation |
| url | https://www.mdpi.com/2227-7390/9/20/2558 |
| work_keys_str_mv | AT samehaskar complexinvestigationsofapiecewisesmoothremanufacturingbertrandduopolygame |