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The dynamics of a duopoly Stackelberg game with marginal costs among heterogeneous players
One of the famous economic models in game theory is the duopoly Stackelberg model, in which a leader and a follower firm manufacture a single product in the market. Their goal is to obtain the maximum profit while competing with each other. The desired dynamics for a firm in a market is the converge...
Autores principales: | , , , , , |
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Formato: | Online Artículo Texto |
Lenguaje: | English |
Publicado: |
Public Library of Science
2023
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Materias: | |
Acceso en línea: | https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10075410/ https://www.ncbi.nlm.nih.gov/pubmed/37018231 http://dx.doi.org/10.1371/journal.pone.0283757 |
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author | Ahmadi, Atefeh Roy, Sourav Mehrabbeik, Mahtab Ghosh, Dibakar Jafari, Sajad Perc, Matjaž |
author_facet | Ahmadi, Atefeh Roy, Sourav Mehrabbeik, Mahtab Ghosh, Dibakar Jafari, Sajad Perc, Matjaž |
author_sort | Ahmadi, Atefeh |
collection | PubMed |
description | One of the famous economic models in game theory is the duopoly Stackelberg model, in which a leader and a follower firm manufacture a single product in the market. Their goal is to obtain the maximum profit while competing with each other. The desired dynamics for a firm in a market is the convergence to its Nash equilibrium, but the dynamics of real-world markets are not always steady and can result in unpredictable market changes that exhibit chaotic behaviors. On the other hand, to approach reality more, the two firms in the market can be considered heterogeneous. The leader firm is bounded rationale, and the follower firm is adaptable. Modifying the cost function that affects the firms’ profit by adding the marginal cost term is another step toward reality. We propose a Stackelberg model with heterogeneous players and marginal costs, which exhibits chaotic behavior. This model’s equilibrium points, including the Nash equilibrium, are calculated by the backward induction method, and their stability analyses are obtained. The influence of changing each model parameter on the consequent dynamics is investigated through one-dimensional and two-dimensional bifurcation diagrams, Lyapunov exponents spectra, and Kaplan-Yorke dimension. Eventually, using a combination of state feedback and parameter adjustment methods, the chaotic solutions of the model are successfully tamed, and the model converges to its Nash equilibrium. |
format | Online Article Text |
id | pubmed-10075410 |
institution | National Center for Biotechnology Information |
language | English |
publishDate | 2023 |
publisher | Public Library of Science |
record_format | MEDLINE/PubMed |
spelling | pubmed-100754102023-04-06 The dynamics of a duopoly Stackelberg game with marginal costs among heterogeneous players Ahmadi, Atefeh Roy, Sourav Mehrabbeik, Mahtab Ghosh, Dibakar Jafari, Sajad Perc, Matjaž PLoS One Research Article One of the famous economic models in game theory is the duopoly Stackelberg model, in which a leader and a follower firm manufacture a single product in the market. Their goal is to obtain the maximum profit while competing with each other. The desired dynamics for a firm in a market is the convergence to its Nash equilibrium, but the dynamics of real-world markets are not always steady and can result in unpredictable market changes that exhibit chaotic behaviors. On the other hand, to approach reality more, the two firms in the market can be considered heterogeneous. The leader firm is bounded rationale, and the follower firm is adaptable. Modifying the cost function that affects the firms’ profit by adding the marginal cost term is another step toward reality. We propose a Stackelberg model with heterogeneous players and marginal costs, which exhibits chaotic behavior. This model’s equilibrium points, including the Nash equilibrium, are calculated by the backward induction method, and their stability analyses are obtained. The influence of changing each model parameter on the consequent dynamics is investigated through one-dimensional and two-dimensional bifurcation diagrams, Lyapunov exponents spectra, and Kaplan-Yorke dimension. Eventually, using a combination of state feedback and parameter adjustment methods, the chaotic solutions of the model are successfully tamed, and the model converges to its Nash equilibrium. Public Library of Science 2023-04-05 /pmc/articles/PMC10075410/ /pubmed/37018231 http://dx.doi.org/10.1371/journal.pone.0283757 Text en © 2023 Ahmadi et al https://creativecommons.org/licenses/by/4.0/This is an open access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0/) , which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. |
spellingShingle | Research Article Ahmadi, Atefeh Roy, Sourav Mehrabbeik, Mahtab Ghosh, Dibakar Jafari, Sajad Perc, Matjaž The dynamics of a duopoly Stackelberg game with marginal costs among heterogeneous players |
title | The dynamics of a duopoly Stackelberg game with marginal costs among heterogeneous players |
title_full | The dynamics of a duopoly Stackelberg game with marginal costs among heterogeneous players |
title_fullStr | The dynamics of a duopoly Stackelberg game with marginal costs among heterogeneous players |
title_full_unstemmed | The dynamics of a duopoly Stackelberg game with marginal costs among heterogeneous players |
title_short | The dynamics of a duopoly Stackelberg game with marginal costs among heterogeneous players |
title_sort | dynamics of a duopoly stackelberg game with marginal costs among heterogeneous players |
topic | Research Article |
url | https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10075410/ https://www.ncbi.nlm.nih.gov/pubmed/37018231 http://dx.doi.org/10.1371/journal.pone.0283757 |
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