Cargando…

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...

Descripción completa

Detalles Bibliográficos
Autores principales: Ahmadi, Atefeh, Roy, Sourav, Mehrabbeik, Mahtab, Ghosh, Dibakar, Jafari, Sajad, Perc, Matjaž
Formato: Online Artículo Texto
Lenguaje:English
Publicado: Public Library of Science 2023
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
_version_ 1785019920455565312
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
work_keys_str_mv AT ahmadiatefeh thedynamicsofaduopolystackelberggamewithmarginalcostsamongheterogeneousplayers
AT roysourav thedynamicsofaduopolystackelberggamewithmarginalcostsamongheterogeneousplayers
AT mehrabbeikmahtab thedynamicsofaduopolystackelberggamewithmarginalcostsamongheterogeneousplayers
AT ghoshdibakar thedynamicsofaduopolystackelberggamewithmarginalcostsamongheterogeneousplayers
AT jafarisajad thedynamicsofaduopolystackelberggamewithmarginalcostsamongheterogeneousplayers
AT percmatjaz thedynamicsofaduopolystackelberggamewithmarginalcostsamongheterogeneousplayers
AT ahmadiatefeh dynamicsofaduopolystackelberggamewithmarginalcostsamongheterogeneousplayers
AT roysourav dynamicsofaduopolystackelberggamewithmarginalcostsamongheterogeneousplayers
AT mehrabbeikmahtab dynamicsofaduopolystackelberggamewithmarginalcostsamongheterogeneousplayers
AT ghoshdibakar dynamicsofaduopolystackelberggamewithmarginalcostsamongheterogeneousplayers
AT jafarisajad dynamicsofaduopolystackelberggamewithmarginalcostsamongheterogeneousplayers
AT percmatjaz dynamicsofaduopolystackelberggamewithmarginalcostsamongheterogeneousplayers