The stability of the Nash equilibrium in a single-leader multi-follower game with strategic adjustment costs

Luping Liu; Zhaonan Mu; Chun Wang; Luping Liu; Zhaonan Mu; Chun Wang

doi:10.3934/math.2026815

AIMS Mathematics

2026, Volume 11, Issue 7: 20068-20084. doi: 10.3934/math.2026815

Previous Article Next Article

Research article

The stability of the Nash equilibrium in a single-leader multi-follower game with strategic adjustment costs

1.
College of Big Data and Intelligent Engineering, Guizhou University of Commerce, Guiyang 550014, Guizhou, China
2.
Guizhou Open University and Guizhou Vocational Technology Institute, Guiyang 550023, Guizhou, China

Received: 20 April 2026 Revised: 24 June 2026 Accepted: 03 July 2026 Published: 08 July 2026
MSC : 34D20, 91A25, 91A65

In a single-leader multi-follower game (SLMFG), leaders or followers often incur additional costs while revising their decisions, such as suppliers versus retailers, central government versus local government, and conglomerates versus subsidiaries; however, these costs are usually ignored. To focus on this issue, using quadratic cost functions, these strategic adjustment costs are introduced into a dynamic SLMFG with adaptive expectations, thereby yielding a new dynamic model. Furthermore, we obtain some necessary and sufficient conditions for local asymptotic stability of the SLMFG. We also prove that both strategic adjustment costs and the leader have increased the stable region of the original Nash equilibrium. By comparing interval increments, the leader and strategy adjustment cost contribute equally to the Nash equilibrium, which, presented in the paper, develop corresponding conclusions to previous research.
- single-leader multi-follower game,
- quadratic cost functions,
- strategic adjustment costs,
- equilibrium stability analysis
Citation: Luping Liu, Zhaonan Mu, Chun Wang. The stability of the Nash equilibrium in a single-leader multi-follower game with strategic adjustment costs[J]. AIMS Mathematics, 2026, 11(7): 20068-20084. doi: 10.3934/math.2026815

Related Papers:

Abstract

In a single-leader multi-follower game (SLMFG), leaders or followers often incur additional costs while revising their decisions, such as suppliers versus retailers, central government versus local government, and conglomerates versus subsidiaries; however, these costs are usually ignored. To focus on this issue, using quadratic cost functions, these strategic adjustment costs are introduced into a dynamic SLMFG with adaptive expectations, thereby yielding a new dynamic model. Furthermore, we obtain some necessary and sufficient conditions for local asymptotic stability of the SLMFG. We also prove that both strategic adjustment costs and the leader have increased the stable region of the original Nash equilibrium. By comparing interval increments, the leader and strategy adjustment cost contribute equally to the Nash equilibrium, which, presented in the paper, develop corresponding conclusions to previous research.

References

[1]	H. V. Stackelberg, The theory of the market economy, Oxford University Press, 1952.
[2]	A. Bagchi, Stackelberg differential games in economic models, Springer Berlin, Heidelberg, 1984. https://doi.org/10.1007/BFb0009151
[3]	G. Papavassilopoulos, J. Cruz, Nonclassical control problems and Stackelberg games, IEEE Trans. Automat. Contr., 24 (1979), 155–166. https://doi.org/10.1109/TAC.1979.1101986 doi: 10.1109/TAC.1979.1101986
[4]	T. Li, S. P. Sethi, A review of dynamic Stackelberg game models, Discrete Cont. Dyn. Syst. Ser.-B, 22 (2017), 125–159. https://doi.org/10.3934/dcdsb.2017007 doi: 10.3934/dcdsb.2017007
[5]	J. Shi, G. Wang, J. Xiong, Leader–follower stochastic differential game with asymmetric information and applications, Automatica, 63 (2016), 60–73. https://doi.org/10.1016/j.automatica.2015.10.011 doi: 10.1016/j.automatica.2015.10.011
[6]	A. Bensoussan, S. Chen, S. P. Sethi, The maximum principle for global solutions of stochastic Stackelberg differential games, SIAM J. Control Optim., 53 (2015), 1956–1981. https://doi.org/10.1137/140958906 doi: 10.1137/140958906
[7]	L. Liu, W. Jia, An intelligent algorithm for solving the efficient Nash equilibrium of a single-leader multi-follower game, Mathematics, 9 (2021), 454. https://doi.org/10.3390/math9050454 doi: 10.3390/math9050454
[8]	H. Yan, H. Hou, M. Deng, L. Si, X. Wang, E. Hu, et al., Stackelberg game theory based model to guide users' energy use behavior, with the consideration of flexible resources and consumer psychology, for an integrated energy system, Energy, 288 (2024), 129806. https://doi.org/10.1016/j.energy.2023.129806 doi: 10.1016/j.energy.2023.129806
[9]	Ö. Erol, Ü. B. Filik, A Stackelberg game-based dynamic pricing and robust optimization strategy for microgrid operations, Int. J. Electr. Power Energy Syst., 155 (2024), 109574. https://doi.org/10.1016/j.ijepes.2023.109574 doi: 10.1016/j.ijepes.2023.109574
[10]	J. P. Gould, Adjustment costs in the theory of investment of the Firm, Rev. Econ. Stud., 35 (1968), 47–55. https://doi.org/10.2307/2974406 doi: 10.2307/2974406
[11]	F. Hayashi, Tobin's marginal $q$ and average $q$: a neoclassical interpretation, Econometrica, 50 (1982), 213–224. https://doi.org/10.2307/1912538 doi: 10.2307/1912538
[12]	G. Bischi, M. Kopel, Equilibrium selection in a nonlinear duopoly game with adaptive expectations, J. Econ. Behav. Organ., 46 (2001), 73–100. https://doi.org/10.1016/S0167-2681(01)00188-3 doi: 10.1016/S0167-2681(01)00188-3
[13]	A. A. Elsadany, Dynamics of a Cournot duopoly game with bounded rationality based on relative profit maximization, Appl. Math. Comput., 294 (2017), 253–263. https://doi.org/10.1016/j.amc.2016.09.018 doi: 10.1016/j.amc.2016.09.018
[14]	C. Wang, W. Tang, J. Pi, Demerit of the information concerning decision-making methodology: a dynamic Cournot game with adaptive expectation, Phys. Lett. A, 527 (2024), 129995. https://doi.org/10.1016/j.physleta.2024.129995 doi: 10.1016/j.physleta.2024.129995
[15]	M. T. Yassen, H. N. Agiza, Analysis of a duopoly game with delayed bounded rationality, Appl. Math. Comput., 138 (2003), 387–402. https://doi.org/10.1016/S0096-3003(02)00143-1 doi: 10.1016/S0096-3003(02)00143-1
[16]	L. C. Baiardi, A. K. Naimzada, Experimental oligopolies modeling: a dynamic approach based on heterogeneous behaviors, Commun. Nonlinear Sci. Numer. Simul., 58 (2018), 47–61. https://doi.org/10.1016/j.cnsns.2017.05.010 doi: 10.1016/j.cnsns.2017.05.010
[17]	C. Wang, J. Pi, D. Zhou, W. Tang, G. Yang, Dynamics of $n$-person Cournot games with asymmetric information and heterogeneous expectations, Phys. A, 618 (2023), 128691. https://doi.org/10.1016/j.physa.2023.128691 doi: 10.1016/j.physa.2023.128691
[18]	F. Tramontana, A. A. Elsadany, B. Xin, H. N. Agiza, Local stability of the Cournot solution with increasing heterogeneous competitors, Nonlinear Anal., 26 (2015), 150–160. https://doi.org/10.1016/j.nonrwa.2015.05.005 doi: 10.1016/j.nonrwa.2015.05.005
[19]	F. Szidarovszky, J. Yen, Dynamic Cournot oligopolies with production adjustment costs, J. Math. Econ., 24 (1995), 95–101. https://doi.org/10.1016/0304-4068(94)00661-S doi: 10.1016/0304-4068(94)00661-S
[20]	M. Kopel, Simple and complex adjustment dynamics in Cournot duopoly models, Chaos Soliton. Fract., 7 (1996), 2031–2048. https://doi.org/10.1016/S0960-0779(96)00070-7 doi: 10.1016/S0960-0779(96)00070-7
[21]	T. D. Howroyd, J. A. Rickard, Cournot oligopoly and output adjustment costs, Econ. Lett., 7 (1981), 113–117. https://doi.org/10.1016/0165-1765(87)90104-2 doi: 10.1016/0165-1765(87)90104-2
[22]	A. Zhang, Y. Zhang, Stability of a Cournot-Nash equilibrium: the multi-product case, J. Math. Econ., 26 (1996), 441–462. https://doi.org/10.1016/0304-4068(95)00760-1 doi: 10.1016/0304-4068(95)00760-1
[23]	C. Wang, W. Tang, J. Pi, The stability of dynamic $n$-person Cournot games with strategic adjustment costs, Chaos Soliton. Fract., 179 (2024), 114409. https://doi.org/10.1016/j.chaos.2023.114409 doi: 10.1016/j.chaos.2023.114409
[24]	L. Schoonbeek, A dynamic Stackelberg model with production-adjustment costs, J. Econ., 66 (1997), 271–282. https://doi.org/10.1007/BF01226829 doi: 10.1007/BF01226829
[25]	P. Y. Nie, A note on dynamic Stackelberg games with leaders in turn, Nonlinear Anal., 13 (2012), 85–90. https://doi.org/10.1016/j.nonrwa.2011.07.015 doi: 10.1016/j.nonrwa.2011.07.015
[26]	W. Hu, C. Bao, T. Dong, Dynamics of a quantum duopoly Stackelberg game model with marginal costs, Quantum Inf. Process., 22 (2023), 445. https://doi.org/10.1007/s11128-023-04196-7 doi: 10.1007/s11128-023-04196-7
[27]	A. Ahmadi, S. Roy, M. Mehrabbeik, D. Ghosh, S. Jafari, M. Perc, The dynamics of a duopoly Stackelberg game with marginal costs among heterogeneous players, PLOS One, 18 (2023), e0283757. https://doi.org/10.1371/journal.pone.0283757 doi: 10.1371/journal.pone.0283757
[28]	J. Ahmed, M. A. Razzaque, M. M. Rahman, S. A. Alqahtani, M. M. Hassan, A Stackelberg game-based dynamic resource allocation in edge federated 5G network, IEEE Access, 10 (2022), 10460–10471. https://doi.org/10.1109/ACCESS.2022.3144960 doi: 10.1109/ACCESS.2022.3144960
[29]	N. Xu, S. Yue, N. Zhao, B. Niu, X. Zhao, SMS-Based optimal control of completely unknown nonlinear systems with unknown actuator saturation, Int. J. Adapt. Control Signal Process., 2026, 1–16. https://doi.org/10.1002/acs.70109 doi: 10.1002/acs.70109
[30]	Z. Li, S. Ding, L. Zhang, G. Song, Filter-based predefined-time optimal fault-tolerant consensus control for nonlinear multi-agent systems via reinforcement learning, Neurocomputing, 679 (2026), 133224. https://doi.org/10.1016/j.neucom.2026.133224 doi: 10.1016/j.neucom.2026.133224
[31]	G. Gandolfo, Economic dynamics: methods and models, North-Holland, 1997.
[32]	R. A. Horn, C. R. Johnson, Matrix analysis, Cambridge University Press, 2012. https://doi.org/10.1017/CBO9780511810817
[33]	R. D. Theocharis, On the stability of the Cournot solution on the oligopoly problem, Rev. Econ. Stud., 27 (1960), 133–134. https://doi.org/10.2307/2296135 doi: 10.2307/2296135

Reader Comments

Your name:*

Email:*
© 2026 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)