The Cooperative Mean Field Game for Production Control with Sticky Price

Abstract

We study a cooperative mean field game problem arising from the production control for multiple firms with price stickiness in the commodity market. The price dynamics for each firm is described as a controlled jump-diffusion process with mean-field interaction. Each firm aims to maximize the so-called social rewards which is defined by the average of individual rewards for all firms. By solving the limiting control problem for the representative firm and an associated fixed-point problem, we construct an explicit approximating optimal strategy when the number of firms grows large.

Keywords

Production control / Sticky price / Social rewards / Cooperative mean field game

References

[1] Bo, L., Wang, S., Yu, X.: Mean field game of optimal relative investment with contagious risk. Preprint (2021) arXiv:2108.00799

[2] Carmona R, Fouque JP, Sun LH. Mean field games and systemic risk. Commun. Math. Sci.. 2015, 13 4 911-933

[3] Carmona R, Dayanikli G, Lauriére M. Mean field models to regulate carbon emissions in electricity production. Dyn. Games Appl.. 2022

[4] Cellini R, Lambertini L. Dynamic oligopoly with sticky prices: closed-loop, feedback, and open-loop solutions. J. Dyn. Contr. Syst.. 2004, 10 3 303-314

[5] Dorf RC, Bishop RH. Modern Control Systems. 2010 New Jersey: Prentice Hall

[6] Élie R, Hubert E, Mastrolia T, Possamaï D. Mean-field moral hazard for optimal energy demand response management. Math. Finance. 2021, 31 1 399-473

[7] Esfahani H. Profitability of horizontal mergers in the presence of price stickiness. Eur. J. Oper. Res.. 2019, 279 941-950

[8] Fershtman C, Kamien MI. Dynamic duopolistic competition with sticky prices. Econometrica. 1987, 55 5 1151-1164

[9] Gu HT, Guo X, Wei XL, Xu RY. Mean-field controls with Q-learning for cooperative MARL: convergence and complexity analysis. SIAM J. Math. Data Sci.. 2021, 3 4 1168-1196

[10] Huang JH, Wang BC, Yong JM. Social optima in mean field linear-quadratic-gaussian control with volatility uncertainty. SIAM J. Control. Optim.. 2021, 59 2 825-856

[11] Jiang CM, Li TQ, Yu J. Approximating Nash equilibrium for production control with sticky price. Oper. Res. Lett.. 2022, 50 335-342

[12] Lacker D, Soret A. Many-player games of optimal consumption and investment under relative performance criteria. Math. Financ. Econ.. 2020, 14 2 263-281

[13] Protter PE. Stochastic Integration and Differential Equations. 2005 New York: Springer-Verlag

[14] Simaan M, Takayama T. Game theory applied to dynamic duopoly problems with production constraints. Automatica. 1978, 14 161-166

[15] Wang BC, Huang MY. Mean field production output control with sticky price: nash and social solutions. Automatica. 2019, 100 90-98

[16] Wang BC, Zhang JF. Social optima in mean field linear-quadratic-Gaussian models with Markov jump parameters. SIAM J. Control. Optim.. 2017, 55 1 429-456

[17] Xin BG, Sun MH. A differential oligopoly game for optimal production planning and water savings. Euro. J. Oper. Res.. 2018, 269 206-217

Funding

National Natural Science Foundation of China(11971368)