Simultaneous Profit Optimization and Market Price Agreement in Networks of LTI Duopoly Systems

Document Type : Research article

Author

Department of Mathematics, Payame Noor University, Tehran, Iran

Abstract

This paper studies networks of dynamic duopoly systems with the objectives of maximization of firms' discounted profits and of convergence of market prices in all markets. We model each duopoly as a dynamical process of price adjustment, driven by production decisions of firms. The profit maximization problem is reformulated as a quadratic optimal control problem by representing the dynamics of the duopoly in a linear time-invariant (LTI) state-space form. In order to impose the price convergence between the markets, a consensus term is added to the cost function. We use the Hamilton--Jacobi--Bellman (HJB) approach to derive a control law that encourages profit maximization and market price convergence over a finite time horizon. The proposed framework reformulates the dynamic duopoly problem in networked markets within an optimal control framework, allowing simultaneous profit optimization and market price coordination.

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[1] S. Ahmed and M. O. Sayin, Dynamic Feedback Strategies for Duopolies Over Partially Observed Consumer Networks, Dynamic Games and Applications, 1–40, (2025).
[2] T. Basar and G. J. Olsder, Dynamic noncooperative game theory, Vol. 23, Siam, 1999.
[3] A. Bhaya and E. Kaszkurewicz Dynamic advertising games in duopolies under one-step-ahead optimal control, Dynamic Games and Applications, 13(3), 721-749, (2023).
[4] C. Cheng, B. Yang, B. Li, Z. Han and F. Peng, Model predictive coordinated cooperative control mechanism for multiagent systems based on priority negotiation, Journal of Process Control, 136, 103182, (2024).
[5] G. A. De Castro and F. Paganini, Convex synthesis of controllers for consensus, Proceedings of American Control Conference IEEE, 6, 4933–4938, (2024).
[6] M. T. Devine and V. Bertsch, Analysing the interactions between demand side and supply side investment decisions in an oligopolistic electricity market using a stochastic mixed complementarity problem, European Journal of Operational Research, (2026).
[7] P. A. A. Driessen, R. M. Hermans and P. P. J. Van Den Bosch, Distributed economic model predictive control of networks in competitive environments, 51st IEEE Conference on Decision and Control (CDC), 266–271, (2012).
[8] J. Engwerda, LQ dynamic optimization and differential games, John Wiley & Sons, 2005.
[9] G. Feichtinger and E. Dockner, Optimal pricing in a duopoly: A noncooperative differential games solution. Journal of Optimization Theory and Applications, Journal of Optimization Theory and Applications, 45(2), 199-218, (1985).
[10] L. Ge and S. Li, Price and product innovation competition with network effects and consumers adaptive learning: A differential game approach, Computers & Industrial Engineering, 193, 110298, (2024).
[11] K. Kok Multi-agent coordination in the electricity grid, from concept towards market introduction, Proceedings of the 9th International Conference on Autonomous Agents and Multiagent Systems: Industry track, 1681–1688, (2010).
[12] V. Nalepova and M. Lampart, Stabilizing energy markets: A dynamic analysis of price cap regulation in oligopolistic environments, Energy Economics 151, 108879, (2025).
[13] R. Olfati-Saber, J. A. Fax and R. M. Murray, Consensus and cooperation in networked multi-agent systems, Proceedings of the IEEE, 95, 215–233, (2007).
[14] J. Ran and Y. Zhou, A stochastic discrete fractional cournot duopoly game: modeling, stability, and optimal control, Complexity, 2024(1), 6680399, (2024).
[15] V. H. P. Rodrigues, T. R. Oliveira, M. Krstic and T. Basar, Nash equilibrium seeking for noncooperative duopoly games via event-triggered control, IEEE 63rd Conference on Decision and Control (CDC), 5248–5255, (2024).
[16] E. Semsar-Kazerooni and K. Khorasani, Multi-agent team cooperation: A game theory approach, Automatica, 45(10), 2205–2213, (2009).
[17] T. A. Weber, Optimal control theory with applications in economics, MIT press, 2011.
[18] M. Ye, L. Yin, G. Wen and Y. Zheng, On distributed Nash equilibrium computation: Hybrid games and a novel consensus-tracking perspective, IEEE Transactions on Cybernetics, 51(10), 5021–5031, (2020). 
Volume 11, Issue 2
September 2026
Pages 352-362
  • Receive Date: 01 June 2026
  • Revise Date: 29 June 2026
  • Accept Date: 23 July 2026
  • Publish Date: 04 September 2026