IDEAS home Printed from https://ideas.repec.org/p/arx/papers/2207.05303.html
   My bibliography  Save this paper

The Inverse Problem of Linear-Quadratic Differential Games: When is a Control Strategies Profile Nash?

Author

Listed:
  • Yunhan Huang
  • Tao Zhang
  • Quanyan Zhu

Abstract

This paper aims to formulate and study the inverse problem of non-cooperative linear quadratic games: Given a profile of control strategies, find cost parameters for which this profile of control strategies is Nash. We formulate the problem as a leader-followers problem, where a leader aims to implant a desired profile of control strategies among selfish players. In this paper, we leverage frequency-domain techniques to develop a necessary and sufficient condition on the existence of cost parameters for a given profile of stabilizing control strategies to be Nash under a given linear system. The necessary and sufficient condition includes the circle criterion for each player and a rank condition related to the transfer function of each player. The condition provides an analytical method to check the existence of such cost parameters, while previous studies need to solve a convex feasibility problem numerically to answer the same question. We develop an identity in frequency-domain representation to characterize the cost parameters, which we refer to as the Kalman equation. The Kalman equation reduces redundancy in the time-domain analysis that involves solving a convex feasibility problem. Using the Kalman equation, we also show the leader can enforce the same Nash profile by applying penalties on the shared state instead of penalizing the player for other players' actions to avoid the impression of unfairness.

Suggested Citation

  • Yunhan Huang & Tao Zhang & Quanyan Zhu, 2022. "The Inverse Problem of Linear-Quadratic Differential Games: When is a Control Strategies Profile Nash?," Papers 2207.05303, arXiv.org, revised Jul 2022.
    • Handle: RePEc:arx:papers:2207.05303
    as

    Download full text from publisher

    File URL: http://arxiv.org/pdf/2207.05303
    File Function: Latest version
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Timothy C Reluga, 2010. "Game Theory of Social Distancing in Response to an Epidemic," PLOS Computational Biology, Public Library of Science, vol. 6(5), pages 1-9, May.
    2. Engwerda, J.C. & van den Broek, W.A. & Schumacher, J.M., 2000. "Feedback Nash equilibria in uncertain infinite time horizon differential games," Other publications TiSEM c431993d-ee67-4a93-9e2d-f, Tilburg University, School of Economics and Management.
    3. Dockner,Engelbert J. & Jorgensen,Steffen & Long,Ngo Van & Sorger,Gerhard, 2000. "Differential Games in Economics and Management Science," Cambridge Books, Cambridge University Press, number 9780521637329, January.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Nikooeinejad, Z. & Heydari, M. & Loghmani, G.B., 2022. "A numerical iterative method for solving two-point BVPs in infinite-horizon nonzero-sum differential games: Economic applications," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 200(C), pages 404-427.
    2. Masahiko Hattori & Yasuhito Tanaka, 2019. "General analysis of dynamic oligopoly with sticky price," Economics Bulletin, AccessEcon, vol. 39(4), pages 2990-2998.
    3. Luca Lambertini & Arsen Palestini & Alessandro Tampieri, 2016. "CSR in an Asymmetric Duopoly with Environmental Externality," Southern Economic Journal, John Wiley & Sons, vol. 83(1), pages 236-252, July.
    4. Açıkgöz, Ömer T. & Benchekroun, Hassan, 2017. "Anticipated international environmental agreements," European Economic Review, Elsevier, vol. 92(C), pages 306-336.
    5. Xiaochi Wu, 2022. "Existence of value for a differential game with asymmetric information and signal revealing," International Journal of Game Theory, Springer;Game Theory Society, vol. 51(1), pages 213-247, March.
    6. Régis Chenavaz & Corina Paraschiv & Gabriel Turinici, 2017. "Dynamic Pricing of New Products in Competitive Markets: A Mean-Field Game Approach," Working Papers hal-01592958, HAL.
    7. Reinhard Neck & Dmitri Blueschke, 2014. "“Haircuts” for the EMU periphery: virtue or vice?," Empirica, Springer;Austrian Institute for Economic Research;Austrian Economic Association, vol. 41(2), pages 153-175, May.
    8. Guillaume Bataille & Benteng Zou, 2024. "International Fisheries Agreements: Endogenous Exits, Shapley Values, and Moratorium Fishing Policy," DEM Discussion Paper Series 24-06, Department of Economics at the University of Luxembourg.
    9. Benchekroun, Hassan & Ray Chaudhuri, Amrita & Tasneem, Dina, 2020. "On the impact of trade in a common property renewable resource oligopoly," Journal of Environmental Economics and Management, Elsevier, vol. 101(C).
    10. A. J. Novak & G. Feichtinger & G. Leitmann, 2010. "A Differential Game Related to Terrorism: Nash and Stackelberg Strategies," Journal of Optimization Theory and Applications, Springer, vol. 144(3), pages 533-555, March.
    11. Gerhard Sorger, 2005. "A dynamic common property resource problem with amenity value and extraction costs," International Journal of Economic Theory, The International Society for Economic Theory, vol. 1(1), pages 3-19, March.
    12. Mukherjee, Arka & Carvalho, Margarida, 2021. "Dynamic decision making in a mixed market under cooperation: Towards sustainability," International Journal of Production Economics, Elsevier, vol. 241(C).
    13. Smala Fanokoa, Pascaux & Telahigue, Issam & Zaccour, Georges, 2011. "Buying cooperation in an asymmetric environmental differential game," Journal of Economic Dynamics and Control, Elsevier, vol. 35(6), pages 935-946, June.
    14. Hilli, Amal & Laussel, Didier & Van Long, Ngo, 2013. "Large shareholders, monitoring, and ownership dynamics: Toward pure managerial firms?," Journal of Economic Dynamics and Control, Elsevier, vol. 37(3), pages 666-679.
    15. Ceddia, M.G. & Bardsley, N.O. & Goodwin, R. & Holloway, G.J. & Nocella, G. & Stasi, A., 2013. "A complex system perspective on the emergence and spread of infectious diseases: Integrating economic and ecological aspects," Ecological Economics, Elsevier, vol. 90(C), pages 124-131.
    16. Denis Claude & Charles Figuières & Mabel Tidball, 2008. "Short-run stick and long-run carrot policy: the role of initial conditions," Working Papers 08-04, LAMETA, Universtiy of Montpellier, revised Feb 2008.
    17. Anna Castañer & Jesús Marín-Solano & Carmen Ribas, 2021. "A time consistent dynamic bargaining procedure in differential games with hterogeneous discounting," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 93(3), pages 555-584, June.
    18. Günther Rehme, 2023. "Capital depreciation allowances, redistributive taxation, and economic growth," Journal of Public Economic Theory, Association for Public Economic Theory, vol. 25(1), pages 168-195, February.
    19. Stergios Athanassoglou & Glenn Sheriff & Tobias Siegfried & Woonghee Huh, 2012. "Optimal Mechanisms for Heterogeneous Multi-Cell Aquifers," Environmental & Resource Economics, Springer;European Association of Environmental and Resource Economists, vol. 52(2), pages 265-291, June.
    20. Wirl, Franz, 2014. "Taxes versus permits as incentive for the intertemporal supply of a clean technology by a monopoly," Resource and Energy Economics, Elsevier, vol. 36(1), pages 248-269.

    More about this item

    NEP fields

    This paper has been announced in the following NEP Reports:

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:arx:papers:2207.05303. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: arXiv administrators (email available below). General contact details of provider: http://arxiv.org/ .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.