IDEAS home Printed from https://ideas.repec.org/a/wsi/igtrxx/v16y2014i02ns0219198914400076.html
   My bibliography  Save this article

Solution Of A Singular Zero-Sum Linear-Quadratic Differential Game By Regularization

Author

Listed:
  • JOSEF SHINAR

    (Technion — Israel Institute of Technology, Haifa 32000, Israel)

  • VALERY Y. GLIZER

    (Department of Applied Mathematics, Ort Braude College of Engineering, P.O. Box 78, Karmiel 21982, Israel)

  • VLADIMIR TURETSKY

    (Department of Applied Mathematics, Ort Braude College of Engineering, P.O. Box 78, Karmiel 21982, Israel)

Abstract

A linear-quadratic zero-sum singular differential game, where the cost functional does not contain the minimizer's control cost, is considered. Due to the singularity, the game cannot be solved either by applying the MinMax principle of Isaacs, or by using the Bellman–Isaacs equation method. In this paper, the solution of the singular game is obtained by using an auxiliary differential game with the same equation of dynamics and with a similar cost functional augmented by an integral of the square of the minimizer's control multiplied by a small positive weighting coefficient. This auxiliary game is a regular cheap control zero-sum differential game. For the analysis of such a cheap control differential game, in the present paper a singular perturbation technique is applied. Based on this analysis, the minimizing control sequence and the maximizer's optimal strategy in the original (singular) game are derived. Moreover, the existence of the value of the original game is established and its expression is derived. The solution is illustrated by an interception example.

Suggested Citation

  • Josef Shinar & Valery Y. Glizer & Vladimir Turetsky, 2014. "Solution Of A Singular Zero-Sum Linear-Quadratic Differential Game By Regularization," International Game Theory Review (IGTR), World Scientific Publishing Co. Pte. Ltd., vol. 16(02), pages 1-32.
    • Handle: RePEc:wsi:igtrxx:v:16:y:2014:i:02:n:s0219198914400076
    DOI: 10.1142/S0219198914400076
    as

    Download full text from publisher

    File URL: http://www.worldscientific.com/doi/abs/10.1142/S0219198914400076
    Download Restriction: Access to full text is restricted to subscribers
    File URL: https://libkey.io/10.1142/S0219198914400076?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Richard Bellman, 1957. "On a Dynamic Programming Approach to the Caterer Problem--I," Management Science, INFORMS, vol. 3(3), pages 270-278, April.
    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. Pierre Bernhard & Marc Deschamps, 2017. "Kalman on dynamics and contro, Linear System Theory, Optimal Control, and Filter," Working Papers 2017-10, CRESE.
    2. Jones, Randall E. & Cacho, Oscar J., 2000. "A Dynamic Optimisation Model of Weed Control," 2000 Conference (44th), January 23-25, 2000, Sydney, Australia 123685, Australian Agricultural and Resource Economics Society.
    3. Voelkel, Michael A. & Sachs, Anna-Lena & Thonemann, Ulrich W., 2020. "An aggregation-based approximate dynamic programming approach for the periodic review model with random yield," European Journal of Operational Research, Elsevier, vol. 281(2), pages 286-298.
    4. Pam Norton & Ravi Phatarfod, 2008. "Optimal Strategies In One-Day Cricket," Asia-Pacific Journal of Operational Research (APJOR), World Scientific Publishing Co. Pte. Ltd., vol. 25(04), pages 495-511.
    5. Aghayi, Nazila & Maleki, Bentolhoda, 2016. "Efficiency measurement of DMUs with undesirable outputs under uncertainty based on the directional distance function: Application on bank industry," Energy, Elsevier, vol. 112(C), pages 376-387.
    6. Tan, Madeleine Sui-Lay, 2016. "Policy coordination among the ASEAN-5: A global VAR analysis," Journal of Asian Economics, Elsevier, vol. 44(C), pages 20-40.
    7. D. W. K. Yeung, 2008. "Dynamically Consistent Solution For A Pollution Management Game In Collaborative Abatement With Uncertain Future Payoffs," International Game Theory Review (IGTR), World Scientific Publishing Co. Pte. Ltd., vol. 10(04), pages 517-538.
    8. Crutchfield, Stephen R. & Brazee, Richard J., 1990. "An Integrated Model of Surface and Ground Water Quality," 1990 Annual meeting, August 5-8, Vancouver, Canada 271011, American Agricultural Economics Association (New Name 2008: Agricultural and Applied Economics Association).
    9. Hanafi, Said & Freville, Arnaud, 1998. "An efficient tabu search approach for the 0-1 multidimensional knapsack problem," European Journal of Operational Research, Elsevier, vol. 106(2-3), pages 659-675, April.
    10. Schön, Cornelia & König, Eva, 2018. "A stochastic dynamic programming approach for delay management of a single train line," European Journal of Operational Research, Elsevier, vol. 271(2), pages 501-518.
    11. Eric D. Gould, 2008. "Marriage and Career: The Dynamic Decisions of Young Men," Journal of Human Capital, University of Chicago Press, vol. 2(4), pages 337-378.
    12. Renato Cordeiro Amorim, 2016. "A Survey on Feature Weighting Based K-Means Algorithms," Journal of Classification, Springer;The Classification Society, vol. 33(2), pages 210-242, July.
    13. Dmitri Blueschke & Ivan Savin, 2015. "No such thing like perfect hammer: comparing different objective function specifications for optimal control," Jena Economics Research Papers 2015-005, Friedrich-Schiller-University Jena.
    14. Sieniutycz, Stanislaw, 2015. "Synthesizing modeling of power generation and power limits in energy systems," Energy, Elsevier, vol. 84(C), pages 255-266.
    15. Miller, Marcus & Papi, Laura, 1997. "The 'laissez faire' bias of managed floating," Journal of International Money and Finance, Elsevier, vol. 16(6), pages 989-1000, December.
    16. Changming Ji & Chuangang Li & Boquan Wang & Minghao Liu & Liping Wang, 2017. "Multi-Stage Dynamic Programming Method for Short-Term Cascade Reservoirs Optimal Operation with Flow Attenuation," Water Resources Management: An International Journal, Published for the European Water Resources Association (EWRA), Springer;European Water Resources Association (EWRA), vol. 31(14), pages 4571-4586, November.
    17. Ghassan, Hassan B. & Al-Jefri, Essam H., 2015. "الحساب الجاري في المدى البعيد عبر نموذج داخلي الزمن [The Current Account in the Long Run through the Intertemporal Model]," MPRA Paper 66527, University Library of Munich, Germany.
    18. David W. K. Yeung & Leon A. Petrosyan, 2014. "Subgame Consistent Cooperative Solutions For Randomly Furcating Stochastic Dynamic Games With Uncertain Horizon," International Game Theory Review (IGTR), World Scientific Publishing Co. Pte. Ltd., vol. 16(02), pages 1-29.
    19. John Stachurski, 2009. "Economic Dynamics: Theory and Computation," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262012774, December.
    20. Mercedes Esteban-Bravo & Jose M. Vidal-Sanz & Gökhan Yildirim, 2014. "Valuing Customer Portfolios with Endogenous Mass and Direct Marketing Interventions Using a Stochastic Dynamic Programming Decomposition," Marketing Science, INFORMS, vol. 33(5), pages 621-640, September.

    More about this item

    Keywords

    Singular differential game; regularization; cheap control; minimizing control sequence; 49N70; 91A23; 93C70;
    All these keywords.

    JEL classification:

    • B4 - Schools of Economic Thought and Methodology - - Economic Methodology
    • C0 - Mathematical and Quantitative Methods - - General
    • C6 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling
    • C7 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory
    • D5 - Microeconomics - - General Equilibrium and Disequilibrium
    • D7 - Microeconomics - - Analysis of Collective Decision-Making
    • M2 - Business Administration and Business Economics; Marketing; Accounting; Personnel Economics - - Business Economics

    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:wsi:igtrxx:v:16:y:2014:i:02:n:s0219198914400076. 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: Tai Tone Lim (email available below). General contact details of provider: http://www.worldscinet.com/igtr/igtr.shtml .

    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.