IDEAS home Printed from https://ideas.repec.org/a/spr/annopr/v295y2020i2d10.1007_s10479-020-03750-1.html
   My bibliography  Save this article

On Lemke processibility of LCP formulations for solving discounted switching control stochastic games

Author

Listed:
  • N. Krishnamurthy

    (Indian Institute of Management (IIM) Indore)

  • S. K. Neogy

    (Indian Statistical Institute, Delhi Centre)

Abstract

Schultz (J Optim Theory Appl 73(1):89–99, 1992) formulated 2-person, zero-sum, discounted switching control stochastic games as a linear complementarity problem (LCP) and discussed computational results. It remained open to prove or disprove Lemke-processibility of this LCP. We settle this question by providing a counter example to show that Lemke’s algorithm does not always successfully process this LCP.We propose a new LCP formulation with the aim of making the underlying matrix belong to the classes R $$_{0}$$ 0 and E $$_{0}$$ 0 , which would imply Lemke processibility. While the underlying matrix in the new formulation is not $$E_0$$ E 0 , we show that it is an R $$_{0}$$ 0 -matrix. Successful processing of Lemke’s algorithm depends on the choice of the initial vector d. Because of the special structure of the LCP in our context, we may, in fact, be able to find a suitable d such that our LCPs are processible by Lemke’s algorithm. We leave this open.

Suggested Citation

  • N. Krishnamurthy & S. K. Neogy, 2020. "On Lemke processibility of LCP formulations for solving discounted switching control stochastic games," Annals of Operations Research, Springer, vol. 295(2), pages 633-644, December.
    • Handle: RePEc:spr:annopr:v:295:y:2020:i:2:d:10.1007_s10479-020-03750-1
    DOI: 10.1007/s10479-020-03750-1
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10479-020-03750-1
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s10479-020-03750-1?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. S. R. Mohan & S. K. Neogy & T. Parthasarathy, 2001. "Pivoting Algorithms For Some Classes Of Stochastic Games: A Survey," International Game Theory Review (IGTR), World Scientific Publishing Co. Pte. Ltd., vol. 3(02n03), pages 253-281.
    2. C. E. Lemke, 1965. "Bimatrix Equilibrium Points and Mathematical Programming," Management Science, INFORMS, vol. 11(7), pages 681-689, May.
    3. Matthew J. Sobel, 1981. "Myopic Solutions of Markov Decision Processes and Stochastic Games," Operations Research, INFORMS, vol. 29(5), pages 995-1009, October.
    4. Nowak, Andrzej S. & Szajowski, Krzysztof, 1998. "Nonzero-sum Stochastic Games," MPRA Paper 19995, University Library of Munich, Germany, revised 1999.
    5. M. Seetharama Gowda & Jong-Shi Pang, 1992. "On Solution Stability of the Linear Complementarity Problem," Mathematics of Operations Research, INFORMS, vol. 17(1), pages 77-83, February.
    6. T. E. S. Raghavan & Zamir Syed, 2002. "Computing Stationary Nash Equilibria of Undiscounted Single-Controller Stochastic Games," Mathematics of Operations Research, INFORMS, vol. 27(2), pages 384-400, May.
    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. Prasenjit Mondal, 2018. "Completely mixed strategies for single controller unichain semi-Markov games with undiscounted payoffs," Operational Research, Springer, vol. 18(2), pages 451-468, July.
    2. S. K. Neogy & Prasenjit Mondal & Abhijit Gupta & Debasish Ghorui, 2018. "On Solving Mean Payoff Games Using Pivoting Algorithms," Asia-Pacific Journal of Operational Research (APJOR), World Scientific Publishing Co. Pte. Ltd., vol. 35(05), pages 1-26, October.
    3. S. R. Mohan, 1997. "Degeneracy Subgraph of the Lemke Complementary Pivot Algorithm and Anticycling Rule," Journal of Optimization Theory and Applications, Springer, vol. 94(2), pages 409-423, August.
    4. van der Laan, G. & Talman, A.J.J. & Yang, Z.F., 2005. "Computing Integral Solutions of Complementarity Problems," Other publications TiSEM b8e0c74e-2219-4ab0-99a2-0, Tilburg University, School of Economics and Management.
    5. Hanna Sumita & Naonori Kakimura & Kazuhisa Makino, 2015. "The Linear Complementarity Problems with a Few Variables per Constraint," Mathematics of Operations Research, INFORMS, vol. 40(4), pages 1015-1026, October.
    6. P. Herings & Ronald Peeters, 2010. "Homotopy methods to compute equilibria in game theory," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 42(1), pages 119-156, January.
    7. Ting Zhang & Yong Wang & Zheng-Hai Huang, 2024. "Projected fixed-point method for vertical tensor complementarity problems," Computational Optimization and Applications, Springer, vol. 89(1), pages 219-245, September.
    8. Dipti Dubey & S. K. Neogy & Debasish Ghorui, 2017. "Completely Mixed Strategies for Generalized Bimatrix and Switching Controller Stochastic Game," Dynamic Games and Applications, Springer, vol. 7(4), pages 535-554, December.
    9. van der Laan, G. & Talman, A.J.J. & Yang, Z.F., 2005. "Solving Discrete Zero Point Problems with Vector Labeling," Other publications TiSEM 9bd940ee-3fe6-4201-aede-7, Tilburg University, School of Economics and Management.
    10. M. Seetharama Gowda, 2019. "Weighted LCPs and interior point systems for copositive linear transformations on Euclidean Jordan algebras," Journal of Global Optimization, Springer, vol. 74(2), pages 285-295, June.
    11. Christian Bidard, 2015. "An oddity property for cross-dual games," Working Papers hal-04141427, HAL.
    12. Senlai Zhu & Jie Ma & Tianpei Tang & Quan Shi, 2020. "A Combined Modal and Route Choice Behavioral Complementarity Equilibrium Model with Users of Vehicles and Electric Bicycles," IJERPH, MDPI, vol. 17(10), pages 1-18, May.
    13. R. B. Bapat & S. K. Neogy, 2016. "On a quadratic programming problem involving distances in trees," Annals of Operations Research, Springer, vol. 243(1), pages 365-373, August.
    14. Rahul Savani & Bernhard von Stengel, 2016. "Unit vector games," International Journal of Economic Theory, The International Society for Economic Theory, vol. 12(1), pages 7-27, March.
    15. Christian Bidard, 2010. "Complementarity Problems and General Equilibrium," Working Papers hal-04140923, HAL.
    16. K. Ahmad & K. R. Kazmi & N. Rehman, 1997. "Fixed-Point Technique for Implicit Complementarity Problem in Hilbert Lattice," Journal of Optimization Theory and Applications, Springer, vol. 93(1), pages 67-72, April.
    17. Hanna Sumita & Naonori Kakimura & Kazuhisa Makino, 2019. "Total dual integrality of the linear complementarity problem," Annals of Operations Research, Springer, vol. 274(1), pages 531-553, March.
    18. Jugal Garg & Ruta Mehta & Vijay V. Vaziranic, 2018. "Substitution with Satiation: A New Class of Utility Functions and a Complementary Pivot Algorithm," Mathematics of Operations Research, INFORMS, vol. 43(3), pages 996-1024, August.
    19. Christian Bidard, 2014. "The Ricardian rent theory two centuries after," Working Papers hal-04141289, HAL.
    20. Zheng-Hai Huang & Liqun Qi, 2019. "Tensor Complementarity Problems—Part III: Applications," Journal of Optimization Theory and Applications, Springer, vol. 183(3), pages 771-791, December.

    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:spr:annopr:v:295:y:2020:i:2:d:10.1007_s10479-020-03750-1. 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: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    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.