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

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  • 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
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    References listed on IDEAS

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    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. Nowak, Andrzej S. & Szajowski, Krzysztof, 1998. "Nonzero-sum Stochastic Games," MPRA Paper 19995, University Library of Munich, Germany, revised 1999.
    4. 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.
    5. 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.
    6. Matthew J. Sobel, 1981. "Myopic Solutions of Markov Decision Processes and Stochastic Games," Operations Research, INFORMS, vol. 29(5), pages 995-1009, October.
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