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Stationary, completely mixed and symmetric optimal and equilibrium strategies in stochastic games

Author

Listed:
  • Sujatha Babu

    (Indian Institute of Technology Madras)

  • Nagarajan Krishnamurthy

    (Indian Institute of Management Indore)

  • T. Parthasarathy

    (Indian Statistical Institute, Chennai Centre)

Abstract

In this paper, we address various types of two-person stochastic games—both zero-sum and nonzero-sum, discounted and undiscounted. In particular, we address different aspects of stochastic games, namely: (1) When is a two-person stochastic game completely mixed? (2) Can we identify classes of undiscounted zero-sum stochastic games that have stationary optimal strategies? (3) When does a two-person stochastic game possess symmetric optimal/equilibrium strategies? Firstly, we provide some necessary and some sufficient conditions under which certain classes of discounted and undiscounted stochastic games are completely mixed. In particular, we show that, if a discounted zero-sum switching control stochastic game with symmetric payoff matrices has a completely mixed stationary optimal strategy, then the stochastic game is completely mixed if and only if the matrix games restricted to states are all completely mixed. Secondly, we identify certain classes of undiscounted zero-sum stochastic games that have stationary optima under specific conditions for individual payoff matrices and transition probabilities. Thirdly, we provide sufficient conditions for discounted as well as certain classes of undiscounted stochastic games to have symmetric optimal/equilibrium strategies—namely, transitions are symmetric and the payoff matrices of one player are the transpose of those of the other. We also provide a sufficient condition for the stochastic game to have a symmetric pure strategy equilibrium. We also provide examples to show the sharpness of our results.

Suggested Citation

  • Sujatha Babu & Nagarajan Krishnamurthy & T. Parthasarathy, 2017. "Stationary, completely mixed and symmetric optimal and equilibrium strategies in stochastic games," International Journal of Game Theory, Springer;Game Theory Society, vol. 46(3), pages 761-782, August.
    • Handle: RePEc:spr:jogath:v:46:y:2017:i:3:d:10.1007_s00182-016-0555-5
    DOI: 10.1007/s00182-016-0555-5
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    References listed on IDEAS

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    1. Parthasarathy, T & Sinha, S, 1989. "Existence of Stationary Equilibrium Strategies in Non-zero Sum Discounted Stochastic Games with Uncountable State Space and State-Independent Transitions," International Journal of Game Theory, Springer;Game Theory Society, vol. 18(2), pages 189-194.
    2. Peter Duersch & Jörg Oechssler & Burkhard Schipper, 2012. "Pure strategy equilibria in symmetric two-player zero-sum games," International Journal of Game Theory, Springer;Game Theory Society, vol. 41(3), pages 553-564, August.
    3. János Flesch & Thiruvenkatachari Parthasarathy & Frank Thuijsman & Philippe Uyttendaele, 2013. "Evolutionary Stochastic Games," Dynamic Games and Applications, Springer, vol. 3(2), pages 207-219, June.
    4. Partha Dasgupta & Eric Maskin, 1986. "The Existence of Equilibrium in Discontinuous Economic Games, I: Theory," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 53(1), pages 1-26.
    5. Philip J. Reny, 1999. "On the Existence of Pure and Mixed Strategy Nash Equilibria in Discontinuous Games," Econometrica, Econometric Society, vol. 67(5), pages 1029-1056, September.
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    Cited by:

    1. K. C. Sivakumar & M. S. Gowda & G. Ravindran & Usha Mohan, 2020. "Preface: International conference on game theory and optimization, June 6–10, 2016, Indian Institute of Technology Madras, Chennai, India," Annals of Operations Research, Springer, vol. 287(2), pages 565-572, April.

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