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Ordered Field Property for Semi-Markov Games when One Player Controls Transition Probabilities and Transition Times

Author

Listed:
  • Prasenjit Mondal

    (Mathematics Department, Jadavpur University, Kolkata 700032, India)

  • Sagnik Sinha

    (Mathematics Department, Jadavpur University, Kolkata 700032, India)

Abstract

Two-person finite semi-Markov games (SMGs) are studied when the transition probabilities and the transition times are controlled by one player at all states. For the discounted games in this class, we prove that the ordered field property holds and there exist optimal/Nash equilibrium stationary strategies for the players. We illustrate that the zero-sum SMGs where only transition probabilities are controlled by one player, do not necessarily satisfy the ordered field property. An algorithm along with a numerical example for the discounted one player control zero-sum SMGs is given via linear programming. For the undiscounted version of such games, we exhibit with an example that if the game ceases to be unichain, an optimal stationary or Markov strategy need not exist, (though in this example of a one-player game we exhibit a semi-stationary optimal strategy/policy). Lastly, we prove that if such games are unichain, then they possess the ordered field property for the undiscounted case as well.

Suggested Citation

  • Prasenjit Mondal & Sagnik Sinha, 2015. "Ordered Field Property for Semi-Markov Games when One Player Controls Transition Probabilities and Transition Times," International Game Theory Review (IGTR), World Scientific Publishing Co. Pte. Ltd., vol. 17(02), pages 1-26.
    • Handle: RePEc:wsi:igtrxx:v:17:y:2015:i:02:n:s0219198915400228
    DOI: 10.1142/S0219198915400228
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    Cited by:

    1. Prasenjit Mondal, 2020. "Computing semi-stationary optimal policies for multichain semi-Markov decision processes," Annals of Operations Research, Springer, vol. 287(2), pages 843-865, April.
    2. 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.
    3. Prasenjit Mondal, 2015. "Linear Programming and Zero-Sum Two-Person Undiscounted Semi-Markov Games," Asia-Pacific Journal of Operational Research (APJOR), World Scientific Publishing Co. Pte. Ltd., vol. 32(06), pages 1-20, December.
    4. Prasenjit Mondal, 2016. "On undiscounted semi-Markov decision processes with absorbing states," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 83(2), pages 161-177, April.

    More about this item

    Keywords

    Semi-Markov games; semi-Markov decision processes; semi-Markov policies; discounted and limiting average payoffs; ordered field property; multichain structure; finite step algorithms for semi-Markov games; 22E46; 90C40; 53C35; 57S20;
    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

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