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A Potential Reduction Algorithm for Two-Person Zero-Sum Mean Payoff Stochastic Games

Author

Listed:
  • Endre Boros

    (Rutgers University)

  • Khaled Elbassioni

    (Masdar Institute of Science and Technology)

  • Vladimir Gurvich

    (Rutgers University
    Higher School of Economics (HSE))

  • Kazuhisa Makino

    (Kyoto University)

Abstract

We suggest a new algorithm for two-person zero-sum undiscounted stochastic games focusing on stationary strategies. Given a positive real $$\varepsilon $$ ε , let us call a stochastic game $$\varepsilon $$ ε -ergodic, if its values from any two initial positions differ by at most $$\varepsilon $$ ε . The proposed new algorithm outputs for every $$\varepsilon >0$$ ε > 0 in finite time either a pair of stationary strategies for the two players guaranteeing that the values from any initial positions are within an $$\varepsilon $$ ε -range, or identifies two initial positions u and v and corresponding stationary strategies for the players proving that the game values starting from u and v are at least $$\varepsilon /24$$ ε / 24 apart. In particular, the above result shows that if a stochastic game is $$\varepsilon $$ ε -ergodic, then there are stationary strategies for the players proving $$24\varepsilon $$ 24 ε -ergodicity. This result strengthens and provides a constructive version of an existential result by Vrieze (Stochastic games with finite state and action spaces. PhD thesis, Centrum voor Wiskunde en Informatica, Amsterdam, 1980) claiming that if a stochastic game is 0-ergodic, then there are $$\varepsilon $$ ε -optimal stationary strategies for every $$\varepsilon > 0$$ ε > 0 . The suggested algorithm is based on a potential transformation technique that changes the range of local values at all positions without changing the normal form of the game.

Suggested Citation

  • Endre Boros & Khaled Elbassioni & Vladimir Gurvich & Kazuhisa Makino, 2018. "A Potential Reduction Algorithm for Two-Person Zero-Sum Mean Payoff Stochastic Games," Dynamic Games and Applications, Springer, vol. 8(1), pages 22-41, March.
  • Handle: RePEc:spr:dyngam:v:8:y:2018:i:1:d:10.1007_s13235-016-0199-x
    DOI: 10.1007/s13235-016-0199-x
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    References listed on IDEAS

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    1. Endre Boros & Khaled Elbassioni & Vladimir Gurvich & Kazuhisa Makino, 2013. "On Canonical Forms for Zero-Sum Stochastic Mean Payoff Games," Dynamic Games and Applications, Springer, vol. 3(2), pages 128-161, June.
    2. Awi Federgruen, 1980. "Successive Approximation Methods in Undiscounted Stochastic Games," Operations Research, INFORMS, vol. 28(3-part-ii), pages 794-809, June.
    3. Krishnendu Chatterjee & Rupak Majumdar & Thomas Henzinger, 2008. "Stochastic limit-average games are in EXPTIME," International Journal of Game Theory, Springer;Game Theory Society, vol. 37(2), pages 219-234, June.
    4. A. J. Hoffman & R. M. Karp, 1966. "On Nonterminating Stochastic Games," Management Science, INFORMS, vol. 12(5), pages 359-370, January.
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