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The Complexity of Solving Stochastic Games on Graphs

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Algorithms and Computation (ISAAC 2009)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 5878))

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Abstract

We consider some well-known families of two-player zero-sum perfect-information stochastic games played on finite directed graphs. The families include stochastic parity games, stochastic mean payoff games, and simple stochastic games. We show that the tasks of solving games in each of these classes (quantitiatively or strategically) are all polynomial time equivalent. In addition, we exhibit a linear time algorithm that given a simple stochastic game and the values of all positions of that game, computes a pair of optimal strategies.

Work supported by Center for Algorithmic Game Theory, funded by the Carlsberg Foundation. A large fraction of the results of this paper appeared in a preprint [10], co-authored by Vladimir Gurvich and the second author of this paper. Vladimir Gurvich’s contributions to that preprint will appear elsewhere.

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References

  1. Andersson, D., Hansen, K.A., Miltersen, P.B., Sørensen, T.B.: Deterministic graphical games revisited. In: Beckmann, A., Dimitracopoulos, C., Löwe, B. (eds.) CiE 2008. LNCS, vol. 5028, pp. 1–10. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  2. Chatterjee, K., Henzinger, T.A.: Reduction of stochastic parity to stochastic mean-payoff games. Inf. Process. Lett. 106(1), 1–7 (2008)

    Article  MathSciNet  Google Scholar 

  3. Chatterjee, K., Jurdziński, M., Henzinger, T.: Simple stochastic parity games. In: Baaz, M., Makowsky, J.A. (eds.) CSL 2003. LNCS, vol. 2803, pp. 100–113. Springer, Heidelberg (2003)

    Google Scholar 

  4. Chatterjee, K., Jurdziński, M., Henzinger, T.A.: Quantitative stochastic parity games. In: SODA 2004: Proceedings of the fifteenth annual ACM-SIAM symposium on Discrete algorithms, pp. 121–130 (2004)

    Google Scholar 

  5. Condon, A.: The complexity of stochastic games. Information and Computation 96, 203–224 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  6. Emerson, E.A., Jutla, C.S., Sistla, A.P.: On model checking for the mu-calculus and its fragments. Theoretical Computer Science 258(1-2), 491–522 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  7. Gillette, D.: Stochastic games with zero stop probabilities. In: Contributions to the Theory of Games III. Annals of Math. Studies, vol. 39, pp. 179–187. Princeton University Press, Princeton (1957)

    Google Scholar 

  8. Gimbert, H., Horn, F.: Simple Stochastic Games with Few Random Vertices are Easy to Solve. In: Amadio, R.M. (ed.) FOSSACS 2008. LNCS, vol. 4962, pp. 5–19. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  9. Gurvich, V., Karzanov, A., Khachiyan, L.: Cyclic games and an algorithm to find minimax cycle means in directed graphs. USSR Computational Mathematics and Mathematical Physics 28, 85–91 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  10. Gurvich, V., Miltersen, P.B.: On the computational complexity of solving stochastic mean-payoff games. arXiv:0812.0486v1 [cs.GT] (2008)

    Google Scholar 

  11. Halman, N.: Simple stochastic games, parity games, mean payoff games and discounted payoff games are all LP-type problems. Algorithmica 49(1), 37–50 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  12. Howard, R.A.: Dynamic Programming and Markov Processes. MIT Press, Cambridge (1960)

    MATH  Google Scholar 

  13. Liggett, T.M., Lippman, S.A.: Stochastic games with perfect information and time average payoff. SIAM Review 11(4), 604–607 (1969)

    Article  MATH  MathSciNet  Google Scholar 

  14. Littman, M.L.: Algorithms for sequential decision making. PhD thesis, Brown University, Department of Computer Science (1996)

    Google Scholar 

  15. McNaughton, R.: Infinite games played on finite graphs. An. Pure and Applied Logic 65, 149–184 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  16. Shapley, L.: Stochastic games. Proc. Nat. Acad. Science 39, 1095–1100 (1953)

    Article  MATH  MathSciNet  Google Scholar 

  17. Zwick, U., Paterson, M.: The complexity of mean payoff games on graphs. Theor. Comput. Sci. 158(1-2), 343–359 (1996)

    Article  MATH  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Computer Science, Aarhus University, Denmark

    Daniel Andersson & Peter Bro Miltersen

Authors
  1. Daniel Andersson
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  2. Peter Bro Miltersen
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Editor information

Editors and Affiliations

  1. Dept. of Electrical and Computer Engineering, University of Hawaii, Holmes Hall 483, 2540 Dole Street, 96822, Honolulu, HI, USA

    Yingfei Dong

  2. Department of Computer Science, University of Texas at Dallas, 75083, Dallas, TX, USA

    Ding-Zhu Du

  3. Department of Computer Science, University of California, 93106, Santa Barbara, CA, USA

    Oscar Ibarra

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Andersson, D., Miltersen, P.B. (2009). The Complexity of Solving Stochastic Games on Graphs. In: Dong, Y., Du, DZ., Ibarra, O. (eds) Algorithms and Computation. ISAAC 2009. Lecture Notes in Computer Science, vol 5878. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-10631-6_13

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  • DOI: https://doi.org/10.1007/978-3-642-10631-6_13

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-10630-9

  • Online ISBN: 978-3-642-10631-6

  • eBook Packages: Computer ScienceComputer Science (R0)

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