Dynamic Games and Applications

, Volume 9, Issue 4, pp 1026–1041 | Cite as

Zero-Sum Stochastic Games over the Field of Real Algebraic Numbers

  • K. Avrachenkov
  • V. Ejov
  • J. A. FilarEmail author
  • A. Moghaddam


We consider a finite state, finite action, zero-sum stochastic games with data defining the game lying in the ordered field of real algebraic numbers. In both the discounted and the limiting average versions of these games, we prove that the value vector also lies in the same field of real algebraic numbers. Our method supplies finite construction of univariate polynomials whose roots contain these value vectors. In the case where the data of the game are rational, the method also provides a way of checking whether the entries of the value vectors are also rational.


Stochastic games Ordered field property Algebraic numbers Algebraic variety Gröbner basis polynomial equations 

Mathematics Subject Classification




  1. 1.
    Bewley T, Kohlberg E (1976) The asymptotic theory of stochastic games. Math Oper Res 1:197–208MathSciNetCrossRefGoogle Scholar
  2. 2.
    Filar JA (1981) Ordered field property for stochastic games when the player who controls transitions changes from state to state. J Optim Theory Appl 34:503–515MathSciNetCrossRefGoogle Scholar
  3. 3.
    Frederiksen S (2015) Semi-algebraic tools for stochastic games. Ph.D. thesis, Aarhus Universitet, Aarhus, DenmarkGoogle Scholar
  4. 4.
    Gillette D (1957) Stochastic games with zero stop probabilities. Contrib Theory Games 3:179–187MathSciNetzbMATHGoogle Scholar
  5. 5.
    Hansen K, Koucky M, Lauritzen N, Miltersen P, Tsigaridas E (2011) Exact algorithms for solving stochastic games: extended abstract. In: STOC 11 proceedings of the forty-third annual ACM symposium on theory of computing, pp 205–214Google Scholar
  6. 6.
    Jaśkiewicz A, Nowak AS (2018) Nonzero-sum stochastic games. Springer, Berlin, pp 281–344Google Scholar
  7. 7.
    Jaśkiewicz A, Nowak AS (2018) Zero-sum stochastic games. Springer, Berlin, pp 215–280Google Scholar
  8. 8.
    Kaplansky I (1945) A contribution to von Neumann’s theory of games. Ann Math 46:474–479MathSciNetCrossRefGoogle Scholar
  9. 9.
    Loustaunau P, Adams WW (1994) An introduction to Gröbner bases. American Mathematical Society, ProvidencezbMATHGoogle Scholar
  10. 10.
    MacLane S, Birkhoff G (1967) Algebra. The Macmillan Company, New YorkzbMATHGoogle Scholar
  11. 11.
    Mertens JF, Neyman A (1981) Stochastic games. Int J Game Theory 10:53–66MathSciNetCrossRefGoogle Scholar
  12. 12.
    Mertens JF, Zamir S (1981) Incomplete information games with transcendental values. Math Oper Res 6:313–318MathSciNetCrossRefGoogle Scholar
  13. 13.
    Milman E (2002) The semi-algebraic theory of stochastic games. Math Oper Res 27:401–418MathSciNetCrossRefGoogle Scholar
  14. 14.
    Neyman A (2003) Real algebraic tools in stochastic games. Springer, Dordrecht, pp 57–75zbMATHGoogle Scholar
  15. 15.
    Parthasarathy T, Raghavan TES (1981) An orderfield property for stochastic games when one player controls transition probabilities. J Optim Theory Appl 33:375–392MathSciNetCrossRefGoogle Scholar
  16. 16.
    Parthasarathy T, Tijs SH, Vrieze OJ (1984) Stochastic games with state independent transitions and separable rewards. Springer, Berlin, pp 262–271zbMATHGoogle Scholar
  17. 17.
    Raghavan TES, Filar JA (1991) Algorithms for stochastic games: a survey. Z Oper Res 35:437–472MathSciNetzbMATHGoogle Scholar
  18. 18.
    Raghavan TES, Tijs SH, Vrieze OJ (1985) On stochastic games with additive reward and transition structure. J Optim Theory Appl 47:451–464MathSciNetCrossRefGoogle Scholar
  19. 19.
    Rotman J (1998) Galois theory. Universitext. Springer, New York (Berlin. Print)CrossRefGoogle Scholar
  20. 20.
    Shapley LS (1953) Stochastic games. Proc Natl Acad Sci 39:1095–1100MathSciNetCrossRefGoogle Scholar
  21. 21.
    Shapley LS, Snow RN (1952) Basic solutions of discrete games. Princeton University Press, Princeton, pp 27–36Google Scholar
  22. 22.
    Sobel MJ (1981) Myopic solutions of markov decision processes and stochastic games. Oper Res 29:995–1009MathSciNetCrossRefGoogle Scholar
  23. 23.
    Szczechla WW, Connell SA, Filar JA, Vrieze OJ (1997) On the Puiseux series expansion of the limit discount equation of stochastic games. SIAM J Control Optim 35:860–875MathSciNetCrossRefGoogle Scholar
  24. 24.
    Tarski A (1951) A decision method for elementary algebra and geometry. University of California Press, BerkeleyzbMATHGoogle Scholar
  25. 25.
    von Neumann J (1928) Zur theorie der gesellschaftsspiele. Math Ann 100:295–320MathSciNetCrossRefGoogle Scholar
  26. 26.
    Weyl H (1952) Elementary proof of a minimax theorem due to Von Neumann. Princeton University Press, Princeton, pp 19–26Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Inria Sophia AntipolisBiotFrance
  2. 2.College of Science and EngineeringFlinders University of South AustraliaBedford ParkAustralia
  3. 3.Faculty of Mechanics and MathematicsMSUMoscowRussia
  4. 4.Centre for Applications in Natural Resource Mathematics, School of Mathematics and PhysicsThe University of QueenslandSt LuciaAustralia

Personalised recommendations