Ordered field property for stochastic games when the player who controls transitions changes from state to state

  • J. A. Filar
Contributed Papers


In this paper, we consider a zero-sum stochastic game with finitely many states restricted by the assumption that the probability transitions from a given state are functions of the actions of only one of the players. However, the player who thus controls the transitions in the given state will not be the same in every state. Further, we assume that all payoffs and all transition probabilities specifying the law of motion are rational numbers. We then show that the values of both a β-discounted game, for rational β, and of a Cesaro-average game are in the field of rational numbers. In addition, both games possess optimal stationary strategies which have only rational components. Our results and their proofs form an extension of the results and techniques which were recently developed by Parthasarathy and Raghavan (Ref. 1).

Key Words

Stochastic games discounting undiscounted stochastic games stationary strategies Cesaro-average payoff probability transitions Archimedean field 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Parthasarathy, T., andRaghavan, T. E. S.,An Order Field Property for Stochastic Games when One Player Controls Transitions, Journal of Optimization Theory and Applications, Vol. 33, No. 3, 1981.Google Scholar
  2. 2.
    Filar, J. A., andRaghavan, T. E. S.,An Algorithm for Solving an Undiscounted Stochastic Game in Which One Player Controls Transitions, Research Memorandum, University of Illinois, Chicago, Illinois, 1979.Google Scholar
  3. 3.
    Bewley, T., andKohlberg, E.,On Stochastic Games with Stationary Strategies, Mathematics of Operations Research, Vol. 3, pp. 104–125, 1978.Google Scholar
  4. 4.
    Blackwell, D.,Discrete Dynamic Programming, Annals of Mathematical Statistics, Vol. 33, pp. 719–726, 1962.Google Scholar
  5. 5.
    Shapley, L. S.,Stochastic Games, Proceedings of the National Academy of Science, Vol. 39, pp. 1095–1100, 1953.Google Scholar
  6. 6.
    Shapley, L. S., andSnow, R. N.,Basic Solutions of Discrete Games, Annals of Mathematics Studies, Princeton University Press, Princeton, New Jersey, Vol. 24, pp. 27–37, 1950.Google Scholar

Copyright information

© Plenum Publishing Corporation 1981

Authors and Affiliations

  • J. A. Filar
    • 1
  1. 1.Department of Applied Statistics, School of StatisticsUniversity of MinnesotaSt. Paul

Personalised recommendations