International Journal of Game Theory

, Volume 5, Issue 4, pp 187–197 | Cite as

The normal distribution and repeated games

  • J. -F. Mertens
  • S. Zamir
Papers

Abstract

For a reperated zero-sum two-person game with incomplete information discussed byZamir, it is proved here that\(\mathop {\lim }\limits_{n \to \infty } \sqrt n v_n (p) = \phi (p)\) whereφ (p) is the normal density function evaluated at itsp-quantile (i.e.\(\phi (p) = \frac{1}{{\sqrt {2\pi } }}e^{ - ({1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2})x^2 } p\) where\(\frac{1}{{\sqrt {2\pi } }}\mathop {\smallint ^p }\limits_{ - \infty }^x e^{ - ({1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2})x^2 } dx = p\). Here for 0⩽p⩽1, (p, 1 −p) is the a priori probability distribution on two states of nature, the actual state of nature is known to the maximizer but not to the minimizer.v n (p) is the minimax value of the game withn stages.

Keywords

Normal Distribution Probability Distribution Density Function Actual State Economic Theory 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Aumann, R.J., andM. Maschler: Game Theoretic Aspects of Gradual Disarmament. Report of the U.S. Arms Control and Disarmament Agency. Washington D.C. Final report on Contract ACDA ST-80, prepared by MATHEMATICA, Princeton N.J. Chapter V, June 1966.Google Scholar
  2. Mertens, J.F., andS. Zamir: The Maximal Variation of a Bounded Martingale, The Hebrew University, Center for Research in Mathematical Economics and Game Theory, Research Memorandum No. 7. June 1975.Google Scholar
  3. Zamir, S.: On the Relation Between Finitely and Infinitely Repeated Games with Incomplete information, International Journal of Game Theory,1 (3), 1971–1972, 179–198.Google Scholar

Copyright information

© Physica-Verlag 1976

Authors and Affiliations

  • J. -F. Mertens
    • 1
  • S. Zamir
    • 2
  1. 1.Universitc Catholique de LouvainLouvainBelgium
  2. 2.The Hebrew University of JerusalemJerusalemIsreal

Personalised recommendations