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


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.


Normal Distribution Probability Distribution Density Function Actual State Economic Theory 
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  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

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