# The normal distribution and repeated games

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## Abstract

For a reperated zero-sum two-person game with incomplete information discussed by*Zamir*, it is proved here that\(\mathop {\lim }\limits_{n \to \infty } \sqrt n v_n (p) = \phi (p)\) where*φ* (*p*) is the normal density function evaluated at its*p*-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 with*n* stages.

## Keywords

Normal Distribution Probability Distribution Density Function Actual State Economic Theory## Preview

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## References

*Aumann, R.J.*, and*M. 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*Mertens, J.F.*, and*S. 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*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