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Journal of Theoretical Probability

, Volume 26, Issue 2, pp 557–567 | Cite as

The Maximal Variation of Martingales of Probabilities and Repeated Games with Incomplete Information

  • Abraham Neyman
Article

Abstract

The variation of a martingale \(p_{0}^{k}=p_{0},\ldots,p_{k}\) of probabilities on a finite (or countable) set X is denoted \(V(p_{0}^{k})\) and defined by
$$ V\bigl(p_0^k\bigr)=E\Biggl(\sum_{t=1}^k\|p_t-p_{t-1}\|_1\Biggr). $$
It is shown that \(V(p_{0}^{k})\leq\sqrt{2kH(p_{0})}\), where H(p) is the entropy function H(p)=−∑ x p(x)logp(x), and log stands for the natural logarithm. Therefore, if d is the number of elements of X, then \(V(p_{0}^{k})\leq\sqrt{2k\log d}\). It is shown that the order of magnitude of the bound \(\sqrt{2k\log d}\) is tight for d≤2 k : there is C>0 such that for all k and d≤2 k , there is a martingale \(p_{0}^{k}=p_{0},\ldots,p_{k}\) of probabilities on a set X with d elements, and with variation \(V(p_{0}^{k})\geq C\sqrt{2k\log d}\). An application of the first result to game theory is that the difference between v k and lim j v j , where v k is the value of the k-stage repeated game with incomplete information on one side with d states, is bounded by \(\|G\|\sqrt{2k^{-1}\log d}\) (where ∥G∥ is the maximal absolute value of a stage payoff). Furthermore, it is shown that the order of magnitude of this game theory bound is tight.

Keywords

Maximal martingale variation Posteriors variation Repeated games with incomplete information 

Mathematics Subject Classification (2000)

60G42 91A20 

Notes

Acknowledgements

This research was supported in part by Israel Science Foundation grants 1123/06 and 1596/10.

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Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Institute of Mathematics, and Center for the Study of RationalityThe Hebrew University of JerusalemJerusalemIsrael

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