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



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.


Maximal martingale variation Posteriors variation Repeated games with incomplete information 

Mathematics Subject Classification (2000)

60G42 91A20 



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


  1. 1.
    Aumann, R.J., Maschler, M.: Repeated Games with Incomplete Information. MIT Press, Cambridge (1995), with the collaboration of R. Stearns MATHGoogle Scholar
  2. 2.
    Cover, T.M., Thomas, J.A.: Elements of Information Theory. Wiley Series in Telecommunications. Wiley, New York (1991) MATHCrossRefGoogle Scholar
  3. 3.
    Mertens, J.-F., Zamir, S.: The value of two-person zero-sum repeated games with lack of information on both sides. Int. J. Game Theory 1, 39–64 (1971) MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Mertens, J.-F., Zamir, S.: The maximal variation of a bounded martingale. Isr. J. Math. 27, 252–276 (1977) MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Zamir, S.: On the relation between finitely and infinitely repeated games with incomplete information. Int. J. Game Theory 1, 179–198 (1972) MathSciNetMATHCrossRefGoogle Scholar

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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