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

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## Abstract

The variation of a martingale \(p_{0}^{k}=p_{0},\ldots,p_{k}\) of probabilities on a finite (or countable) set It is shown that \(V(p_{0}^{k})\leq\sqrt{2kH(p_{0})}\), where

*X*is denoted \(V(p_{0}^{k})\) and defined by$$ V\bigl(p_0^k\bigr)=E\Biggl(\sum_{t=1}^k\|p_t-p_{t-1}\|_1\Biggr). $$

*H*(*p*) is the entropy function*H*(*p*)=−∑_{ x }*p*(*x*)log*p*(*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.

## References

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