Journal of Theoretical Probability

, Volume 26, Issue 2, pp 557–567

# 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

60G42 91A20

## Notes

### Acknowledgements

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

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