AbstractThe 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)=−∑xp(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≤2k: there is C>0 such that for all k and d≤2k, 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 vk and limjvj, where vk 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.
[1]
Jérôme Renault,et al.
Repeated Games with Incomplete Information
,
2009,
Encyclopedia of Complexity and Systems Science.
[2]
Thomas M. Cover,et al.
Elements of Information Theory (Wiley Series in Telecommunications and Signal Processing)
,
2006
.
[3]
E D MARTINGALE,et al.
The Maximal Variation of a Bounded Martingale
,
2007
.
[4]
Shmuel Zamir,et al.
On the relation between finitely and infinitely repeated games with incomplete information
,
1971
.
[5]
Jean-François Mertens,et al.
The value of two-person zero-sum repeated games with lack of information on both sides
,
1971
.
[6]
Thomas M. Cover,et al.
Elements of Information Theory
,
2005
.