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

Abstract

The variation of a martingale $p_0^k=p_0,...,p_k$ of probabilities on a finite (or countable) set $X$ is denoted $V(p_0^k)$ and defined by $V(p_0^k)=E(\sum_{t=1}^k|p_t-p_{t-1}|_1)$. It is shown that $V(p_0^k)\leq \sqrt{2kH(p_0)}$, where $H(p)$ is the entropy function $H(p)=-\sum_xp(x)\log p(x)$ 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 $d\leq 2^k$: there is $C>0$ such that for every $k$ and $d\leq 2^k$ there is a martingale $p_0^k=p_0,...,p_k$ 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 $\lim_kv_k$, 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.