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

The variation of a martingale p₀^k = p₀,...,p_k of probabilities on a finite (or countable) set X is denoted V(p₀^k) and defined by, It is shown that V(p₀^k ≤ √2k H (p₀), 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(p₀^k) ≤ √2k log d. It is shown that the order of magnitude of the bound √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^k₀ = p₀, ..., P_k of probabilities on a set X with d elements, and with variation V(p₀^k)≥ C√2k log d. An application of the first result to game theory is that the difference between v_k and lim_jv_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 {double pipe}G{double pipe}√2k^-1 log d (where {double pipe}G{double pipe} 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.