Abstract: Blackwell (1956a) proved a minimax theorem for games with a vector loss and characterized sets such that a player has a strategy under which, whatever strategy the other player uses, the average payoff approaches the set. Based on this result, Blackwell (1956b) described a strategy for a sequence of plays of a game, under which the average loss approaches the Bayes risk with respect to the relative frequencies of the opponent's actions. In both cases, the distance of the average loss from a set, in repeated plays of the game, was proved to converge to 0 with probability one. In both cases, we show that the rate of the convergence is better than (n/log1+tn)-1/2 for every positive t, obtain bounds for the L2 norm of the distance and extend the results to cases in which past losses and actions are only estimated.
Key words and phrases: Almost uniform convergence, Bayes risk, game, L2(E) convergence, sequence-compound statistical methods, sequence of plays, stochastic approximation, strategy, vector payoff.