ON BLACKWELL'S MINIMAX THEOREM AND THE COMPOUND DECISION METHOD

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/ log 1+t n) �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.

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