REPETITIVE PLAY OF AN UNKNOWN GAME AGAINST NATURE

A repetitive play of a game against Nature is considered under the assumption that the player knows nothing about the game except his own set of strategies. After each play, he is told the value of the random loss incurred by him. A strategic rule for the player is defined with the property that the average loss achieves asymptotically the minimum functional of the game in probability and uniformly in all sequences of Natures strategies. The rate of convergence of expected average losses is shown as well.