Zero-Sum Games for Two Asymptotically Optimal Sequences of Automata

The author considers zero-sum games for two automata whose memory grows without limit. It is proved that for a relatively large class of asymptotically optimal sequences of automata, the limiting payoff always lies between the upper and lower values of the game. Inside these limits, the payoff depends only weakly on the matrix of the game and is determined primarily by the structures of the playing automata.