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A many-step two-person game is studied with a fixed sequence of moves under aggregated information on every move at the decision making instant and on the choice of player 2 at the current move. Player 1, knowing this information at every step i, first chooses a strategy xi(·) = (x1(·),…, xn(·)), i = , n, and informs it for n moves to player 2 at the beginning of the game. His maximal guaranteed result and the corresponding optimal (ε-optimal) strategy are determined. Such games under complete (nonaggregated) information are formulated and a compact expression for the strategy of player 1 is derived.