Taking Brouwer's intuitionistic standpoint, we examine finite and infinite games of perfect information for players I and II. If one understands the disjunction occurring in the classical notion of determinacy constructively, even finite games are not always determinate. We therefore suggest an intuitionistically different notion of determinacy and prove that every subset of Cantor space is determinate in the proposed sense. Our notion is biased and considers games from the viewpoint of player I. In Cantor space, both player I and player II have two alternative possibilities for each move. It turns out that every subset of a space, where player II has, for each one of his moves, no more than a finite number of alternative possibilities while player I perhaps has infinitely many choices, is determinate in the proposed sense from the viewpoint of player I.