Games with Finite Resources as defined by Gale are two-person zero-sum N-stage games in which each player has N resources and may use each resource once and only once in the N stages. Gale's theorem on these games is generalized in several directions. First the payoff is allowed to be any symmetric function of the stage payoffs. Second, the players are allowed some latitude in choosing which game is being played. Applications are given to some open questions in the area of Inspection Games. Finally the payoff is allowed to be random, thus incorporating a result of Ross on Goofspiel. Application is made to a game-theoretic version of the Generalized House Selling Problem.
