Abraham Wald presents a revised theory of statistical decision functions, addressing limitations in his earlier work. He introduces weaker conditions for the theory to hold, particularly removing the restrictive assumption of compactness of the space of distribution functions, which was previously necessary for the theory. Instead, he uses separability in the continuous case and discrete conditions in the discrete case. He also relaxes the continuity assumption of the weight function, allowing for discontinuous weight functions that are more realistic in many statistical problems. Wald discusses the concept of statistical decision functions as a zero-sum two-person game, where the statistician and Nature are the two players. The goal is to find a decision rule that minimizes the expected loss or risk. He proves several theorems about the strict determinateness of such games, showing that under certain conditions, the game is strictly determined, meaning the maximum of the minimum and the minimum of the maximum of the payoff function are equal. He introduces the concept of minimax solutions, which are decision rules that minimize the maximum possible loss. He shows that such solutions exist under relatively weak conditions. The paper also discusses the cost function of experimentation, allowing for costs that depend not only on the number of observations but also on the decision rule used. In the discrete case, Wald shows that the existence of minimax solutions can be established under weaker conditions. He also provides a detailed analysis of the conditions required for the strict determinateness of the game, including the boundedness of the weight function and the continuity of the cost function. The paper concludes with the main theorem, which states that under the conditions outlined, the statistical decision problem is strictly determined. This has significant implications for the theory of statistical decision functions, as it provides a foundation for the existence of optimal decision rules in various statistical problems.Abraham Wald presents a revised theory of statistical decision functions, addressing limitations in his earlier work. He introduces weaker conditions for the theory to hold, particularly removing the restrictive assumption of compactness of the space of distribution functions, which was previously necessary for the theory. Instead, he uses separability in the continuous case and discrete conditions in the discrete case. He also relaxes the continuity assumption of the weight function, allowing for discontinuous weight functions that are more realistic in many statistical problems. Wald discusses the concept of statistical decision functions as a zero-sum two-person game, where the statistician and Nature are the two players. The goal is to find a decision rule that minimizes the expected loss or risk. He proves several theorems about the strict determinateness of such games, showing that under certain conditions, the game is strictly determined, meaning the maximum of the minimum and the minimum of the maximum of the payoff function are equal. He introduces the concept of minimax solutions, which are decision rules that minimize the maximum possible loss. He shows that such solutions exist under relatively weak conditions. The paper also discusses the cost function of experimentation, allowing for costs that depend not only on the number of observations but also on the decision rule used. In the discrete case, Wald shows that the existence of minimax solutions can be established under weaker conditions. He also provides a detailed analysis of the conditions required for the strict determinateness of the game, including the boundedness of the weight function and the continuity of the cost function. The paper concludes with the main theorem, which states that under the conditions outlined, the statistical decision problem is strictly determined. This has significant implications for the theory of statistical decision functions, as it provides a foundation for the existence of optimal decision rules in various statistical problems.