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The application of decision theory on the basis of the expected utility criterion is discussed. Cases are considered where the form of the utility function is not known except for certain of its characteristics such as increasing concavity or increasing concavity with positive third derivatives, and/or the probability of the individual state of nature is not known exactly, and only certain relations between the probabilities of different states are available. First the reformulation of statistical dominance is carried out by assuming a limited amount of information on the probability of the state. Then a new method of dominance relation, referred to as S-S dominance, is introduced; here limited information on both the utility function and the probability of the state is assumed to be available, as is complete knowledge of the conditional probability distribution of the consequences for each state. Finally, the feasibility of S-S dominance is demonstrated through examples.