An Imbedding Approach to Additive Value Zero-One Problems

Existing approaches to the determination of the parameters for zero-one additive value functions can involve a large amount of work. This paper considers an alternative imbedding approach which allows the parameters to be determined to within an error range 1/K within K steps. In considering the method itself specific elements of theory are developed relating to the existence of additive values for imbedding sets X · Y*, X · Ỹ where X is a subset of zero-one vectors, Y* is a countable subset of numbers in the range −∞ < y < ∞, and Ỹ is a connected open interval of the range −∞ < y < ∞. In seeking further approximation methods, it is necessary to investigate properties of the component value, u(·) over Y* and Ỹ. The coefficients of the zero-one variables are shown to be unique for Y* once two values have been preset. Under certain circumstances it can be demonstrated that a continuous u(·) over Ỹ exists, and that considerable variations in u(·) may be compatible with the specified preference structures, although in special circumstances u(·) is unique over Ỹ.