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A Fractional Hypercube Decomposition Theorem for Multiattribute Utility Functions

This paper establishes a fundamental decomposition theorem in multiattribute utility theory. The methodology uses fractional hypercubes to generate a variety of attribute independence conditions that are necessary and sufficient for various decompositions: the additive, Keeney's quasi-additive, Fishburrt's diagonal, and others. These other nonaddirive utility decompositions contain some nonseparable interaction terms and are therefore applicable to decision problems not covered by earlier models. The paper defines a fractional hypercube and introduces the corresponding multiple element conditional preference order. The main theorem is produced from the solution of equations that are derived from transformations of linear functions that preserve these conditional preference orders. The computations and scaling required in implementing the main result are demonstrated by obtaining four utility decompositions on three attributes: apex, diagonal, quasi-pyramid, and semicube. We illustrate the methodology with geometric structures that correspond to the fractional hypercubes.