## Fractional hypercube decompositions of multiattribute utility functionsSome new results in multiattribute utility theory are presented in this report. It is shown how fractional hypercubes induce the multiple element conditional preference orders used to specify attribute independence conditions, and how they also provide a system of equations used to produce the corresponding multiattribute utility decomposition. Fractional hypercube decompositions include most of the previous forms (additive, Keeney's quasi-additive, and Fishburn's diagonal) and give many new utility decompositons (e.g., pyramid, bipyramid, semicube). These new forms model nonseparable attribute interactions, so they are applicable to decision problems in systems analysis, resource allocation, and bundle evaluation, among others. (Author). |

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### Contents

FRACTIONAL HYPERCUBES AND CONDITIONAL ORDERS | 24 |

THE FUNDAMENTAL MULTIATTRIBUTE UTILITY DECOMPOSITION | 37 |

HOW FRACTIONAL DECOMPOSITIONS INCORPORATE SOME PREVIOUS | 61 |

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### Common terms and phrases

ai-l aliasing alternatives apex AQ(x attribute interactions attributes at base base level bipartite order bundle evaluation problems chapter common reference element composite interactions computations conditional elements conditional orders conditional preference orders conditional utility functions convex combinations decision analyst decision problems decomposition on three defined Definition denoted design of experiments diagonal decomposition diagonal independence example expected utility factors Fishburn fractional decompositions fractional hypercubes fractional independence axioms fractional orders given hybrid decomposition hypercube slices independence conditions inter interaction terms isotonic k-factor interaction Keeney lemma linear lottery main effects main theorem marginal probability MAUT multiattribute utility decomposition multiattribute utility theory nonadditive nonseparable interactions null function outcome quasi-pyramid decomposition residual coefficients riskless semicube decomposition simple probability measures specified standard coefficients star operator strict weak order subsets summation Suppose theorem 3.2 three attributes tion u(xj universal fraction unnormalized utility independence verify vertices