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A Fractional Hypercube Decomposition Theorem for Multiattribute Utility Functions

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  • Peter H. Farquhar

    (Northwestern University, Evanston, Illinois)

Abstract

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.

Suggested Citation

  • Peter H. Farquhar, 1975. "A Fractional Hypercube Decomposition Theorem for Multiattribute Utility Functions," Operations Research, INFORMS, vol. 23(5), pages 941-967, October.
    • Handle: RePEc:inm:oropre:v:23:y:1975:i:5:p:941-967
    DOI: 10.1287/opre.23.5.941
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    Cited by:

    1. Han Bleichrodt & Ulrich Schmidt & Horst Zank, 2009. "Additive Utility in Prospect Theory," Management Science, INFORMS, vol. 55(5), pages 863-873, May.
    2. Corrente, Salvatore & Greco, Salvatore & Ishizaka, Alessio, 2016. "Combining analytical hierarchy process and Choquet integral within non-additive robust ordinal regression," Omega, Elsevier, vol. 61(C), pages 2-18.
    3. L. Robin Keller & Jay Simon, 2019. "Preference Functions for Spatial Risk Analysis," Risk Analysis, John Wiley & Sons, vol. 39(1), pages 244-256, January.
    4. Ali E. Abbas, 2009. "Multiattribute Utility Copulas," Operations Research, INFORMS, vol. 57(6), pages 1367-1383, December.
    5. Hauser, John R. & Urban, Glen L., 1975. "A normative methodology for modeling consumer response to innovation," Working papers 785-75., Massachusetts Institute of Technology (MIT), Sloan School of Management.
    6. Ali E. Abbas & Ronald A. Howard, 2005. "Attribute Dominance Utility," Decision Analysis, INFORMS, vol. 2(4), pages 185-206, December.
    7. Ali E. Abbas & Zhengwei Sun, 2019. "Archimedean Utility Copulas with Polynomial Generating Functions," Decision Analysis, INFORMS, vol. 16(3), pages 218-237, September.
    8. Ali E. Abbas & Zhengwei Sun, 2015. "Multiattribute Utility Functions Satisfying Mutual Preferential Independence," Operations Research, INFORMS, vol. 63(2), pages 378-393, April.
    9. Andrea C. Hupman & Jay Simon, 2023. "The Legacy of Peter Fishburn: Foundational Work and Lasting Impact," Decision Analysis, INFORMS, vol. 20(1), pages 1-15, March.
    10. Ali E. Abbas, 2011. "The Multiattribute Utility Tree," Decision Analysis, INFORMS, vol. 8(3), pages 180-205, September.
    11. Ali E. Abbas & David E. Bell, 2015. "Ordinal One-Switch Utility Functions," Operations Research, INFORMS, vol. 63(6), pages 1411-1419, December.
    12. Ali E. Abbas, 2013. "Utility Copula Functions Matching All Boundary Assessments," Operations Research, INFORMS, vol. 61(2), pages 359-371, April.
    13. Jiehua Xie & Zhengyong Zhou, 2022. "Patchwork Constructions of Multiattribute Utility Functions," Decision Analysis, INFORMS, vol. 19(2), pages 141-169, June.

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