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Decomposing the Cross Derivatives of a Multiattribute Utility Function into Risk Attitude and Value

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  • Ali E. Abbas

    (Department of Industrial and Enterprise Systems Engineering, College of Engineering, University of Illinois at Urbana--Champaign, Urbana, Illinois 61801)

Abstract

The cross derivatives of a multiattribute utility function play an important role in the choice between multivariate lotteries and in multiattribute Taylor expansions of the utility function. This paper decomposes the cross derivatives into two components: the derivatives of a single-attribute utility function over value and the cross derivatives of the value function. This approach provides a simple method for reasoning about the signs of the cross derivatives of a multiattribute utility function using derivatives of a univariate utility function and the properties of the value function. To illustrate the approach, we relate the multivariate risk aversion concept, which involves the mixed partial derivative of the utility function, to the Arrow--Pratt risk aversion function. We show that for additive value functions, a decision maker is multivariate risk averse if and only if he is risk averse over value in the Arrow--Pratt sense. For other value functions, however, a decision maker can be risk averse or risk seeking over value and still exhibit multivariate risk aversion. The approach also derives the conditions on the value function that relate two important classes of utility functions: single attribute utility functions whose derivatives alternate in sign and multiattribute utility functions whose cross derivatives alternate in sign. These two classes are widely used in practice and form the basis of univariate and multivariate stochastic dominance. Several examples illustrate the approach.

Suggested Citation

  • Ali E. Abbas, 2011. "Decomposing the Cross Derivatives of a Multiattribute Utility Function into Risk Attitude and Value," Decision Analysis, INFORMS, vol. 8(2), pages 103-116, June.
  • Handle: RePEc:inm:ordeca:v:8:y:2011:i:2:p:103-116
    DOI: 10.1287/deca.1110.0205
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    Cited by:

    1. L. Robin Keller & Ali Abbas & J. Eric Bickel & Vicki M. Bier & David V. Budescu & John C. Butler & Enrico Diecidue & Robin L. Dillon-Merrill & Raimo P. Hämäläinen & Kenneth C. Lichtendahl & Jason R. W, 2012. "From the Editors ---Brainstorming, Multiplicative Utilities, Partial Information on Probabilities or Outcomes, and Regulatory Focus," Decision Analysis, INFORMS, vol. 9(4), pages 297-302, December.
    2. Andrew Grant & Steve Satchell, 2019. "Endogenous divorce risk and investment," Journal of Population Economics, Springer;European Society for Population Economics, vol. 32(3), pages 845-876, July.
    3. L. Robin Keller, 2011. "From the Editor ---Multiattribute and Intertemporal Preferences, Probability, and Stochastic Processes: Models and Assessment," Decision Analysis, INFORMS, vol. 8(3), pages 165-169, September.
    4. William B. Haskell & J. George Shanthikumar & Z. Max Shen, 2017. "Aspects of optimization with stochastic dominance," Annals of Operations Research, Springer, vol. 253(1), pages 247-273, June.
    5. Ali E. Abbas, 2011. "The Multiattribute Utility Tree," Decision Analysis, INFORMS, vol. 8(3), pages 180-205, September.
    6. L. Robin Keller & Kelly M. Kophazi, 2011. "From the Editors---Deterrence, Multiattribute Utility, and Probability and Bayes' Updating," Decision Analysis, INFORMS, vol. 8(2), pages 83-87, June.
    7. L. Robin Keller & Ali Abbas & J. Eric Bickel & Vicki M. Bier & David V. Budescu & John C. Butler & Philippe Delquié & Kenneth C. Lichtendahl & Jason R. W. Merrick & Ahti Salo & George Wu, 2011. "From the Editors ---Probability Scoring Rules, Ambiguity, Multiattribute Terrorist Utility, and Sensitivity Analysis," Decision Analysis, INFORMS, vol. 8(4), pages 251-255, December.
    8. Gabriele Marconi, 2013. "Rankings, accreditations, and international exchange students," IZA Journal of European Labor Studies, Springer;Forschungsinstitut zur Zukunft der Arbeit GmbH (IZA), vol. 2(1), pages 1-22, December.
    9. Wong, Eugene Y.C. & Tai, Allen H. & Lau, Henry Y.K. & Raman, Mardjuki, 2015. "An utility-based decision support sustainability model in slow steaming maritime operations," Transportation Research Part E: Logistics and Transportation Review, Elsevier, vol. 78(C), pages 57-69.
    10. Argyris, Nikolaos & French, Simon, 2017. "Nuclear emergency decision support: A behavioural OR perspective," European Journal of Operational Research, Elsevier, vol. 262(1), pages 180-193.

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