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Invariant Utility Functions and Certain Equivalent Transformations

Author

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  • Ali E. Abbas

    (Department of Industrial and Enterprise System Engineering, College of Engineering, University of Illinois at Urbana-Champaign, 104 South Mathews Avenue, Urbana, Illinois 61801)

Abstract

This paper defines invariant utility functions to continuous monotonic transformations. We also define transformation invariance as the condition in which the certain equivalent of a lottery follows a continuous monotonic transformation that is applied to its outcomes. We show that invariant utility functions uniquely satisfy transformation invariance, and we illustrate how knowledge of an invariance criterion determines the functional form of the utility function. This formulation extends the widely used notions of invariance to shift and scale transformations on the outcomes of a lottery to more general monotonic transformations. Moreover, we interpret any continuous and strictly monotonic utility function as an invariant utility function to a composite monotonic transformation. Furthermore, we show how this composite transformation uniquely characterizes the utility function up to a linear transformation. We derive the invariance formulations that lead to the assignment of hyperbolic absolute risk-averse (HARA) utility functions, linear plus exponential utility functions, and a two-parameter power-logarithmic utility function that generalizes the logarithmic utility function. We work through several examples to illustrate the approach.

Suggested Citation

  • Ali E. Abbas, 2007. "Invariant Utility Functions and Certain Equivalent Transformations," Decision Analysis, INFORMS, vol. 4(1), pages 17-31, March.
    • Handle: RePEc:inm:ordeca:v:4:y:2007:i:1:p:17-31
    DOI: 10.1287/deca.1060.0083
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    References listed on IDEAS

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    1. David E. Bell, 1988. "One-Switch Utility Functions and a Measure of Risk," Management Science, INFORMS, vol. 34(12), pages 1416-1424, December.
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    Cited by:

    1. Ali E. Abbas, 2012. "Valuing Changes in Investment Opportunities," Operations Research, INFORMS, vol. 60(6), pages 1451-1460, December.
    2. L. Robin Keller, 2009. "From the Editor..," Decision Analysis, INFORMS, vol. 6(1), pages 1-3, March.
    3. L. Robin Keller & Ali Abbas & Manel Baucells & Vicki M. Bier & David Budescu & John C. Butler & Philippe Delquié & Jason R. W. Merrick & Ahti Salo & George Wu, 2010. "From the Editors..," Decision Analysis, INFORMS, vol. 7(4), pages 327-330, December.
      • L. Robin Keller & Manel Baucells & Kevin F. McCardle & Gregory S. Parnell & Ahti Salo, 2007. "From the Editors..," Decision Analysis, INFORMS, vol. 4(4), pages 173-175, December.
      • L. Robin Keller & Manel Baucells & John C. Butler & Philippe Delquié & Jason R. W. Merrick & Gregory S. Parnell & Ahti Salo, 2008. "From the Editors..," Decision Analysis, INFORMS, vol. 5(4), pages 173-176, December.
      • L. Robin Keller & Manel Baucells & John C. Butler & Philippe Delquié & Jason R. W. Merrick & Gregory S. Parnell & Ahti Salo, 2009. "From the Editors ..," Decision Analysis, INFORMS, vol. 6(4), pages 199-201, December.
    4. Manel Baucells & Samuel E. Bodily, 2024. "The Discount Rate for Investment Analysis Applying Expected Utility," Decision Analysis, INFORMS, vol. 21(2), pages 125-141, June.
    5. Ali E. Abbas & Zhengwei Sun, 2019. "Archimedean Utility Copulas with Polynomial Generating Functions," Decision Analysis, INFORMS, vol. 16(3), pages 218-237, September.
    6. L. Robin Keller, 2010. "From the Editor..," Decision Analysis, INFORMS, vol. 7(3), pages 235-237, September.
    7. 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.
    8. Ali Abbas, 2011. "Risk-adjusted martingales and the design of “indifference” gambles," Theory and Decision, Springer, vol. 71(4), pages 643-668, October.
    9. Kenneth C. Lichtendahl & Samuel E. Bodily, 2010. "Preferences for Consumption Streams: Scale Invariance, Correlation Aversion, and Delay Aversion Under Mortality Risk," Operations Research, INFORMS, vol. 58(4-part-1), pages 985-997, August.
    10. Robert F. Bordley & Elena Katok & L. Robin Keller, 2010. "Honoring Michael H. Rothkopf's Legacy of Rigor and Relevance in Auction Theory: From the Editors," Decision Analysis, INFORMS, vol. 7(1), pages 1-4, March.
    11. 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.
    12. Ali E. Abbas & János Aczél, 2010. "The Role of Some Functional Equations in Decision Analysis," Decision Analysis, INFORMS, vol. 7(2), pages 215-228, June.
    13. Kenneth C. Lichtendahl & Raul O. Chao & Samuel E. Bodily, 2012. "Habit Formation from Correlation Aversion," Operations Research, INFORMS, vol. 60(3), pages 625-637, June.
    14. Ali Abbas, 2010. "Invariant multiattribute utility functions," Theory and Decision, Springer, vol. 68(1), pages 69-99, February.
    15. Ali E. Abbas & David E. Bell, 2015. "Ordinal One-Switch Utility Functions," Operations Research, INFORMS, vol. 63(6), pages 1411-1419, December.
    16. L. Robin Keller & Kelly M. Kophazi, 2010. "From the Editors..," Decision Analysis, INFORMS, vol. 7(2), pages 151-154, June.

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