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Preferences, Expected Utility, Risk Aversion and Prudence

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Medical Decision Making

Abstract

We introduce von Neumann and Morgenstern’s expected utility theory, which is the basic normative theory for decisions under uncertainty. The concept of quality-adjusted life years (QALYs), which is an application of this theory, is increasingly used as a measure for determining the public funding of medical interventions. We present the concept of risk aversion and conclude with the notion of prudence. Both concepts are important for an understanding of physician behavior.

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Notes

  1. 1.

    Von Neumann, J. & Morgenstern, O.—Theory of Games and Economic Behavior, 2nd Edition—Princeton University Press, Princeton—1947, p. 17.

  2. 2.

    We ought, also, for formal completeness, to assume explicitly that any state is at least as good as itself: \( \left({x}_1,{x}_2\right)\underset{\sim }{\succ}\left({x}_1,{x}_2\right) \) (axiom of reflexivity).

  3. 3.

    Note that individuals need not belong to any of these three categories. An individual may accept a coin toss where he receives €1 if heads comes up and pays €1 if tails comes up, but reject a gamble where the amount is raised to €1000. This individual is a risk lover for small bets and risk-averse for large bets. There have been different explanations for this puzzle. The Friedman-Savage Hypothesis (Friedman and Savage 1948) suggests that the preference scaling function is doubly inflected. Another explanation is that most individuals regard gambling as a recreational activity, so the pure joy of gambling will lead them to accept a fair gamble even though they are risk-averse regarding income.

  4. 4.

    First order stochastic dominance implies that shifting probability mass from an outcome to a preferred outcome should lead to a preferred prospect. In other words, the more health the better.

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Author information

Authors and Affiliations

  1. Department of Business and Economics, University of Basel, Basel, Switzerland

    Stefan Felder

  2. School of Business Studies, Stralsund University of Applied Sciences, Stralsund, Germany

    Thomas Mayrhofer

Authors
  1. Stefan Felder
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  2. Thomas Mayrhofer
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Exercises

Exercises

  1. 1.

    Prove that indifference curves can never cross!

  2. 2.

    A decision maker with a QALY utility function is often said to be risk-neutral. Under which assumptions is this true, under which, wrong?

  3. 3.

    An individual has a utility function according to the QALY model. His health can take on the states 1 and 2 with respective longevities T 1 and T 2 and respective probabilities p and \( 1-p \). The following table summarizes the initial situation, including utility weights for the two health states:

    y

    u(h i )

    p i

    T y

    1

    0.6

    0.8

    10

    2

    0.9

    0.2

    15

    1. (a)

      Calculate the QALYs and expected life years for the initial situation.

    2. (b)

      Which lifespan with perfect health is equivalent to the initial situation?

    3. (c)

      Assume that life expectancy with perfect health is 8 years. Calculate the probability of death \( \left(1-\pi \right) \) that is equivalent to the initial situation.

    4. (d)

      Calculate the change in QALYs if remaining life expectancy increases by 1 year in state 1 and by 2 years in state 2.

  4. 4.

    The utility weights in the following table are taken from a study where randomly chosen individuals responded to time trade-off questions (see Sackett and Torrance 1978).

    Health state h i

    Utility weight u(h i ) for one year

    Inability to work due to TBC

    0.68

    Stationary dialysis

    0.65

    Breast amputation due to breast cancer

    0.48

    Use this information to solve the following problems:

    1. (a)

      How many QALYs are produced by a treatment with dialysis which increases life expectancy by 8 years?

    2. (b)

      How many QALYs are gained if one TBC case is prevented which would otherwise last 3 months and require treatment at home?

    3. (c)

      A breast cancer patient undergoes a breast amputation and lives on for another 6 years. Screening 1 year earlier would have detected the cancer, led to an immediate operation and thus increased life expectancy by an additional 2 years (i.e., 9 years after screening) (assume that the utility weight without operation for the first year is 1). Calculate the utility gain from screening.

  5. 5.

    Prudence

    1. (a)

      Show that individuals with decreasing risk aversion \( A(h)= \) \( -u\hbox{'}\hbox{'}(h)/u\hbox{'}(h) \) must also be prudent, i.e., \( u\hbox{'}\hbox{'}\hbox{'}(h)>0 \).

    2. (b)

      Prove that \( \psi >\kappa \) if \( \frac{\partial \pi }{\partial h}<0 \).

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Felder, S., Mayrhofer, T. (2017). Preferences, Expected Utility, Risk Aversion and Prudence. In: Medical Decision Making. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-53432-8_3

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  • DOI: https://doi.org/10.1007/978-3-662-53432-8_3

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