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Medical Decision Making
The final chapter takes a more realistic stance to medical decision making by assuming that physicians are only imperfect agents of their patients. Specifically, we assume that physicians internalize only some share of the patient’s utility and follow a profit motive in their test and treatment decisions. We then analyze the effects of imperfect agency on the thresholds and discuss the role of liability rules and medical guidelines in regulating imperfect agency. Finally, we present nonexpected utility models under risk and uncertainty (i.e., ambiguity). While these models can explain observed test and treatment decisions, they are not suitable for normative analyses aimed at providing guidance on medical decision making.
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1.
Consider a lottery with the following outcomes:
\( {h}_1=10;{h}_2=16;{h}_3=25 \) and the corresponding probabilities
\( {p}_1=0.1,{p}_2=0.5,{p}_3=0.4. \) The utility function is
\( u(h)=h \) the probability weighting function is
\( w(p)=\sqrt{p} \). What is the value of the lottery under EUT and under the dual theory framework, resp.?
2.
Assume the following utility function
\( u(h)=\sqrt{h} \) and the following probability weighting function
\( w(p)=\sqrt{p} \). Moreover, let’s assume a patient with an a priori probability of illness equal to
p = 0.2 and a diagnostic test with characteristics
Se =
Sp = 0.8. Outcomes (measured in life years) are as follows:
\( {h}_s^{}=9;{h}_s^{+}=16;{h}_h^{+}=25;{h}_h^{}=36. \)
(a)
Draw the decision tree in extensive and normal forms (note that the outcomes should be ranked from lowest to highest)
(b)
What is the certainty equivalent, i.e., value attached to the decision (in life years) using the extensive form under
(1)
EUT
(2)
Yaari’s dual theory
(c)
What is the certainty equivalent (in life years) using the normal form under
(1)
EUT
(2)
Yaari’s dual theory
(d)
Compare the results of (b) and (c) and discuss.
3.
Let’s assume two physicians, one is ambiguity neutral and acts according to EUT with
\( u(h)=\sqrt{h} \) and one is ambiguity averse with the same (elementary) utility function but also
\( \varphi (EU)=\sqrt{EU} \). Moreover, both physicians see the same patient. Health outcomes are as follows
\( {h}_s^{}=4;{h}_s^{+}=16;{h}_h^{+}=25;{h}_h^{}=36. \)
(a)
Assume that the a priori probability of illness for that patient is 30 %.
(1)
Calculate
\( E{U}^{+} \) and
\( E{U}^{} \) for the first physician. What will he do?
(2)
Calculate
\( {V}^{+} \) and
\( {V}^{} \) for the second physician. What will he do?
(b)
Assume now that the a priori probability of illness is uncertain and can take two values,
p
_{1} and
p
_{2} with
\( {p}_1=10\% \) and
\( {p}_2=50\% \). Furthermore,
\( \mu =50\% \) denotes the subjective probability of both physicians that
p
_{1} is the true probability of illness and
\( 1\mu =50\% \) denotes the subjective probability with which both physicians believe that
p
_{2} is the true a priori probability of illness (note that the weighted expected a priori probability of illness is the same as before:
\( \overline{p}=\mu {p}_1+\left(1\mu \right){p}_2=30\% \)).
(1)
Calculate
\( E{U}^{+} \) and
\( E{U}^{} \) for the first physician. What will he do?
(2)
Calculate
\( {V}^{+} \) and
\( {V}^{} \) for the second physician. What will he do?
(3)
Compare the results of (a) and (b) and discuss observed behavior vs. “what should a physician do”?
(a)
Draw the decision tree in extensive and normal forms (note that the outcomes should be ranked from lowest to highest)
(b)
What is the certainty equivalent, i.e., value attached to the decision (in life years) using the extensive form under
(1)
EUT
(2)
Yaari’s dual theory
(c)
What is the certainty equivalent (in life years) using the normal form under
(1)
EUT
(2)
Yaari’s dual theory
(d)
Compare the results of (b) and (c) and discuss.
(1)
EUT
(2)
Yaari’s dual theory
(1)
EUT
(2)
Yaari’s dual theory
(a)
Assume that the a priori probability of illness for that patient is 30 %.
(1)
Calculate
\( E{U}^{+} \) and
\( E{U}^{} \) for the first physician. What will he do?
(2)
Calculate
\( {V}^{+} \) and
\( {V}^{} \) for the second physician. What will he do?
(b)
Assume now that the a priori probability of illness is uncertain and can take two values,
p
_{1} and
p
_{2} with
\( {p}_1=10\% \) and
\( {p}_2=50\% \). Furthermore,
\( \mu =50\% \) denotes the subjective probability of both physicians that
p
_{1} is the true probability of illness and
\( 1\mu =50\% \) denotes the subjective probability with which both physicians believe that
p
_{2} is the true a priori probability of illness (note that the weighted expected a priori probability of illness is the same as before:
\( \overline{p}=\mu {p}_1+\left(1\mu \right){p}_2=30\% \)).
(1)
Calculate
\( E{U}^{+} \) and
\( E{U}^{} \) for the first physician. What will he do?
(2)
Calculate
\( {V}^{+} \) and
\( {V}^{} \) for the second physician. What will he do?
(3)
Compare the results of (a) and (b) and discuss observed behavior vs. “what should a physician do”?
(1)
Calculate
\( E{U}^{+} \) and
\( E{U}^{} \) for the first physician. What will he do?
(2)
Calculate
\( {V}^{+} \) and
\( {V}^{} \) for the second physician. What will he do?
(1)
Calculate
\( E{U}^{+} \) and
\( E{U}^{} \) for the first physician. What will he do?
(2)
Calculate
\( {V}^{+} \) and
\( {V}^{} \) for the second physician. What will he do?
(3)
Compare the results of (a) and (b) and discuss observed behavior vs. “what should a physician do”?
1
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3
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7
In a letter to Milton Friedman, dated August 25, 1950, Paul Samuelson, an originally severe critic, wrote that he accepted the expected utility theory after Leonard Savage persuaded him of the normative force of the Independence Axiom (quoted after Moscati
2016).
Note that the parameter
β can also capture potential moral hazard in the case of the patient whose medical costs are covered by health insurance coverage. Instead of paying the full price, this patient only pays a fraction
\( 1\beta \) of the treatment cost.
Note that these conditions were also required in Chap.
5 for the effect of an increase in the willingness to pay for a QALY on the testtreatment threshold being positive.
This explains the classification as rankdependent choice model.
Note the similarity between the threshold analysis under the dual theory and the analysis of the certainty equivalent under EUT.
In the framework of nonEUT models, the value of information relates to differences in outcomes rather than differences in EUT.
Another interpretation is that these are weighting factors for the size of a trial in a metaanalysis of prevalence estimates.
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 Titel
 Imperfect Agency and NonExpected Utility Models
 DOI
 https://doi.org/10.1007/9783662534328_11
 Autoren:

Stefan Felder
Thomas Mayrhofer
 Verlag
 Springer Berlin Heidelberg
 Sequenznummer
 11
 Kapitelnummer
 Chapter 11