A theory of medical decision making under uncertainty

Abstract

This paper presents an axiomatic model of medical decision making and discusses its potential applications. The medical decision problems envisioned concern the choice of a medical treatment following a diagnosis in situations in which data allow construction of an empirical distribution over the potential outcomes associated with the alternative treatments. In its descriptive interpretation, the model is an hypothesis about the patient’s choice behavior. The theory also aims to aid physicians in recommending treatments in a coherent manner.

Appendix

Proof of Theorem 2

$(a)\Rightarrow (b).$ By Theorem 1, ≽  _a is represented by

$$ V_{a}\left( p,f\right) =\sum_{\omega \in \Omega }p\left( \omega \right) U_{a}(f\left( \omega \right) ,\omega ). \label{equ1} $$

(14)

Action-independent risk attitudes, (A.6), and the uniqueness part of Theorem 1 imply that for all a, a′ ∈ A, $U_{a}\left( \cdot ,\omega \right)$ and $U_{a^{\prime }}\left( \cdot ,\omega \right)$ are linear transformations of one another. Moreover, the representation (14) and (A.7) imply that, for all a,a′ ∈ A that are elementarily linked at f, and $\bar{p},\underline{p},\bar{p}^{\prime },\underline{p}^{\prime }\in P$ satisfying $\left( \bar{p},f\right) \succ _{a}\left( \underline{p} ,f\right)$ such that $\left( a,\bar{p},f\right) \sim \left( a^{\prime }, \bar{p}^{\prime },f\right)$ and $\left( a,\underline{p},f\right) \sim \left( a^{\prime },\underline{p}^{\prime },f\right),$

$$ V_{a}\left( \alpha \bar{p}+\left( 1-\alpha \right) \underline{p},f\right) =V_{a^{\prime }}\left( \alpha \bar{p}^{\prime }+\left( 1-\alpha \right) \underline{p}^{\prime },f\right) \label{equ1a} $$

(15)

for all $\alpha \in \left( 0,1\right) .$

Fix a ⁰ and let $U\left( \cdot ,\omega \right) :=U_{a^{0}}\left( \cdot ,\omega \right) \$for all ω ∈ Ω. Then

$$ U_{a}\left( f\left( \omega \right) ,\omega \right) =\lambda (a)U\left( f\left( \omega \right) ,\omega \right) +v\left( a\right) ,\text{ for all } a\in A,f\in F\text{ and }\omega \in \Omega . \label{equ2} $$

(16)

Let a ₀,a ∈ A be elementarily linked at f, and define $\lambda \left( a\right)$ and $v\left( a\right)$ by the unique solution to the following equations:

$$ \lambda \left( a\right) \sum_{\omega \in \Omega }\bar{p}\left( \omega \right) U(f\left( \omega \right) ,\omega )+v\left( a\right) =\sum_{\omega \in \Omega }\bar{p}^{\prime }\left( \omega \right) U(f\left( \omega \right) ,\omega ) \label{equ3a} $$

(17)

and

$$ \lambda \left( a\right) \sum_{\omega \in \Omega }\underline{p}\left( \omega \right) U(f\left( \omega \right) ,\omega )+v\left( a\right) =\sum_{\omega \in \Omega }\underline{p}^{\prime }\left( \omega \right) U(f\left( \omega \right) ,\omega ). \label{equ3b} $$

(18)

Let a and a′ be elementarily linked, and define $\lambda \left( a^{\prime }\right)$ and $v\left( a^{\prime }\right)$ by the unique solution to the equations

$$ \lambda \left( a^{\prime }\right) \sum_{\omega \in \Omega }\bar{p}^{\prime }\left( \omega \right) U(f^{\prime }\left( \omega \right) ,\omega )+v\left( a^{\prime }\right) =\lambda \left( a\right) \sum_{\omega \in \Omega }\bar{p} \left( \omega \right) U(f^{\prime }\left( \omega \right) ,\omega )+v\left( a\right) \label{equ4a} $$

(19)

and

$$ \lambda \left( a^{\prime }\right) \sum_{\omega \in \Omega }\underline{p} ^{\prime }\left( \omega \right) U(f^{\prime }\left( \omega \right) ,\omega )+v\left( a^{\prime }\right) =\lambda \left( a\right) \sum_{\omega \in \Omega }\underline{p}\left( \omega \right) U(f^{\prime }\left( \omega \right) ,\omega )+v\left( a\right) . \label{equ4b} $$

(20)

Because A is finite and linked, repeating this process, it is possible to solve $\left( \lambda \left( a\right) ,v\left( a\right) \right)$ for all a ∈ A.

For every a ∈ A, define:

$$ B_{a}=\left\{\left( p,f\right) \in P\times F\mid \left( p,f\right) \succeq _{a}\left( p^{\prime },f^{\prime }\right) \text{ }\forall \left( p^{\prime },f^{\prime }\right) \in P\times F\right\} $$

and

$$ W_{a}=\left\{\left( p,f\right) \in P\times F\mid \left( p^{\prime },f^{\prime }\right) \succeq _{a}\left( p,f\right) \text{ }\forall \left( p^{\prime },f^{\prime }\right) \in P\times F\right\}. $$

By the compactness of P×F and continuity of ≽  _a , the sets B _a and W _a are closed and nonempty. Moreover, since $\left( p,f\right) =\left( \Sigma _{\omega \in \Omega }p\left( \omega \right) e^{\omega },f\right) ,$ constraint independence and transitivity imply that there are ω,ω′ ∈ Ω such that $\left( e^{\omega },f\right) \in B_{a}$ and $\left( e^{\omega ^{\prime }},f^{\prime }\right) \in W_{a}.$ Define

$$ B_{a}^{0}=\{\left( p,f\right) \in B_{a}\mid p=e^{\omega }\text{ for some } \omega \in \Omega \} $$

and

$$ W_{a}^{0}=\{\left( p,f\right) \in W_{a}\mid p=e^{\omega }\text{ for some } \omega \in \Omega \}. $$

By coordinate essentiality, for all a ∈ A, $B_{a}^{0}\cap W_{a}^{0}$ is empty.

By Theorem 1 $V_{a}\left( e^{\omega },f\right) =U_{a}\left( f\left( \omega \right) ,\omega \right) ,$ for all a ∈ A, and, by Eq. 16, $U_{a}\left( f\left( \omega \right) ,\omega \right) =\lambda (a)U\left( f\left( \omega \right) ,\omega \right) +v\left( a\right) ,$ where λ(a) > 0. Similarly, $V_{a}\left( e^{\omega ^{\prime }},f^{\prime }\right) =U_{a}\left( f^{\prime }\left( \omega ^{\prime }\right) ,\omega ^{\prime }\right) =\lambda (a)U\left( f^{\prime }\left( \omega ^{\prime }\right) ,\omega ^{\prime }\right) +v\left( a\right) .$ Consequently, $\left( e^{\omega },f\right) \in B_{a}^{0}$ and $\left( e^{\omega ^{\prime }},f^{\prime }\right) \in W_{a}^{0}$ if and only if $\left( e^{\omega },f\right) \in B_{a^{\prime }}^{0}$ and $\left( e^{\omega ^{\prime }},f^{\prime }\right) \in W_{a^{\prime }}^{0},$ for all a,a′ ∈ A. Given $\left( e^{\omega },f\right) \in B_{a}^{0},$ rearrange the set A letting j > i if $\left( a_{j},e^{\omega },f\right) \succ \left( a_{i},e^{\omega },f\right) .$ (If $\left( a_{j},e^{\omega },f\right) \sim \left( a_{i},e^{\omega },f\right) ,$ then the order is arbitrary.) Hence A can be written as an n −tuple $\left( a_{1},...,a_{n}\right) ,$ and a _i and a _i + 1 are elementarily linked, i = 1,...,n − 1.

Let f,f′ ∈ F and ω,ω′ ∈ Ω be such that $\left( e^{\omega },f\right) \in B_{a}^{0}$ and $\left( e^{\omega ^{\prime }},f^{\prime }\right) \in W_{a}^{0}$. Define $\hat{f}=\left( f\left( \omega \right) ,\left( f^{\prime }\left( \omega ^{\prime }\right) ,f_{-\omega ^{\prime }}\right) _{-\omega }\right) .$ Because $f\left( \omega \right)$ is the ω −th coordinate of both f and $\hat{f},$ and $f^{\prime }\left( \omega ^{\prime }\right)$ is the ω′ −th coordinate of both f′ and $\hat{f},$ the certainty principle implies $\left( e^{\omega },\hat{f}\right) \in B_{a}^{0}$ and $\left( e^{\omega ^{\prime }},\hat{f}\right) \in W_{a}^{0}.$

Let a and a′ be elementarily linked at f ^∗ ∈ F with $\bar{p},\underline{p},\bar{p}^{\prime },\underline{p}^{\prime }\in P$ satisfying $\left( \bar{p},f^{\ast }\right) \succ _{a}\big( \underline{p} ,f^{\ast }\big)$ such that $\left( a,\bar{p},f^{\ast }\right) \sim \left( a^{\prime },\bar{p}^{\prime },f^{\ast }\right)$ and $\big( a,\underline{p} ,f^{\ast }\big) \sim \big( a^{\prime },\underline{p}^{\prime },f^{\ast }\big).$ There are then $\bar{\alpha}_{a},\underline{\alpha }_{a},\bar{ \alpha}_{a^{\prime }},\underline{\alpha }_{a^{\prime }}\in \left[ 0,1\right]$ such that $\left( \bar{p},f^{\ast }\right) \sim _{a}\left(\left( \bar{\alpha} _{a}e^{\omega }+\left( 1-\bar{\alpha}_{a}\right) e^{\omega ^{\prime }}\right) ,\hat{f}\right)$, $\big( \underline{p},f^{\ast }\big) \sim _{a}\left(\left( \underline{\alpha }_{a}e^{\omega }+\left( 1-\underline{\alpha } _{a}\right) e^{\omega ^{\prime }}\right) ,\hat{f}\right),$ $\left( \bar{p}^{\prime },f^{\ast }\right) \sim _{a^{^{\prime }}}\left(\left( \bar{\alpha}_{a^{\prime }}e^{\omega }+\left( 1-\bar{\alpha}_{a^{\prime }}\right) e^{\omega ^{\prime }}\right) ,\hat{f}\right),$ and $\big( \underline{p}^{\prime },f^{\ast }\big) \sim _{a^{\prime }}\left(\left( \underline{\alpha }_{a^{\prime }}e^{\omega }+\left( 1-\underline{\alpha }_{a^{\prime }}\right) e^{\omega ^{\prime }}\right) ,\hat{f}\right).$ By transitivity

$$ \left(a,\left( \bar{\alpha}_{a}e^{\omega }+\left( 1-\bar{\alpha}_{a}\right) e^{\omega ^{\prime }}\right) ,\hat{f}\right)\sim \left(a^{\prime },\left( \bar{\alpha} _{a^{\prime }}e^{\omega }+\left( 1-\bar{\alpha}_{a^{\prime }}\right) e^{\omega ^{\prime }}\right) ,\hat{f}\right) \label{equ4c} $$

(21)

and

$$ \left(a^{\prime },\left( \underline{\alpha }_{a}e^{\omega }+\left( 1-\underline{ \alpha }_{a}\right) e^{\omega ^{\prime }}\right) ,\hat{f}\right)\sim \left(a^{\prime },\left( \underline{\alpha }_{a^{\prime }}e^{\omega }+\left( 1-\underline{ \alpha }_{a^{\prime }}\right) e^{\omega ^{\prime }}\right) ,\hat{f}\right). \label{equ4d} $$

(22)

To simplify the notation, let $\left( \alpha e^{\omega }+\left( 1-\alpha \right) e^{\omega ^{\prime }}\right) \equiv \hat{q}\left( \alpha \right) .$ Then, by Eqs. 21 and 22, a and a′ are elementarily linked at $\hat{f}\in F$ and $\hat{q}\left( \bar{\alpha}_{a}\right) ,\hat{q} \left( \underline{\alpha }_{a}\right) ,\hat{q}\left( \bar{\alpha}_{a^{\prime }}\right) ,\hat{q}\left( \underline{\alpha }_{a^{\prime }}\right) \in P.$

Consider next $\left( a,p,f\right) .$ By definition $\big( a,e^{\omega }, \hat{f}\big) \succeq \big( a,p,f\big) \succeq \big( a,e^{\omega ^{\prime }},\hat{f}\big) .$ Thus, by (A.2) and (A.5), there is a unique α _p such that $\left( a,p,f\right) \sim \left( a,\hat{q}\left( \alpha _{p}\right) ,\hat{f}\right)$.

Consider the alternatives $\left( a,p,f\right) \$and$\ \left( a^{\prime },p^{\prime },f^{\prime }\right) \$and, without loss of generality, suppose that $\left( a^{\prime },p^{\prime },f^{\prime }\right) \succeq \left( a,p,f\right)$. Three cases need to be considered:

Case 1: $\left( a^{\prime },p^{\prime },f^{\prime }\right) \!\succeq\! \left(\! a,e^{\omega },\hat{f}\right) .$ Then $V_{a^{\prime }}\left( p^{\prime },f^{\prime }\right) \!\geq\! U_{a}\!\left( \hat{f}\left( \omega \right) ,\omega \right) \!\geq\! V_{a}\left( p,f\right) .$

Case 2: $\left(\!a^{\prime },e^{\omega ^{\prime }},\hat{f}\right) \!\succeq\! \left(a,p,f\right) .$ Then $V_{a^{\prime }}\left( p^{\prime },f^{\prime }\right) \!\geq\! U_{a^{\prime }}\left(\!\hat{f}\!\left( \omega ^{\prime }\right)\!,\omega ^{\prime }\right) \!\geq\! V_{a}\left( p,f\right) .$

Case 3: $\left( a^{\prime },e^{\omega },\hat{f}\right) \succ \left( a^{\prime },p^{\prime },f^{\prime }\right) \succeq \left( a,p,f\right) \succ \left( a^{\prime },e^{\omega ^{\prime }},\hat{f}\right) .$ Then, by (A.2) and (A.5), there are unique $\hat{q}\left( \alpha _{p^{\prime }}\right)$ and $\hat{q}\left( \alpha _{p}\right)$ such that $\left( a^{\prime },p^{\prime },f^{\prime }\right) \sim \left( a^{\prime },\hat{q}\left( \alpha _{p^{\prime }}\right) ,\hat{f}\right)$ and $\left( a,p,f\right) \sim \left( a^{\prime },\hat{q}\left( \alpha _{p}^{\prime }\right) ,\hat{f} \right) ,$ respectively. Moreover, by the same argument, there is a unique $\hat{q}\left( \alpha _{p}\right)$ satisfying $\left( a,p,f\right) \sim \left( a,\hat{q}\left( \alpha _{p}\right) ,\hat{f}\right)$. By transitivity, $\left( a^{\prime },\hat{q}\left( \alpha _{p}^{\prime }\right) ,\hat{f}\right) \sim \left( a,\hat{q}\left( \alpha _{p}\right) ,\hat{f} \right) .$ Hence, by transitivity,

$$ \left( a^{\prime },\hat{q}\left( \alpha _{p^{\prime }}\right) ,\hat{f} \right) \sim \left( a^{\prime },p^{\prime },f^{\prime }\right) \succeq \left( a,p,f\right) \sim \left( a^{\prime },\hat{q}\left( \alpha _{p}^{\prime }\right) ,\hat{f}\right) \sim \left( a,\hat{q}\left( \alpha _{p}\right) ,\hat{f}\right) . \label{equ4e} $$

(23)

By Eqs. 14 and 15,

$$ \begin{array}{c} V_{a^{\prime }}\left( p^{\prime },f^{\prime }\right) =V_{a^{\prime }}\left( \hat{q}\left( \alpha _{p^{\prime }}\right) ,\hat{f}\right) =\\ \lambda \left( a^{\prime }\right) \left[ \alpha _{p^{\prime }}U(f\left( \omega \right) ,\omega )+\left( 1-\alpha _{p^{\prime }}\right) U\left(f\left( \omega ^{\prime }\right) ,\omega ^{\prime }\right)\right] +v\left( a^{\prime }\right) \geq \\ \lambda \left( a^{\prime }\right) \left[ \alpha _{p}^{\prime }U(f\left( \omega \right) ,\omega )+\left( 1-\alpha _{p}^{\prime }\right) U\left(f\left( \omega ^{\prime }\right) ,\omega ^{\prime }\right)\right] +v\left( a^{\prime }\right) = \\ \lambda \left( a\right) \left[ \alpha _{p}U(f\left( \omega \right) ,\omega )+\left( 1-\alpha _{p}\right) U\left(f\left( \omega ^{\prime }\right) ,\omega ^{\prime }\right)\right] +v\left( a\right) = \\ V_{a}\left( \hat{q}\left( \alpha _{p}\right) ,\hat{f}\right) =V_{a}\left( p,f\right) \label{equ4f} \end{array} $$

(24)

Thus, by Theorem 1 and Eqs. 19 and 20,

$$ \lambda \left( a^{\prime }\right) \sum_{\omega \in \Omega }p^{\prime }\left( \omega \right) U(f^{\prime }\left( \omega \right) ,\omega )+v\left( a^{\prime }\right) \geq \lambda \left( a\right) \sum_{\omega \in \Omega }p\left( \omega \right) U(f\left( \omega \right) ,\omega )+v\left( a\right). $$

(25)

If a and a′ are not elementarily linked, then, a and a′ are linked since A is linked. Define α ₁,...,α _n − 1 by $\left( a_{i},\alpha _{i}\bar{p}_{i}+\left( 1-\alpha _{i}\right) \underline{p}_{i},f^{\ast }\right) \sim \left( a_{i+1},\alpha _{i}\bar{p}_{i}^{\prime }+\left( 1-\alpha _{i}\right) \underline{p}_{i}^{\prime },f^{\ast }\right) ,$ where a = a ₁ and $a^{\prime }=a_{n-1}.$ The conclusion follows by repeated application of the representation.

$\left( b\right) \Rightarrow \left( a\right) .$ That (b) implies (A.1) – (A.5) follows from Theorem 1. That it implies (A.6) and (A.7) is immediate.

To prove the uniqueness part, note that, by Theorem 1, the functions U(·,ω),ω ∈ Ω, are unique up to a uniform positive linear transformation. Given U(·,ω),ω ∈ Ω, the uniqueness of λ(·) and v(·) follow from Eqs. 17–20. □