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-3emHippocratic Utility
\maketitle
\begin{abstract}
A utility function has been proposed that values more those lives that are saved by not imposing a harmful treatment and values less those lives that could be saved by treating people who would otherwise die. I do not dispute the ethical motivation behind this kind of asymmetry. However, as my example illustrates, the scope of applicability of such a decision criterion may be limited.
\end{abstract}
\section{A Decision Criterion}
Imagine the following scenario: There is a disease. Randomized controlled trials show that the ``old'' and default drug, developed in 1926, saves 10\% percent of lives. According to randomized controlled trials, a ``new'' drug, just developed in 2026, saves 20\% of lives. A decision maker has to decide whether to give the default drug ($d=0)$ or the new drug ($d=1$) to a patient who has just developed the disease.
This situation can be captured by the potential outcomes model \citep{neyman1923application,rubin1974estimating}. We divide the population into four principal strata. Each patient has deterministic responses $y$ to both drugs: $y^d=0$ if the patient dies and $y^d=1$ if they survive, for $d=0, 1$. The type of the patient is unknown to the decision maker, who only knows that $P(y^0=1)=10\%$ and $P(y^1=1)=20\%$. That is, the marginals of the distribution are given, but the joint distribution $P$ is unknown. The marginals are the most information we can hope to obtain from a randomized controlled trial.
Suppose that the decision maker maximizes expected utility:
$\max_{d\in \{0, 1\}} \mathbb{E}_P \ u(d, y^0, y^1),$
where $\mathbb{E}_P$ is the expectation under probability $P$ and $u:D\times Y\times Y\to \mathbb{R}$ is the utility function that depends on the decision $d\in D$ and the vector potential outcomes $y\in Y\times Y$.\footnote{Or equivalently, as is often done in statistics, minimizes the loss function $\ell=-u$.}
Crucially, utility depends on the whole vector of potential outcomes $(y^0, y^1)$. Such utility functions have recently been proposed in the medical decision making literature \citep{ben2024policy,christy2024starting}. Motivated by the Hippocratic Oath ``first, do no harm,'' the literature proposes what I will call the Hippocratic utility function, where the utility difference between $d=1$ and $d=0$ is given by:
\begin{equation}
u(d=1, y^0, y^1)-u(d=0, y^0, y^1)=\begin{cases}
0 &\text{ if }y^0=y^1\\
1 &\text{ if }y^0<y^1\\
-\lambda &\text{ if }y^0>y^1,\\
\end{cases}
\end{equation}
where $\lambda>1$. That is, the loss of a life caused by the new drug is weighted more heavily than the loss of a life caused by the old drug. The parameter $\lambda>1$ measures the degree of the asymmetry in the utility function. Formally this is similar to loss aversion of \cite{KT79}, except they were using this formalism to provide a descriptive theory of behavior, whereas we are now in the normative realm. Many scholars have a deep conviction that this is the normatively correct utility function. The purpose of this note is not to try to change their deep conviction, but to point out that the scope of applications of such utility functions may be very limited.
\section{Some Properties of this Criterion}
I will discuss three properties of this utility function. The first two are very well known. The main argument of this note revolves around the third property.
\subsection{Choosing the Dominated Option}
Much has been discussed about the fact that if $\lambda$ is high enough, then the old drug will prescribed even though the new one saves twice as many lives. More formally, for any $P\in \mathcal{P}$ that puts a positive probability on all four strata there exists $\lambda>1$ such that the optimal decision is $d=0$. To see that, let the probabilities of the four strata under $P$ be $P^{00} :=P(y^0=0, y^1=0), P^{01}:=P(y^0=0, y^1=1), P^{10}:=P(y^0=1, y^1=0),P^{11}:=P(y^0=1, y^1=1)$. The expected utility difference between $d=1$ and $d=0$ is $P^{01}-\lambda P^{10}$. We know that $P^{01}+P^{11}=20\%$ and $P^{10}+P^{11}=10\%$. Thus, the expected utility difference is $10\%+(1-\lambda)P^{10}$. This is less than zero for $\lambda$ large enough, provided that $P^{10}>0$.\footnote{This is bigger than zero for all values of $\lambda>1$ when $P^{10}=0$, which \cite{pearl2022probabilities} calles the monotonicity assumption. This is related but distinct from the monotonicity assumption of \cite{ImbensGuidoW1994IAEO}.}
This violation of stochastic dominance is caused by the fact that Hippocratic utility penalizes deaths of people in different strata differently. The proponents of Hippocratic utility argue that this is what they indeed mean to do and claim that counting lives of some people more than other people is ethically well-motivated.
Based on this violation of dominance, \cite{gelman2024russian} argue that utility should depend only on the realized outcome, or only on the marginals (what is sometimes called ``stochastic potential outcomes''). Both of those proposals rule out Hippocratic utility because they lead to the decision $d=1$.
\subsection{Choice Indeterminacy}
It has also been noticed that if only marginals of $P$ are known, the decision criterion does not offer any clear advice to the decision maker. As we showed in the example above, not all probability measures $P\in \mathcal{P}$ lead to the same decision because expected utility depends on the correlation structure of $P$. In our example, if $P\in \mathcal{P}$ and $P^{01}=0$, then no matter how high $\lambda$ is, it is optimal to choose $d=1$, while for all other $P\in \mathcal{P}$ the decision $d=0$ is taken for high enough values of $\lambda$.
Thus, the decision will depend on a ``free parameter.'' To proceed, we could choose $P\in \mathcal{P}$ to minimize regret, like \cite{ben2024policy}. Or we could instead use maxmin expected utility: $\max_{d\in \{0, 1\}} \min_{P\in \mathcal{P}} \mathbb{E}_P \ u(d, y^0, y^1).$ We could also assume that $P$ is a product measure. Or we could use maxmax utility. There are infinitely many possibilities\ldots
Each of these ways to choose the free parameter will lead to a different decision rule. And the choice between these rules is not based on the data at hand, and not even on the value of $\lambda$, but rather on some other reasoning.
Another solution to this problem is to only use utility functions that lead to the same decision for all $P\in \mathcal{P}$. \cite{koch2025statistical} show that the value of expected utility depends only on the marginals if and only if the utility is an additive function of the vector of potential outcomes. This rules out Hippocratic utility (and coincides with \citeauthor{gelman2024russian}'s \citeyearpar{gelman2024russian} proposal in the binary case).
\subsection{Status Quo Bias}
To be fair to the literature, the proponents of the utility function with $\lambda>1$ focus on slightly different situations than the one described here. My thought experiment corresponds to decision problem 1 in Table \ref{tab:problems}, where $d=0$ is one chemical compound (call it $A$) and $d=1$ is another chemical compound (call it $B$). In the original application we are facing decision problem 2, where choosing $d=0$ corresponds to not giving a drug at all.
\begin{table}[htb!]
\centering
\begin{tabular}{c|cc}
\hline
decision problem & $d=0$ & $d=1$ \\
\hline
1 & compound $A$ & compound $B$\\
2 & do not treat & compound $B$\\
3 & compound $B$ & compound $A$\\
4 & do not treat, patient self-medicates with $A$ & compound $B$\\
\hline
\end{tabular}
\caption{Decision Problems} \label{tab:problems}
\end{table}
I am making the modification to the original example to illustrate an important point. Consider now decision problem 3, where compound $B$ was discovered in 1926 and compound $A$ was discovered in 2026. Now a decision maker who has $\lambda$ high enough will prescribe compound $B$ to the patient. Thus, by comparing decision problems 1 and 3, we conclude that Hippocratic utility suffers from status-quo bias \citep{samuelson1988status}: the decision whether to prescribe compound $A$ or compound $B$ depends on a completely random historical fact (which compound was discovered first).
I have a hard time believing that a decision maker who wants to do ``no harm'' would want to base their decisions on random historical facts. I hope that the reader agrees with me here! I am not arguing against using Hippocratic utility in decision problem 2, just that it does not apply to decision problems 1 and 3.
In principle, there is a way to apply Hippocratic utility to decision problems 1 and 3 without introducing status quo bias. Here, the researcher would define the utility function directly over objects like compound $A$ and compound $B$, not objects like $d=0$ and $d=1$. This leads to internally consistent choices that satisfy completeness and transitivity. But asymmetric loss still leads to violations of stochastic dominance. The only difference from before is that without status-quo bias the decision maker prefers compound $A$ over compound $B$ independently of which one is the ``default action'' or which one was discovered first. So now we are sacrificing lives not because of these things, but rather because of the chemical formula of compound $A$ and compound $B$. Is this defensible?\footnote{Of course, we could violate dominance because of the production costs of different compounds. But those costs do not depend on strata, so they cannot lead to Hippocratic Utility.}
\section{Discussion}
Using Hippocratic utility in medical decision-making requires a principled account of what the status quo is. When it is ``no intervention,'' there's a case to be made. When it's ``whichever drug was discovered first,'' there isn't.
But what does ``no intervention'' really mean? Consider decision problem 4, which is a slight modification of decision problem 2. Now compound $A$ is sold over-the-counter. The decision maker thinks that if decision $d=0$ ``no treatment at all'' is taken, the patient will start self-medicating with compound $A$ with some probability. If that probability is high, the effective value of $\lambda$ should be close to one, so the same decision is made as with $\lambda=1$. Thus, I hope that if you like Hippocratic utility because of your ethical conviction, you agree with me that we should choose $d=1$ in decision problem 4.
There are also legal arguments for using Hippocratic utility. When a treated patient dies, it is much easier for their family to sue the hospital, compared to the family of a dead, untreated patient \citep{tian2000probabilities}. In such applications the value of $\lambda$ is proportional to the probability of the lawsuit. This motivation for Hippocratic Utility is distinct from the ethical one, because here we would choose $d=0$ in both decision problems 2 and 4.
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