Information can sometimes be worse than useless

The idea of this model is that the result of the test is public information, capturing patient autonomy and informed consent established in the 2015 Montgomery case.

We assume the screening itself is exogenous, e.g. it’s offered as part of a routine checkup and the patient doesn’t consider whether to forgo it.

Nature draws a health state $\theta \in \set{S, H}$ with $\Pr(S) = p$ (small chance of sickness). A scan produces a public signal $\sigma \in \set{+, -}$ with:

• $\Pr(+\mid S) = 1$ (perfect sensitivity, i.e. never misses the condition if present)
• $\Pr(+\mid H) = f > 0$ (some false positives)

The subgame where $\sigma = -$ is trivial, since we then know for sure that $\theta=H$. Following a scan result $\sigma = +$, the doctor D chooses an action $a$, either investigating $I$ or waiting $W$:

• If $a=I$ then the game ends (we suppose that further tests are ordered to reveal whether the patient actually is sick).
• If $a=W$ then the true state $\theta$ is eventually revealed.
  • If $\theta = H$, the game ends (the patient is perfectly healthy)
  • If $\theta = S$, P chooses $l \in \set{L, N}$ (Litigate or Not, for having not been treated sooner).

The payoffs are defined as follows:

Outcome	Doctor	Patient
$(I, H)$	$-k$	$-c$
$(I, S)$	$-k$	$B - c - d$
$(W, H)$	$0$	$0$
$(W, S, N)$	$0$	$-d$
$(W, S, L)$	$-M$	$-d + R$

The motivation behind these payoffs is that:

• The doctor has some administrative costs $k$ involved with investigating
• The investigation also involves some costs to the patient $c$ (e.g. discomfort, anxiety, time), but means they secure some benefit $B$ from treatment when sick
• The disease has cost $d$
• If the doctor waited and the patient turned out to be sick, then a court will rule in favour of the patient should she choose to litigate. This results in the doctor suffering a malpractice penalty of $M$, and the patient getting a reward net of litigation expenses $R$.

We focus on the subgame where $\sigma = +$, and solve for equilibrium by backward induction.

First, note that Bayesian updating gives $q := P(S \mid +) = \frac{p}{p + f(1-p)}$.

Clearly at the final node, P will choose to litigate, since $R\gt 0$. Anticipating this, D compares $v_D(I) = -k$ to $v_D(W) = -qM$, thus choosing to investigate iff $q \gt k/M$. We can assume the malpractice penalty $M$ is much larger than the administrative costs $k$, so the threshold $k/M$ is small – i.e., D investigates for almost any posterior $q$.

The first-best outcome (i.e., that which a utilitarian social planner would implement) is that we investigate iff $qB \gt c + k \iff q \gt \frac{c + k}{B}$, and regardless of the outcome no litigation is allowed. So, there’s distortion when $\frac{k}{M} < q < \frac{c+k}{B}$: the doctor investigates even though it’s suboptimal.

Notably, when $\frac{k}{M} < q < \frac{c}{B}$, the outcome of the game is Pareto-dominated by the first-best outcome. It’s not merely an administrative hassle for the doctor – the patient’s interests are also harmed. And this is the region we’re likely in for mass asymptomatic testing, since the posterior for illness following a positive scan $q$ is indeed small (though not so small as $\frac{k}{M}$), due to low prevalence.

Even if the result from the scan is publicly known, the doctor might have special expertise that means they have more information about the patient’s health state than from the scan alone. This turns the setup into a signalling game. We introduce an intrinsic motivation parameter $\gamma$ to capture the doctor’s desire to treat genuine illnesses absent legal pressure.

Again, we’ll focus on the subgame where $\sigma = +$. Now, the doctor forms an expert assessment $e \in \set{h, l}$ before choosing $a \in \set{I, W}$ with prior $\Pr(e = h) = \alpha$, where $q_h := \Pr(S \mid {+}, h) > q_l := \Pr(S \mid {+}, l)$. Intuitively, the doctor’s experience means that they have various additional heuristics for assessing a positive-scan patient as high or low risk, but these can’t be easily explained to the patient or verified by them.

The payoffs are defined as follows:

Outcome	Doctor	Patient
$(I, H)$	$-k$	$- c$
$(I, S)$	$-k$	$B - c - d$
$(W, H)$	$0$	$0$
$(W, S, N)$	$-\gamma$	$-d$
$(W, S, L)$	$-\gamma - M_e$	$-d + R_e$

The main differences from before are now that:

• D has an intrinsic penalty $\gamma$ if they fail to investigate a sick patient
  • We assume $q_h \gamma > k > q_l \gamma$: absent litigation, the $h$-type strictly prefers $I$ and the $l$-type strictly prefers $W$.
  • We assume further that $\gamma$ is well-calibrated such that it is indeed socially optimal to investigate only high-risk patients, i.e. that $q_l​<\frac {c+k}B​ < q_h$.
• The malpractice penalty and litigation reward depend on the expert assessment $e$, explained below.

Under a documentation standard, the court ignores D’s private assessment, and will award damages whenever the scan was positive, no investigation occurred, but the patient was sick. We assume that the malpractice penalty $M$ is large enough that $q_l(\gamma + M) > k$, i.e. even for an $l$-type, the combined professional and malpractice cost of failing to investigate a patient who turns out to be sick exceeds the procedural cost.

In this setting, the unique equilibrium is pooling on $I$: both doctor types investigate every positive finding, regardless of their private expert assessment. So, we have unnecessary procedures in cases where $e=l$, similar to the overinvestigation described in the first model. Even though $l$-type doctors know that investigation isn’t worth it, the Montgomery regime means that they have to disclose the positive test result to the patient (rather than withholding it paternalistically) – and this leads to them overinvestigating to avoid litigation.

Prior to the Montgomery ruling, we had the Bolam test, for medical negligence, where courts would only find a doctor at fault if they failed to act in line with reasonable medical practice. Under this legal regime, doctors would’ve been able to withhold the screening result if they thought it was in the interests of the patient (e.g., if they had observed $e=l$ and therefore determined no investigation was worthwhile, even if $\sigma=+$). Modelling the scan result as private information would change the structure of the game, but is similar in spirit to a setup where courts defer to the medical establishment when assessing negligence claims. Under such an expertise standard, a patient who sues will win her case only if the doctor acted unreasonably, i.e. failed to take the optimal action ex ante – which in our model corresponds to if D chose $W$ despite $e=h$.

Here, the unique equilibrium is separating in dominant strategies, where $h$-types investigate and $l$-types wait. What’s interesting is that this is the first-best outcome, yet it arises from shifting autonomy from the patient to the doctor, contrary to recent trends.