Wouldn't they just coordinate on diagnosing all but the most obviously healthy patient as ill?
Treating health as a continuous rather than binary variable does complicate this problem and, I think, breaks my solution. If the doctors agree on an ordinal ranking of all patients from "most ill" to "most hale", they can coordinate their diagnoses much easier, as they can search over a smaller space of ensemble diagnoses. If there are lots of "degenerate cases" (i.e. people with the same degree of illness) this might be harder. Requiring a certain minimum Hamming distance (based on some prior) from the all-ill Schelling point doesn't help at all in the case of nondegenerate ordinal ranking.
I suspect if you model it out, and assign varying strength of illness signal and varying noise of diagnosis for different doctors' ability, you'll find that this only works when the signals are clear enough that you don't need a doctor to tell someone they're ill.
You've also forgotten that doctors don't have to coordinate in detail in order to coordinate on answers. Medical school and diagnostic guides are chock full of coordination mechanisms for measurement->diagnosis. In your scheme, doctors who follow the guides and never use their experience, intuition, and good judgement to think deeper are the ones who get paid. Wilson gets paid, House goes elsewhere. That may be your goal - "prefer a younger doctor" is common advice for a reason. But be aware that you're optimizing for standards, not truth.
They could just coordinate to always mark the first patient as ill, but none of the others
More likely, they'll converge on using easy heuristics for diagnoses, which ruins the purpose of seeing a doctor in the first place, since you could just use the heuristics yourself
They could coordinate based on patient race, or sex.
Whatsmore, if the line between ill and hale is at all ambiguous or hard to measure precisely, this method starts to fail hard.
I'm assuming that race and sex and all other discernable features are independent and identically distributed, and also that the game overseer will not pay the doctors if they coordinate on obvious patterns involving race, sex, etc.
If only a few of the patients are ambiguously ill, it might be possible to discern this by observing an almost perfect agreement between ensemble diagnoses. In the iterated game, this might cause the doctors to preferentially flip diagnoses that they think are more ambiguous or not flip diagnoses that they think are more certain. This is not a perfect guarantee, of course.
A naive solution is to get a second opinion. However, a second doctor is also incentivized to diagnose you ill!
Why is this? They don't get paid more for treatment. What's wrong with paying one doctor to recommend a treatment and then paying a second to enact it?
There was a post a while back (by Eliezer I think) about separating diagnosis and treatment and how that would help align the incentives, but now I can't find it.
The problem with this solution is that the doctors can precommit to always diagnosing ill through acausal trade, so that they are rewarded in the case that they are the one who is randomly chosen for treatment (assuming they know the game).
It may be worthwhile to extend this to doctors who aren't perfect - that is are only correct most of the time - and see what happens.
Feels like we could escape the risk of coordination with many patients distributed over many doctors, and patient and doctor allocation is always random.
The greedy doctor problem is defined as follows: suppose you want a possible illness to be diagnosed correctly, but your doctor is incentivized to diagnose you sick because they can then charge you more for treatment. How do you get an accurate diagnosis and avoid paying more than you need?
A naive solution is to get a second opinion. However, a second doctor is also incentivized to diagnose you ill! What if you leverage adversarial dynamics and pay one doctor for treatment only if both doctors agree you are ill and the other only if both doctors agree you are hale? Unfortunately, this instantiates a form of the "battle of the sexes" game and the doctors randomise their diagnoses according to a mixed-strategy Nash equilibrium. Neither of these approaches reliably gets you diagnosed correctly.
Assuming the doctors cannot coordinate through signalling, I think ensembling might solve the problem in the absence of unforseen acausal coordination. My (totally impractical) strategy is as follows:
There are three obvious Schelling points: diagnosing all patients ill; diagnosing all patients hale; and diagnosing all patients truthfully (assuming that the doctors agree). Likely, at least one patient will be genuinely ill and at least one will be genuinely hale. The doctors can get more money if they diagnose some patients as ill, which biases them away from the all-hale Schelling point. The doctors will not get paid if they diagnose all patients as ill, which biases them away from the all-ill Schelling point. Thus, the only remaining obvious Schelling point is all-truthful.
It is possible that the doctors could acausally coordinate on an alternative Schelling point. For example, there might be mutually exploitable patterns in the data or precommited doctor strategies (e.g. from medical school, the scoundrels) that allow the doctors to diagnose more patients as ill and thus receive a higher pay. If any untruthful Schelling point is known to the game overseer, they can disincentivise coordination on it by refusing to pay the doctors if they coordinate on it. Thus, only unforseen Schelling points are dangerous in single-shot iterations of the greedy doctor game.
In iterated forms of this game, it is important to not tell the doctors why they were or were not paid to prevent signalling and coordination. Even a simple distance measure between the doctors' diagnoses, such as the Hamming distance, might allow the doctors to coordinate over time. If the doctors only receive the "paid/unpaid" signal, in the absence of acausal coordination, they will likely search in the vicinity of a Schelling point until the "paid" signal is received. If the number of game iterations is large and the patient number is small, the doctors will probably coordinate away from the all-truthful Schelling point. Thus, increasing the number of patients is critical to prevent trivial coordination over time. Increasing the number of doctors also can reduce the probability of trivial coordination.
If the game overseer is confident that some minimum percentage of patients is genuinely hale, they can refuse to pay the doctors if their diagnosis ensembles agree within a certain Hamming distance of the known untruthful Schelling points. This makes it harder for the doctors to trivially coordinate, as the set of diagnosis ensembles in the vicinity of the Hamming distance perimeter is large.