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I had the same thought, but I can think of two problems with it.

1. The doctor's stake is pretty limited.

The old royal court doctor model is the perfect model for this. It's based on the doctor having very few (one?) patient. If the patient dies, not only do you risk your substantial income and high status, but you might even lose your life if the royal family gets suspicious enough.

It's a good solution if you're very rich. Much richer than the average doctor.

2. When you have more than one patient, at some point you probably earn more by treating less.

The less work you spend on each patient, the more patients you can take on. The more patients you can take on, the more you earn. If you simply stop treating patients - "you're perfectly healthy, don't worry" - you can basically take on an unlimited amount of patients and collect their fees until they perish. (This sounds like a caricature of the US insurance system, from my western EU perspective.)

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What about this convoluted system:

- The doctors get paid a recurring fee that starts small and increases the longer you keep a patient around. This counters the problem that patients can just be replaced by a younger and healthier patient.

- The doctors get assigned patients randomly. This addresses the problem that doctors would choose young and healthy patients.

- The doctors can only have a certain number of patients at once. This makes each patient more valuable, and the doctor will be incentivized to actually treat patients, instead of having as many as possible.

Oh wait...

Now, the problem is that a young, ambitious, greedy doctor would be incentivized to cull their set of patients and select for the most viable ones in the long run. This actually incentivizes them to kill off patients with minor defects?




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