TL;DR: How to reason about people who are smarter than you. A few proposals, interspersed with reinforcement learning and humorous fiction. Ending on a surprising connection to logical inductors.
The first thing that came to mind to me, from the question asked, is to mirror my own incentives. For me, being alive is good, treating a disease is annoying, but getting a disease really sucks, so just pass those incentives on to the doctor.
The algorithm, then:
1. always pay the doctor, regardless of diagnosis or not, as long as I'm alive
2. deduct a small amount from their account when they diagnose a disease
3. deduct a large amount when they fail to diagnose a disease and we eventually see results in later rounds
This biases towards overtreatment when there is some actual uncertainty (because the punishment to fail to treat is greater than the punishment for treating), but it should cut down on totally fraudulent diagnosis, while guaranteeing we treat real diseases.
However, this relies on having extra information than you posited later on, which is that we know nothing about our actual state. The fact that we can know nothing about our actual state, no matter the time horizon, seems unintuitive, given the problem setup, but maybe you just want to solve this particular problem because it applies to other domains. If so, maybe you would want to adjust the toy problem to say that this is, like, a one-shot cancer diagnosis where you won't know the results for a long time.
I'm aware that I might be missing the whole point of the thought experiment, but after reading the problem statement I just could not stop thinking about the ideal solution: start paying the doctor, asap, a lot of money, and keep paying as long as you're healthy, stopping the payments the moment you get sick. Given that the doctor is greedy (he will always want more, no matter how much he already has), he will do his best to keep you healthy so that the money does not stop flowing.
I don’t know why the current state of the world doesn’t work. Have a salaried independent board that can punish or suspend licences if diagnostic or practice does not meet generally accepted standards. Help, am I missing something?
"While this sounds clever, it falls into the same trap as before. When both doctors are greedy, they will coordinate and both always say that you are either healthy or that you need treatment."
You're walking right past the obvious solution here of simply not allowing the doctors to coordinate.
Have doctor #1 check you out. Then take an anonymized copy of your medical history, travel 500 miles in any direction, and find a doctor #2 to check you out. As long as they don't know each other and have no way of finding or contacting the other one, Scenario Two should work fine. For higher confidence, use more doctors. It's basically like avionics computers at this point.
I suppose it's possible that a set of sufficiently malicious doctors could all meet at a conference and agree that when a rich dude approaches them for a checkup, they'll run a specific hash of the person's name+birthdate+SSN, look up a disease using that number in a standard reference book, and all agree to say that rich dude has that expensive disease. Or if they wanted to be really clever, they could use a specific ordered list of non-fatal diseases which have long-running and expensive treatments.
But in that case, it would be virtually impossible for even the cleverest detective to find a pattern, so your Scenario Three would fail, as well.
Short of going to medical school yourself, I'm not sure there's a perfect solution to "every expert agrees to say the same wrong thing" -- but that's a conspiracy, and the likelihood of it being sustained depends on limiting the number of conspirators. There are millions of physicians in the world, and the conspiracy would need to include a significant fraction of them, so using the Grimes conspiracy equation, we can predict it wouldn't last very long.
The solution is to explain to the doctor that the only reasonable course of action for a person who loves anyone is to love everyone. I love you and I want you to be loved and cared for at all times. I cannot personally ensure that, so I should love everyone and promote everyone loving everyone to increase the chances that you will always be in the company of people who treat you lovingly.
Are we assuming that the doctor is indifferent to whether we learn the diagnosis, or that it's actively trying to prevent us from learning it? If withholding the diagnosis is valuable to the doctor, then what's stopping it from bribing the detective (by letting it win more often than random chance) in exchange for keeping their shared source of randomness a secret?
Let's say our payout is $100 and the doctor assigns a value of $2 to keeping my diagnosis a secret. So the doctor chooses a deterministic pattern which contains no information about ground truth (which the detective can therefore predict perfectly) and the detective intentionally guesses wrong 49% of the time. Now they each expect to get $51 of utility out of each prediction.
I don't think there's an equilibrium, though. They are both better off by colluding, but what's stopping the detective from just taking all of that "new" utility by guessing correctly more often? If the detective starts winning more than 52% of the time, the doctor will just start giving the real diagnosis. But between 50-52% I don't think there's a stable point for them to coordinate on. If they're true clones, though, they could "pre-determine" an arbitrary number in that range, like 51%, and be strictly better off than if they hadn't.
Cool thought experiment!! Speaking from experience, in any negotiation - the party who speaks first loses. The trick is to get the doc to name a price on his services, for say a year. If he wants X, you pay him. In fact, if you like his service, give him a $$ bonus, and stroke his ego by telling him how much he is appreciated. He wanted X, he is getting X+ ... I suspect he will even harder next year.
The issue is that everyone has a different agenda, so why not adjust the payment to match with the the same outcome for both? So you set up a subscription, and pay the same amount regardless of you being healthy or ill. It then incentivises the doctor to give you an earlier diagnosis for serious illnesses, when its cheaper to fix. The doctor is then running a business to keep you healthy for the least amount possible. It does sound like the idea behind the NHS, so it doesn't work when its scaled up, but for a single doctor, it might just work!
Have all doctors become employees of the state and be paid the same yearly income (based on a minimum of consultations/year) whether they result in further treatment or not. Consultations and treatments are "free" for the patient (paid for by public funds held in common). Your entire post is a capitalist problem and just proves how shite capitalism is for most of us.
I have the impression, you are somewhat sloppy with the question of repetition. For distinguishing between the troll doctors and the good doctors, you rely on being eventually able to make a judgement yourself. However, if repetition is foreseen, you can as well tell the doctor, that you will choose a different one / talk negatively about him, when you are posterior able to make a judgement. This gives an incentive to make the right diagnosis, because the doctor will want you to return. A lot of real world decisions are based on this concept.
The real problem is, when you can't make a timely judgement even after the fact - happens in medicine, because people overestimate, what real world medicine actually can do for them - or when it is crucial, you get the call right the first time.
If you solve this, you should advice the Saudi gov't. They desparately try to get scientists to work for them and ultimately diversify the Saudi economy. As they have no clue of anything much, they run into exactly the presented problem.
How about paying the doctor, but only randomly. This would take away the greed part. The doctor would know that keeping me healthy is the best course of action in the long run.
Get a second, highly-esteemed doctor who is paid a high flat rate whose only job is to assess how good a job the first doctor is doing (say, out of 100), and pay the first doctor in proportion to the score he is given by the second doctor?
Could you get get three opinions, chosen randomly and secretly to reduce coordination for a vote. Then you take away a large amount of money from the wrong doctor who disagrees and distribute to the correct 2 doctors. If all 3 agree, they all get paid the standard rate. If we assume that they cannot coordinate except through correct diagnosis, their financial interest is aligned. The issue is preventing collusion among the 3 doctors, which is why it needs a facilities to handle that (central administrator, patient anonymization, etc). Bonus it also eliminates the greedy stupid doctor as they are more likely to disagree with 2 greedy, smart doctors, lose all their money.
Put the doctor on retainer with an increasing payment based on the expectancy of death in any given year.
The first thing that came to mind to me, from the question asked, is to mirror my own incentives. For me, being alive is good, treating a disease is annoying, but getting a disease really sucks, so just pass those incentives on to the doctor.
The algorithm, then:
1. always pay the doctor, regardless of diagnosis or not, as long as I'm alive
2. deduct a small amount from their account when they diagnose a disease
3. deduct a large amount when they fail to diagnose a disease and we eventually see results in later rounds
This biases towards overtreatment when there is some actual uncertainty (because the punishment to fail to treat is greater than the punishment for treating), but it should cut down on totally fraudulent diagnosis, while guaranteeing we treat real diseases.
However, this relies on having extra information than you posited later on, which is that we know nothing about our actual state. The fact that we can know nothing about our actual state, no matter the time horizon, seems unintuitive, given the problem setup, but maybe you just want to solve this particular problem because it applies to other domains. If so, maybe you would want to adjust the toy problem to say that this is, like, a one-shot cancer diagnosis where you won't know the results for a long time.
I'm aware that I might be missing the whole point of the thought experiment, but after reading the problem statement I just could not stop thinking about the ideal solution: start paying the doctor, asap, a lot of money, and keep paying as long as you're healthy, stopping the payments the moment you get sick. Given that the doctor is greedy (he will always want more, no matter how much he already has), he will do his best to keep you healthy so that the money does not stop flowing.
The second solution is similar to Lex's Arguing Machine. https://www.youtube.com/watch?v=YBvcKtLKNAw
Imagine if we can blend second and third solution into something more accurate. Also, is this system tied to Hanson v Yarvin on decision markets?
I don’t know why the current state of the world doesn’t work. Have a salaried independent board that can punish or suspend licences if diagnostic or practice does not meet generally accepted standards. Help, am I missing something?
"While this sounds clever, it falls into the same trap as before. When both doctors are greedy, they will coordinate and both always say that you are either healthy or that you need treatment."
You're walking right past the obvious solution here of simply not allowing the doctors to coordinate.
Have doctor #1 check you out. Then take an anonymized copy of your medical history, travel 500 miles in any direction, and find a doctor #2 to check you out. As long as they don't know each other and have no way of finding or contacting the other one, Scenario Two should work fine. For higher confidence, use more doctors. It's basically like avionics computers at this point.
I suppose it's possible that a set of sufficiently malicious doctors could all meet at a conference and agree that when a rich dude approaches them for a checkup, they'll run a specific hash of the person's name+birthdate+SSN, look up a disease using that number in a standard reference book, and all agree to say that rich dude has that expensive disease. Or if they wanted to be really clever, they could use a specific ordered list of non-fatal diseases which have long-running and expensive treatments.
But in that case, it would be virtually impossible for even the cleverest detective to find a pattern, so your Scenario Three would fail, as well.
Short of going to medical school yourself, I'm not sure there's a perfect solution to "every expert agrees to say the same wrong thing" -- but that's a conspiracy, and the likelihood of it being sustained depends on limiting the number of conspirators. There are millions of physicians in the world, and the conspiracy would need to include a significant fraction of them, so using the Grimes conspiracy equation, we can predict it wouldn't last very long.
The solution is to explain to the doctor that the only reasonable course of action for a person who loves anyone is to love everyone. I love you and I want you to be loved and cared for at all times. I cannot personally ensure that, so I should love everyone and promote everyone loving everyone to increase the chances that you will always be in the company of people who treat you lovingly.
Are we assuming that the doctor is indifferent to whether we learn the diagnosis, or that it's actively trying to prevent us from learning it? If withholding the diagnosis is valuable to the doctor, then what's stopping it from bribing the detective (by letting it win more often than random chance) in exchange for keeping their shared source of randomness a secret?
Let's say our payout is $100 and the doctor assigns a value of $2 to keeping my diagnosis a secret. So the doctor chooses a deterministic pattern which contains no information about ground truth (which the detective can therefore predict perfectly) and the detective intentionally guesses wrong 49% of the time. Now they each expect to get $51 of utility out of each prediction.
I don't think there's an equilibrium, though. They are both better off by colluding, but what's stopping the detective from just taking all of that "new" utility by guessing correctly more often? If the detective starts winning more than 52% of the time, the doctor will just start giving the real diagnosis. But between 50-52% I don't think there's a stable point for them to coordinate on. If they're true clones, though, they could "pre-determine" an arbitrary number in that range, like 51%, and be strictly better off than if they hadn't.
Could you pay the greedy doctor the same amount regardless of her conclusion? She would have no incentive to give inaccurate advice.
Cool thought experiment!! Speaking from experience, in any negotiation - the party who speaks first loses. The trick is to get the doc to name a price on his services, for say a year. If he wants X, you pay him. In fact, if you like his service, give him a $$ bonus, and stroke his ego by telling him how much he is appreciated. He wanted X, he is getting X+ ... I suspect he will even harder next year.
The issue is that everyone has a different agenda, so why not adjust the payment to match with the the same outcome for both? So you set up a subscription, and pay the same amount regardless of you being healthy or ill. It then incentivises the doctor to give you an earlier diagnosis for serious illnesses, when its cheaper to fix. The doctor is then running a business to keep you healthy for the least amount possible. It does sound like the idea behind the NHS, so it doesn't work when its scaled up, but for a single doctor, it might just work!
Have all doctors become employees of the state and be paid the same yearly income (based on a minimum of consultations/year) whether they result in further treatment or not. Consultations and treatments are "free" for the patient (paid for by public funds held in common). Your entire post is a capitalist problem and just proves how shite capitalism is for most of us.
I have the impression, you are somewhat sloppy with the question of repetition. For distinguishing between the troll doctors and the good doctors, you rely on being eventually able to make a judgement yourself. However, if repetition is foreseen, you can as well tell the doctor, that you will choose a different one / talk negatively about him, when you are posterior able to make a judgement. This gives an incentive to make the right diagnosis, because the doctor will want you to return. A lot of real world decisions are based on this concept.
The real problem is, when you can't make a timely judgement even after the fact - happens in medicine, because people overestimate, what real world medicine actually can do for them - or when it is crucial, you get the call right the first time.
If you solve this, you should advice the Saudi gov't. They desparately try to get scientists to work for them and ultimately diversify the Saudi economy. As they have no clue of anything much, they run into exactly the presented problem.
How about paying the doctor, but only randomly. This would take away the greed part. The doctor would know that keeping me healthy is the best course of action in the long run.
Get a second, highly-esteemed doctor who is paid a high flat rate whose only job is to assess how good a job the first doctor is doing (say, out of 100), and pay the first doctor in proportion to the score he is given by the second doctor?
Could you get get three opinions, chosen randomly and secretly to reduce coordination for a vote. Then you take away a large amount of money from the wrong doctor who disagrees and distribute to the correct 2 doctors. If all 3 agree, they all get paid the standard rate. If we assume that they cannot coordinate except through correct diagnosis, their financial interest is aligned. The issue is preventing collusion among the 3 doctors, which is why it needs a facilities to handle that (central administrator, patient anonymization, etc). Bonus it also eliminates the greedy stupid doctor as they are more likely to disagree with 2 greedy, smart doctors, lose all their money.