I don’t understand how you think I’ve changed the problem.
The problem was about dynamic inconsistency in beliefs, while you are talking about a solution to dynamic inconsistency in actions. Your assumption that people act independently from each other, which was not part of the original problem, - it was even explicitly mentioned that people have enough time to discuss the problem and come to a coordinated solution, before the experiment started, - allowed you to ignore this nuance.
Eliezer seems to have endorsed getting rid of the anthropic aspect, as I have done.
As I stated in my post, anthropic and non-anthropic variants are fully isomorphic, so this aspect is indeed irrelevant.
I suspect that you just want to outlaw the whole concept that the different people with green balls might do different things.
The problem is not about what people might do. The problem is about what people should do. What is the correct estimation of expected utility? Based on the P(Heads) being 50% or 90%? Should people just foresee that decision makers will update their probability estimate to 90% and thus agree to take the bet in the first place? What is the probability theory justification for doing/not doing it?
But in actual reality, different people may do different things.
In actual reality all kind of mathematical models happen to be useful. You can make an argument that a specific mathematical model isn’t applicable for some real world case or is applicable to less real world cases than a different model. That doesn’t mean that you have invalidated the currently less useful model or solved its problems. In general, calling a scenarios, that we can easily implement in code right now, to be “unrealistic thought experiments” and assuming that they will never be relevant, is quite misguided. It would be so, even if your more complicated solution, with different assumptions, indeed were more helpful to the current most common real world cases.
Which isn’t actually the case as far as I can tell.
Let’s get as close to the real world as possible and, granted, focus on the problem as you’ve stated it. Suppose we actually did this experiment. What is the optimal strategy?
Your graph shows that expected utility tops when p=1. So does it mean that based on your analysis people should always take the bet? No, this can’t be right, after all then we are loosing the independence condition. But what about having the probability of taking the bet be very high, like 0.9 or even 0.99? Can a group of twenty people with unsynchronized random number generators—arguably not exactly your most common real world case, but let’s grant that too—use this strategy to do better than always refusing the bet?
Your graph shows that expected utility tops when p=1. So does it mean that based on your analysis people should always take the bet?
What this is saying is that if everyone other than you always takes the bet, then you should as well. Which is true; if the other 19 people coordinated to always take the bet, and you get swapped in as the last person and your shirt is green, you should take the bet. Because you’re definitely pivotal and there’s a 9⁄10 chance there are 18 greens.
If 19 always take the bet and one never does, the team gets a worse expected utility than if they all always took the bet. Easy to check this.
Another way of thinking about this is that if green people in general take the bet 99% of the time, that’s worse than if they take the bet 100% of the time. So on the margin taking the bet more often is better at some point.
Globally, the best strategy is for no one to take the bet. That’s what 20 UDTs would coordinate on ahead of time.
I’m having a hard time making sense of what you’re arguing here:
The problem was about dynamic inconsistency in beliefs, while you are talking about a solution to dynamic inconsistency in actions.
I don’t see any inconsistency in beliefs. Initially, everyone thinks that the probability that the urn with 18 green balls is chosen is 1⁄2. After someone picks a green ball, they revise this probability to 9⁄10, which is not an inconsistency, since they have new evidence, so of course they may change their belief. This revision of belief should be totally uncontroversial. If you think a person who picks a green ball shouldn’t revise their probability in this way then you are abandoning the whole apparatus of probability theory developed over the last 250 years. The correct probability is 9⁄10. Really. It is.
I take the whole point of the problem to be about whether people who for good reason initially agreed on some action, conditional on the future event of picking a green ball, will change their mind once that event actually occurs—despite that event having been completely anticipated (as a possibility) when they thought about the problem beforehand. If they do, that would seem like an inconsistency. What is controversial is the decision theory aspect of the problem, not the beliefs.
Your assumption that people act independently from each other, which was not part of the original problem, - it was even explicitly mentioned that people have enough time to discuss the problem and come to a coordinated solution, before the experiment started, - allowed you to ignore this nuance.
As I explain above, the whole point of the problem is whether or not people might change their minds about whether or not to take the bet after seeing that they picked a green ball, despite the prior coordination. If you build into the problem description that they aren’t allowed to change their minds, then I don’t know what you think you’re doing.
My only guess would be that you are focusing not on the belief that the urn with 18 green balls was chosen, but rather on the belief in the proposition “it would be good (in expectation) if everyone with a green ball takes the bet”. Initially, it is rational to think that it would not be good for everyone to take the bet. But someone who picks a green ball should give probability 9⁄10 to the proposition that the urn with 18 balls was chosen, and therefore also to the proposition that everyone taking the bet would result in a gain, not a loss, and one can work out that the expected gain is also positive. So they will also think “if I could, I would force everyone with a green ball to take the bet”. Now, the experimental setup is such that they can’t force everyone with a green ball to take the bet, so this is of no practical importance. But one might nevertheless think that there is an inconsistency.
But there actually is no inconsistency. Seeing that you picked a green ball is relevant evidence, that rightly changes your belief in whether it would be good for everyone to take the bet. And in this situation, if you found some way to cheat and force everyone to take the bet (and had no moral qualms about doing so), that would in fact be the correct action, producing an expected reward of 5.6, rather than zero.
I don’t see any inconsistency in beliefs. Initially, everyone thinks that the probability that the urn with 18 green balls is chosen is 1⁄2. After someone picks a green ball, they revise this probability to 9⁄10, which is not an inconsistency, since they have new evidence, so of course they may change their belief. This revision of belief should be totally uncontroversial. If you think a person who picks a green ball shouldn’t revise their probability in this way then you are abandoning the whole apparatus of probability theory developed over the last 250 years. The correct probability is 9⁄10. Really. It is.
I don’t like this way of argument by authority and sheer repetition.
That said, I feel totally confused about the matter so I can’t say whether I agree or not.
Well, for starters, I’m not sure that Ape in the coat disagrees with my statements above. The disagreement may lie elsewhere, in some idea that it’s not the probability of the urn with 18 green balls being chosen that is relevant, but something else that I’m not clear on. If so, it would be helpful if Ape in the coat would confirm agreement with my statement above, so we could progress onwards to the actual disagreement.
If Ape in the coat does disagree with my statement above, then I really do think that that is in the same category as people who think the “Twin Paradox” disproves special relativity, or that quantum mechanics can’t possibly be true because it’s too weird. And not in the sense of thinking that these well-established physical theories might break down in some extreme situation not yet tested experimentally—the probability calculation above is of a completely mundane sort entirely analogous to numerous practical applications of probability theory. Denying it is like saying that electrical engineers don’t understand how resistors work, or that civil engineers are wrong about how to calculate stresses in bridges.
The problem was about dynamic inconsistency in beliefs, while you are talking about a solution to dynamic inconsistency in actions. Your assumption that people act independently from each other, which was not part of the original problem, - it was even explicitly mentioned that people have enough time to discuss the problem and come to a coordinated solution, before the experiment started, - allowed you to ignore this nuance.
As I stated in my post, anthropic and non-anthropic variants are fully isomorphic, so this aspect is indeed irrelevant.
The problem is not about what people might do. The problem is about what people should do. What is the correct estimation of expected utility? Based on the P(Heads) being 50% or 90%? Should people just foresee that decision makers will update their probability estimate to 90% and thus agree to take the bet in the first place? What is the probability theory justification for doing/not doing it?
In actual reality all kind of mathematical models happen to be useful. You can make an argument that a specific mathematical model isn’t applicable for some real world case or is applicable to less real world cases than a different model. That doesn’t mean that you have invalidated the currently less useful model or solved its problems. In general, calling a scenarios, that we can easily implement in code right now, to be “unrealistic thought experiments” and assuming that they will never be relevant, is quite misguided. It would be so, even if your more complicated solution, with different assumptions, indeed were more helpful to the current most common real world cases.
Which isn’t actually the case as far as I can tell.
Let’s get as close to the real world as possible and, granted, focus on the problem as you’ve stated it. Suppose we actually did this experiment. What is the optimal strategy?
Your graph shows that expected utility tops when p=1. So does it mean that based on your analysis people should always take the bet? No, this can’t be right, after all then we are loosing the independence condition. But what about having the probability of taking the bet be very high, like 0.9 or even 0.99? Can a group of twenty people with unsynchronized random number generators—arguably not exactly your most common real world case, but let’s grant that too—use this strategy to do better than always refusing the bet?
What this is saying is that if everyone other than you always takes the bet, then you should as well. Which is true; if the other 19 people coordinated to always take the bet, and you get swapped in as the last person and your shirt is green, you should take the bet. Because you’re definitely pivotal and there’s a 9⁄10 chance there are 18 greens.
If 19 always take the bet and one never does, the team gets a worse expected utility than if they all always took the bet. Easy to check this.
Another way of thinking about this is that if green people in general take the bet 99% of the time, that’s worse than if they take the bet 100% of the time. So on the margin taking the bet more often is better at some point.
Globally, the best strategy is for no one to take the bet. That’s what 20 UDTs would coordinate on ahead of time.
I’m having a hard time making sense of what you’re arguing here:
I don’t see any inconsistency in beliefs. Initially, everyone thinks that the probability that the urn with 18 green balls is chosen is 1⁄2. After someone picks a green ball, they revise this probability to 9⁄10, which is not an inconsistency, since they have new evidence, so of course they may change their belief. This revision of belief should be totally uncontroversial. If you think a person who picks a green ball shouldn’t revise their probability in this way then you are abandoning the whole apparatus of probability theory developed over the last 250 years. The correct probability is 9⁄10. Really. It is.
I take the whole point of the problem to be about whether people who for good reason initially agreed on some action, conditional on the future event of picking a green ball, will change their mind once that event actually occurs—despite that event having been completely anticipated (as a possibility) when they thought about the problem beforehand. If they do, that would seem like an inconsistency. What is controversial is the decision theory aspect of the problem, not the beliefs.
As I explain above, the whole point of the problem is whether or not people might change their minds about whether or not to take the bet after seeing that they picked a green ball, despite the prior coordination. If you build into the problem description that they aren’t allowed to change their minds, then I don’t know what you think you’re doing.
My only guess would be that you are focusing not on the belief that the urn with 18 green balls was chosen, but rather on the belief in the proposition “it would be good (in expectation) if everyone with a green ball takes the bet”. Initially, it is rational to think that it would not be good for everyone to take the bet. But someone who picks a green ball should give probability 9⁄10 to the proposition that the urn with 18 balls was chosen, and therefore also to the proposition that everyone taking the bet would result in a gain, not a loss, and one can work out that the expected gain is also positive. So they will also think “if I could, I would force everyone with a green ball to take the bet”. Now, the experimental setup is such that they can’t force everyone with a green ball to take the bet, so this is of no practical importance. But one might nevertheless think that there is an inconsistency.
But there actually is no inconsistency. Seeing that you picked a green ball is relevant evidence, that rightly changes your belief in whether it would be good for everyone to take the bet. And in this situation, if you found some way to cheat and force everyone to take the bet (and had no moral qualms about doing so), that would in fact be the correct action, producing an expected reward of 5.6, rather than zero.
I don’t like this way of argument by authority and sheer repetition.
That said, I feel totally confused about the matter so I can’t say whether I agree or not.
Well, for starters, I’m not sure that Ape in the coat disagrees with my statements above. The disagreement may lie elsewhere, in some idea that it’s not the probability of the urn with 18 green balls being chosen that is relevant, but something else that I’m not clear on. If so, it would be helpful if Ape in the coat would confirm agreement with my statement above, so we could progress onwards to the actual disagreement.
If Ape in the coat does disagree with my statement above, then I really do think that that is in the same category as people who think the “Twin Paradox” disproves special relativity, or that quantum mechanics can’t possibly be true because it’s too weird. And not in the sense of thinking that these well-established physical theories might break down in some extreme situation not yet tested experimentally—the probability calculation above is of a completely mundane sort entirely analogous to numerous practical applications of probability theory. Denying it is like saying that electrical engineers don’t understand how resistors work, or that civil engineers are wrong about how to calculate stresses in bridges.