Thanks for the answer, but I am afraid I am more confused than before. In the part of the post which begins, “So, new problem...”, the coin is gone, and instead Omega will decide what to do based on whether an urn contains a red or blue marble, about which you have certain information. There is no coin. Can you restate your explanation in terms of the urn and marble?
I’m not sure what exactly confuses you. Coin, urn, what does it matter? See the original post for the context in which the coin is used. Consider it a rigged coin, one probability of which landing on each side was a topic of that debate MBlume talks about.
Let me try restating the scenario more explicitly, see if I understand that part.
Omega comes to you and says, “There is an urn with a red or blue ball in it. I decided that if the ball were blue, I would come to you and ask you to give me 1000 utilons. Of course, you don’t have to agree. I also decided that if the ball were red, I would come to you and give you 1000 utilons—but only if I predicted that if I asked you to give me the utilons in the blue-ball case, you would have agreed. If I predicted that you would not have agreed to pay in the blue-ball case, then I would not pay you in the red-ball case. Now, as it happens, I looked at the ball and found it blue. Will you pay me 1000 utilons?”
The difference from the usual case is that instead of a coin flip determining which question Omega asks, we have the ball in the urn. I am still confused about the significance of this change.
Is it that the coin flip is a random process, but that the ball may have gotten into the urn by some deterministic method?
Is it that the coin flip is done just before Omega asks the question, while the ball has been sitting in the urn, unchanged, for a long time?
Is it that we have partial information about the urn state, therefore the odds will not be 50-50, but potentially something else?
Is it the presence of a prediction market that gives us more information about what the state of the urn is?
Is it that our previous estimates, and those of the prediction market, have varied over time, rather than being relatively constant? (Are we supposed to give some credence to old views which have been superseded by newer information?)
Another difference is that in the original problem, the positive payoff was much larger than the negative one, while in this case, they are equal. Is that significant?
And once again, if this were not an Omega question, but just some random person offering a deal whose outcome depended on a coin flip vs a coin in an urn, why don’t the same considerations arise?
Is it that the coin flip is a random process, but that the ball may have gotten into the urn by some deterministic method?
Randomness is uncertainty, and determinism doesn’t absolve you of uncertainty. If you find yourself wondering what exactly was that deterministic process that fits your incomplete knowledge, it is a thought about randomness. A coin flip is as random as a pre-placed ball in an urn, both in deterministic and stochastic worlds, so long as you don’t know what the outcome is, based on the given state of knowledge.
Is it that we have partial information about the urn state, therefore the odds will not be 50-50, but potentially something else?
The tricky part is what this “partial information” is, as, for example, looking at the urn after Omega reveals the actual color of the ball doesn’t count.
Another difference is that in the original problem, the positive payoff was much larger than the negative one, while in this case, they are equal. Is that significant?
In the original problem, payoffs differ so much to counteract lack of identity between amount of money and utility, so that the bet does look better than nothing. For example, even if $100*0.5-$100*0.5>0, it doesn’t guarantee that U($100)*0.5+U(-$100)*0.5>0. In this post, the values in utilons are substituted directly to place the 50⁄50 bet exactly at neutral.
And once again, if this were not an Omega question, but just some random person offering a deal whose outcome depended on a coin flip vs a coin in an urn, why don’t the same considerations arise?
They could, you’d just need to compute that tricky answer that is the topic of this post, to close the deal. This question actually appears in legal practice, see hindsight bias.
Thanks for the answer, but I am afraid I am more confused than before. In the part of the post which begins, “So, new problem...”, the coin is gone, and instead Omega will decide what to do based on whether an urn contains a red or blue marble, about which you have certain information. There is no coin. Can you restate your explanation in terms of the urn and marble?
I’m not sure what exactly confuses you. Coin, urn, what does it matter? See the original post for the context in which the coin is used. Consider it a rigged coin, one probability of which landing on each side was a topic of that debate MBlume talks about.
Let me try restating the scenario more explicitly, see if I understand that part.
Omega comes to you and says, “There is an urn with a red or blue ball in it. I decided that if the ball were blue, I would come to you and ask you to give me 1000 utilons. Of course, you don’t have to agree. I also decided that if the ball were red, I would come to you and give you 1000 utilons—but only if I predicted that if I asked you to give me the utilons in the blue-ball case, you would have agreed. If I predicted that you would not have agreed to pay in the blue-ball case, then I would not pay you in the red-ball case. Now, as it happens, I looked at the ball and found it blue. Will you pay me 1000 utilons?”
The difference from the usual case is that instead of a coin flip determining which question Omega asks, we have the ball in the urn. I am still confused about the significance of this change.
Is it that the coin flip is a random process, but that the ball may have gotten into the urn by some deterministic method?
Is it that the coin flip is done just before Omega asks the question, while the ball has been sitting in the urn, unchanged, for a long time?
Is it that we have partial information about the urn state, therefore the odds will not be 50-50, but potentially something else?
Is it the presence of a prediction market that gives us more information about what the state of the urn is?
Is it that our previous estimates, and those of the prediction market, have varied over time, rather than being relatively constant? (Are we supposed to give some credence to old views which have been superseded by newer information?)
Another difference is that in the original problem, the positive payoff was much larger than the negative one, while in this case, they are equal. Is that significant?
And once again, if this were not an Omega question, but just some random person offering a deal whose outcome depended on a coin flip vs a coin in an urn, why don’t the same considerations arise?
Randomness is uncertainty, and determinism doesn’t absolve you of uncertainty. If you find yourself wondering what exactly was that deterministic process that fits your incomplete knowledge, it is a thought about randomness. A coin flip is as random as a pre-placed ball in an urn, both in deterministic and stochastic worlds, so long as you don’t know what the outcome is, based on the given state of knowledge.
The tricky part is what this “partial information” is, as, for example, looking at the urn after Omega reveals the actual color of the ball doesn’t count.
In the original problem, payoffs differ so much to counteract lack of identity between amount of money and utility, so that the bet does look better than nothing. For example, even if $100*0.5-$100*0.5>0, it doesn’t guarantee that U($100)*0.5+U(-$100)*0.5>0. In this post, the values in utilons are substituted directly to place the 50⁄50 bet exactly at neutral.
They could, you’d just need to compute that tricky answer that is the topic of this post, to close the deal. This question actually appears in legal practice, see hindsight bias.