For scenario 7, I think I may have generated a type of situation where the type of coin flip might matter, but I feel like I also may have made an error somewhere. I’ll post what I have so far for verification.
To explain, imagine that Omega knows in advance, that the logical coin flip is going to be tails every time he flips the logical coin, because odd is tails and he is asking about the first digit of pi, which is odd.
Now, in this case, you would also know the first digit of pi is odd, so that wouldn’t be an information asymmetry. You just wouldn’t play if you knew Omega has made a logical decision to use a logical coin that came up tails, because you would never even hypothetically have ever gotten money. It would be as if Omega said: “1=1, so I decided to ask you to give me $100 dollars. Whatever you do in this situation, nothing else will happen differently in reality as a result. Naturally you don’t want to give up your $100. But I’m also telling you that if 0=1, I’d give you $10000, but only if you’d agree to give me $100 if 1=1.” It seems reasonable to not give Omega money in that case.
However, since Omega has more computing power, there are always going to be logical coins that look random to you that Omega can use: Maybe the trillionth digit of pi is unknown to you, but Omega calculated it before hand thinking about making you any offers, and it happens to have been odd/tails.
Omega can even do something which has indexical random components and logical components that ends up being logically calculatable. If Omega rolls an indexical six sided die, adds 761(A ‘random’ seed) to the number, and then chooses to check the even/odd status anywhere from the 762nd digit of pi through the 767th digit of pi from the results of the die roll. If it’s odd, the coin is tails. That is the Feynmann point, http://en.wikipedia.org/wiki/Feynman_point All the digits are 9, which is odd, so the coin is always tails.
If this is a simple indexical coin flip, Omega can’t have that sort of advance knowledge.
However, what I find confusing is that asking about the recorded past result of an indexical coin flip appears to be a type of logical uncertainty. So since this has occurred in the past, what seems to be the exact same reasoning would tell me not to give Omega the money right now, because I can’t trust that Omega won’t exploit the information asymmetry.
This is where I was concerned I had missed something, but I’m not sure what if anything I am missing.
I think that all you are observing here is that your probability that other agents know the result of the coin flip changes between the two situations. However, others can know the result for either type of flip, so this is not really a qualitative difference. It is a way in which other information about the coin flip matters, other than just whether or not it is logical.
You achieve this by making the coin flip correlated with other facts, which is what you did. (I think made more confusing and veiled by the fact that these facts are within the mind of Omega.)
Omega does not have to have advance knowledge of an indexical coin flip. He just needs knowledge, which he can have.
For scenario 7, I think I may have generated a type of situation where the type of coin flip might matter, but I feel like I also may have made an error somewhere. I’ll post what I have so far for verification.
To explain, imagine that Omega knows in advance, that the logical coin flip is going to be tails every time he flips the logical coin, because odd is tails and he is asking about the first digit of pi, which is odd.
Now, in this case, you would also know the first digit of pi is odd, so that wouldn’t be an information asymmetry. You just wouldn’t play if you knew Omega has made a logical decision to use a logical coin that came up tails, because you would never even hypothetically have ever gotten money. It would be as if Omega said: “1=1, so I decided to ask you to give me $100 dollars. Whatever you do in this situation, nothing else will happen differently in reality as a result. Naturally you don’t want to give up your $100. But I’m also telling you that if 0=1, I’d give you $10000, but only if you’d agree to give me $100 if 1=1.” It seems reasonable to not give Omega money in that case.
However, since Omega has more computing power, there are always going to be logical coins that look random to you that Omega can use: Maybe the trillionth digit of pi is unknown to you, but Omega calculated it before hand thinking about making you any offers, and it happens to have been odd/tails.
Omega can even do something which has indexical random components and logical components that ends up being logically calculatable. If Omega rolls an indexical six sided die, adds 761(A ‘random’ seed) to the number, and then chooses to check the even/odd status anywhere from the 762nd digit of pi through the 767th digit of pi from the results of the die roll. If it’s odd, the coin is tails. That is the Feynmann point, http://en.wikipedia.org/wiki/Feynman_point All the digits are 9, which is odd, so the coin is always tails.
If this is a simple indexical coin flip, Omega can’t have that sort of advance knowledge.
However, what I find confusing is that asking about the recorded past result of an indexical coin flip appears to be a type of logical uncertainty. So since this has occurred in the past, what seems to be the exact same reasoning would tell me not to give Omega the money right now, because I can’t trust that Omega won’t exploit the information asymmetry.
This is where I was concerned I had missed something, but I’m not sure what if anything I am missing.
I think that all you are observing here is that your probability that other agents know the result of the coin flip changes between the two situations. However, others can know the result for either type of flip, so this is not really a qualitative difference. It is a way in which other information about the coin flip matters, other than just whether or not it is logical.
You achieve this by making the coin flip correlated with other facts, which is what you did. (I think made more confusing and veiled by the fact that these facts are within the mind of Omega.)
Omega does not have to have advance knowledge of an indexical coin flip. He just needs knowledge, which he can have.