I have a few clarifying questions about the rules that I am trying to understand.
First Question: Are you allowed to tell Omega an integer that you can't actually calculate but that Omega can, if you know that integer exists and can uniquely define it even if you can't calculate it?
Example: You say "The integer calculated as a result of the maximal strategy that you, Omega can construct without falling into a loop." Does that represent an extremely high integer to Omega, even though you couldn't actually calculate it yourself because you have less room for computations than Omega does, or is it in an invalid response?
Second question: If you and Mrs. X both give the exact same response (for instance, if the above answer is valid and you both say it), then what happens?
Edit: I read your linked post and you said:
It should be noted that God, or any being capable of hypercomputation, has real problems in these situations: they actually have infinite options (not a finite options of choosing their future policy), and so don't have any solution available.
Which means that that particular example is probably bad, since there is no integer calculated as a result of the maximal strategy that Omega can construct without falling into a loop.
However, the question "Can you ask Omega to do you calculations that you yourself can't do?" is still relevant for answers like "The number I am about to say, raised to the power of the number I am about to say, the number I am about to say times. (Long pause as you calculate your Maximum integer.) My Maximum integer."
Although, I suppose if you can run calculations on a hypercomputer, and you live indefinitely long while talking to it, You could potentially end up getting caught in a infinite loop of saying "The number I am about to say, raised to the power of the number I am about to say, the number I am about to say times... And that number can be calculated by taking the number I am about to say, raised to the power of the number I am about to say, the number I am about to say times, and that number can be calculated..."
So if you could have the hypercomputer help you with calculations, and wanted the highest finite number you and the hypercomputer together could generate, you would also have to ask the hyper computer to increment your number in such a way that you didn't get caught in an infinite loop while attempting to increment it.
Are you allowed to tell Omega an integer that you can't actually calculate but that Omega can, if you know that integer exists and can uniquely define it even if you can't calculate it?
It doesn't really matter much. It changes the methods you use somewhat, but the basic result is the same. For that matter, you could even allow infinite numbers without doing a whole lot.
If you and Mrs. X both give the exact same response (for instance, if the above answer is valid and you both say it), then what happens?
I'd guess that you both get half a utility.
In an earlier post, I talked about how we could deal with variants of the Heaven and Hell problem - situations where you have an infinite number of options, and none of them is a maximum. The solution for a (deterministic) agent was to try and implement the strategy that would reach the highest possible number, without risking falling into an infinite loop.
Wei Dai pointed out that in the cases where the options are unbounded in utility (ie you can get arbitrarily high utility), then there are probabilistic strategies that give you infinite expected utility. I suggested you could still do better than this. This started a conversation about choosing between strategies with infinite expectation (would you prefer a strategy with infinite expectation, or the same plus an extra dollar?), which went off into some interesting directions as to what needed to be done when the strategies can't sensibly be compared with each other...
Interesting though that may be, it's also helpful to have simple cases where you don't need all these subtleties. So here is one:
Omega approaches you and Mrs X, asking you each to name an integer to him, privately. The person who names the highest integer gets 1 utility; the other gets nothing. In practical terms, Omega will reimburse you all utility lost during the decision process (so you can take as long as you want to decide). The first person to name a number gets 1 utility immediately; they may then lose that 1 depending on the eventual response of the other. Hence if one person responds and the other doesn't, they get the 1 utility and keep it. What should you do?
In this case, a strategy that gives you a number with infinite expectation isn't enough - you have to beat Mrs X, but you also have to eventually say something. Hence there is a duel of (likely probabilistic) strategies, implemented by bounded agents, with no maximum strategy, and each agent trying to compute the maximal strategy they can construct without falling into a loop.