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Dagon comments on The Number Choosing Game: Against the existence of perfect theoretical rationality - Less Wrong Discussion
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If it takes time, that's a cost. In your scenario, an agent can keep going forever instantly, whatever that means. That's the nonsense you need to resolve to have a coherent problem. Add in a time limit and calculation rate, and you're back to normal rationality. As the time limit or rate approach infinity, so does the utility.
"Add in a time limit and calculation rate, and you're back to normal rationality" - I am intentionally modelling a theoretical construct, not reality. Claims that my situation isn't realistic aren't valid, as I have never claimed that this theoretical situation does correspond to reality. I have purposefully left this question open.
Ai-yah. That's fine, but please then be sure to caveat your conclusion with "in this non-world..." rather than generalizing about nonexistence of something.
The perfectly rational agent considers all possible different world-states, determines the utility of each of them, and states "X", where X is the utility of the perfect world.
For the number "X+epsilon" to have been a legal response, the agent would have had to been mistaken about their utility function or what the possible worlds were.
Therefore X is the largest real number.
Note that this is a constructive proof, and any attempt at counterexample should attempt to prove that the specific X discovered by a perfectly rational omniscient abstract agent with a genie. If the general solution is true, it will be trivially true for one number.
That's not how maths works.