You are trying to apply realistic constraints to a hypothetical situation that is not intended to be realistic
Your thought experiment, as you want it to be interpreted, is too unrealistic for it to imply a new and surprising critique of Bayesian rationality in our world. However, the title of your post implies (at least to me) that it does form such a critique.
The gamesmaster has no desire to engage with any of your questions or your attempts to avoid directly naming a number. He simply tells you to just name a number.
If we interpret the thought experiment as happening in a world similar to our own—which I think is more interesting than an incomprehensible world where the 2nd law of thermodynamics does not exist and the Kolmogorov axioms don’t hold by definition—I would be surprised that such a gamesmaster would view Arabic numerals as the only or best way to communicate an arbitrarily large number. This seems, to me, like a primitive human thought that’s very limited in comparison to the concepts available to a superintelligence which can read a human’s source code and take measurements of the neurons and subatomic particles in his brain. As a human playing this game I would, unless told otherwise in no uncertain terms, try to think outside the limited-human box, both because I believe this would allow me to communicate numbers of greater magnitude and because I would expect the gamesmaster’s motive to include something more interesting, and humane and sensible, than testing my ability to recite digits for an arbitrary length of time.
There’s a fascinating tension in the idea that the gamesmaster is an FAI, because he would bestow upon me arbitrary utility, yet he might be so unhelpful as to have me recite a number for billions of years or more. And what if my utility function includes (timeless?) preferences that interfere with the functioning of the gamesmaster or the game itself?
Your thought experiment, as you want it to be interpreted, is too unrealistic for it to imply a new and surprising critique of Bayesian rationality in our world. However, the title of your post implies (at least to me) that it does form such a critique.
If we interpret the thought experiment as happening in a world similar to our own—which I think is more interesting than an incomprehensible world where the 2nd law of thermodynamics does not exist and the Kolmogorov axioms don’t hold by definition—I would be surprised that such a gamesmaster would view Arabic numerals as the only or best way to communicate an arbitrarily large number. This seems, to me, like a primitive human thought that’s very limited in comparison to the concepts available to a superintelligence which can read a human’s source code and take measurements of the neurons and subatomic particles in his brain. As a human playing this game I would, unless told otherwise in no uncertain terms, try to think outside the limited-human box, both because I believe this would allow me to communicate numbers of greater magnitude and because I would expect the gamesmaster’s motive to include something more interesting, and humane and sensible, than testing my ability to recite digits for an arbitrary length of time.
There’s a fascinating tension in the idea that the gamesmaster is an FAI, because he would bestow upon me arbitrary utility, yet he might be so unhelpful as to have me recite a number for billions of years or more. And what if my utility function includes (timeless?) preferences that interfere with the functioning of the gamesmaster or the game itself?
“However, the title of your post”—titles need to be short so they can’t convey all the complexity of the actual situation.
“Which I think is more interesting”—To each their own.