Let’s say the prison guard must pick a sequence from a single equivalence class. Then, the prison guard still has infinitely many choices, and knowing the tails doesn’t tell anything about the heads. Therefore, intuition still says that each prisoner has only a 50% chance to know about his own hat color. However, the now obvious strategy to just agree on one representative, which now does not require AC, still works with the same result.

Some might object that the prisoners are given too easy a problem in my variation for the outcome to be surprising, but in my opinion the paradox in both cases is that we turn 50% “felt” individual chance of infinitely many prisoners into finite failure guarantees (Note that in my variation we still have intuitive independence of the individual chances: the probability that two given prisoners survive is intuitively 25%)

In both cases the 50% are not rigorous, and I think this supports previously stated opinions that the problem is not with AC but with our intuition about chances, which cannot be made rigorous in this problem. In particular, I think that the strategies in both cases require “aligning” the choices of the prisoners in a way that prevents measurability. In the original problem, this aligning happens dependent on the actual pick of the sequence (and leads to non-measurability of the survival-events), in my variation it is built into the problem (and leads to problems even defining the prison guards probability measure)

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Red lines show the factors containing positive roots and blue lines show the factors containing negative roots as they are added. Dotted lines represent the factors accumulated from previous iterations. Light gray line represents the base factor f(x) = x.

Green line shows the approximant, notice the reasonable convergence in the interval [-pi, pi] as more factors are added.

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