> If we don't care about randomness, can't the judge just use the strategy of Always pick Wednesday?
The point is that because the process is underspecified the prisoner is using one interpretation while the judge is using another. (Otherwise the judge would have had to conclude that no choice could be made.) The reader switches between the two specs in an apparent paradox without considering that they are two different specs.
For example, the judge could simply be breaking the rule by considering Friday as a possible choice. That is a different spec from the prisoner, and if such a judge ended up settling on Wednesday then the prisoner would be surprised.
But here's another angle. If the prisoner logically concluded that the judge cannot pick any weekday, the judge can pick Friday and still satisfy the requirement that the prisoner be surprised.
I think that's cheating, though, and the important part is the underspecification.
The point is that because the process is underspecified the prisoner is using one interpretation while the judge is using another. (Otherwise the judge would have had to conclude that no choice could be made.) The reader switches between the two specs in an apparent paradox without considering that they are two different specs.
For example, the judge could simply be breaking the rule by considering Friday as a possible choice. That is a different spec from the prisoner, and if such a judge ended up settling on Wednesday then the prisoner would be surprised.
But here's another angle. If the prisoner logically concluded that the judge cannot pick any weekday, the judge can pick Friday and still satisfy the requirement that the prisoner be surprised.
I think that's cheating, though, and the important part is the underspecification.