One way to think about this paradox is as a game with two players—a prisoner and an executioner. The game progresses over five turns, representing the days of the week.
On each turn, the prisoner secretly chooses whether or not they expect to be killed on that day. Then the executioner chooses whether or not to kill the prisoner. The game ends on the turn the executioner decides to kill the prisoner. If the prisoner wasn't expecting to die, the executioner wins; otherwise, the prisoner wins.
The important detail here (which is obscured by the logical perspective) is when the prisoner is allowed to expect to die. If they're allowed to expect their death on every single day, the prisoner can just do that and win automatically. If they can only expect to die once, there's a situation the usual paradoxical argument doesn't consider: that the prisoner has survived until Thursday but has already used their chance to expect to die. In this case, they know they're going to die on Friday, but there's nothing they can do about it.
In this view of the world (where the prisoner can play "I expect to die today" exactly once, there is actually not a Nash equilibrium where the executioner is guaranteed to win.
In other words the judge's sentence can't be guaranteed to be carried out.
You almost always need mixed strategies for Nash Equilibrium to exist so that's not much of a surprise that it doesn't exist if the prisoner has to choose a pure one (expects to die exactly this day and not other days).
What I find interesting about the paradox is its relation to anxiety.
Quoting Wikipedia:
"Anxiety is an emotion characterized by an unpleasant state of inner turmoil, often accompanied by nervous behaviour such as pacing back and forth, somatic complaints, and rumination. It is the subjectively unpleasant feelings of dread over anticipated events, such as the feeling of imminent death." [1]
As well as -perhaps even more so- depression.
The expected outcome has a related effect on the psyche, but the fact is that in reality the prisoner has no effect on their death. Which means that the most useful way to deal with your last days (how many they are) is akin to the principles of carpe diem. [2] Which is something both anxiety and depression hamper.
On each turn, the prisoner secretly chooses whether or not they expect to be killed on that day. Then the executioner chooses whether or not to kill the prisoner. The game ends on the turn the executioner decides to kill the prisoner. If the prisoner wasn't expecting to die, the executioner wins; otherwise, the prisoner wins.
The important detail here (which is obscured by the logical perspective) is when the prisoner is allowed to expect to die. If they're allowed to expect their death on every single day, the prisoner can just do that and win automatically. If they can only expect to die once, there's a situation the usual paradoxical argument doesn't consider: that the prisoner has survived until Thursday but has already used their chance to expect to die. In this case, they know they're going to die on Friday, but there's nothing they can do about it.