I've always enjoyed puzzling through philosophical paradoxes, and one that I would sometimes mention to students was the so-called pop quiz paradox. The way it works is this: the teacher tells his students that there will be a pop quiz next week, where a pop quiz is defined as one whose specific date cannot be predicted. (The unexpected hanging paradox is the same thing, except that it's a prisoner's being hanged that will be the unpleasant, unpredictable surprise.) It might come on Monday, it might come on Friday, or any day in between. It will happen, it won't be announced beforehand, and it can't be predicted with certainty beforehand.
But with this definition a problematic result ensues. Suppose, the students reason in advance, that it hasn't happened after next Thursday's class. Then it would have to be on Friday, but since Friday is impossible (it would be predictable with certainty) it wouldn't be a pop quiz. (We're assuming here and throughout that both the teacher and the students are fully rational, and are aware that the other partie(s) are fully rational as well.) Therefore, the students know on Wednesday that since it can't happen on Friday, it would have to happen on Thursday. But if it has to happen on Thursday, then once again it's not a pop quiz. And since this reasoning can be repeated, it can't be on Wednesday, Tuesday, or Monday either. In fact, one could have an infinite number of days, and all can be ruled out one by one. This seems nuts, however - obviously if a teacher tells the students that there will be a pop quiz some time during the semester or the school year, it's clear that the students can be surprised, even if the reasoning purports to show that a pop quiz is impossible. So something is going wrong somewhere.
Anyhow, I was thinking about Fabiano Caruana's chances to win the match during the classical portion, and wondering when exactly he should "panic" about the possibility of the rapid (and possibly blitz) playoff. (I'm assuming he's a relatively heavy underdog in the rapid and a serious underdog in the blitz, if it comes to that. And by "panic" I mean that he should take some extra - but not suicidal - risks to increase the possibility of a decisive classical game.) Certainly by game 12 it will be time to take some extra risks, but what about before that?
Now, let's suppose that Magnus Carlsen will know when Caruana is in a panic situation, and that this knowledge can be used to give his chances a serious extra boost. (This may not be true, but let's suppose it is for the sake of the thought experiment.) If that's correct, then it seems that we'll have another version of the pop quiz paradox. Caruana won't want to wait until game 12 to panic, because then Carlsen will know and will have an extra edge. But then after game 10 Carlsen will know that Caruana can't wait until game 12, and must therefore panic in game 11. But then that's bad for Caruana as well...and so on, all the way back to game 1. So how do we resolve this? And speaking outside the bounds of the paradox, when should Caruana panic (in the sense above) if the match is tied?