The Pop Quiz/Unexpected Hanging Paradox, Chess Edition
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?
Reader Comments (8)
There is a twist in this case. Normally, Carlsen may expect Caruana to 'panic' in a game where he (Caruana) has white. Therefore Caruana should 'panic' with black pieces. But in these championships Black has held significant advantages, so that element of surprise has gone away.
I have no idea what this has to do with Notre Dane, which is all I come here for..,
[DM: The students I challenged with the pop quiz paradox were Notre Damers, of course!]
Thanks a lot for the nice puzzle!
"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."
After that definition, a pop quiz can only be announced within an unlimited time line.
Since the teacher restricts the time line from next Monday to Friday, his specific pop quiz* has another definition: "a pop quiz* is defined as one whose specific date cannot be predicted, except it happens on Friday."
Now I would even bet a small amount on Caruana. I could imagine, the pressure for Carlsen, loosing his title is increasing for the last 4 games. I don't think Caruana will 'panic' at any time.
I think "survivorship bias"---playing to maximize the span of time when you are in contention---governs all. In football too. There is more "regret" in going for 2 to win and failing than in kicking for the tie and losing in OT.
[DM: That's true, but it doesn't mean that it's the rational approach. There are some interesting statistical discussions not just about when to go for 2, but whether one should punt at all, and in hockey one should apparently pull the goalie much, much earlier than actually happens in real games.]
I might quibble about whether you are allowed to assign arbitrary values to properties like predictability and how changing information changes those values ... I might really quibble most about your statement referring to infinite days. Do you mean an arbitrary number of days? Infinite days seems like it would indeed be unpredictable since there is no end and the hard stop is what creates the problem.
[DM: Yes, the way you put it is better. What's clear is that one needs a last day, and arguably a first day as well. With those parameters, the intervening time span can be as large (but finite) as one likes. As for probability values, is >0 really all that arbitrary?]
Well It is if you define it as 1 and then set the rules to disallow 1 by fiat.
If I get to incorporate new information a day by day and reevaluate my odds, then as I converge on perfect information, 1 will be the result. But the rules here seem to inexplicably forbid gaining perfect information.
I like to conceptualize it as, when Thursday comes and goes, all predictions have failed, I now have perfect information and knowing the quiz is Friday is not a prediction at all. However, If I predicted Friday before Thursday and choose not to reevaluate the probability space, then I indeed did not know which day it was with certainty.
I should add that I think this is still a good evaluation tool. A bit like playing pot odds in poker. Each day the chance that Fabi panics goes up. The odds of him pressuring prematurely and losing can be evaluated against the odds it's panic time and pressing leads to a win. He probably combines that with draw odds which he might even treat as a win.
The Karjakin match surely encourages the Caruana camp to wait and hope Magnus self-destructs. (Hopefully Magnus has learnt to control his emotions a bit better.)
I'm also assuming no one here thinks what Topalov did in the last classical game against Anand was sensible match strategy.