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Manfred comments on Open Thread, September 15-30, 2012 - Less Wrong
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Maybe this has been discussed here, but I wanted to see what you guys think of the surprise test paradox.
The paradox goes like this: it is impossible to give a surprise test.
Say a teacher tells her class on Monday that there will be a surprise test this week. The test cannot be on Friday, because when no test has been given by end-of-class Thrusday, everyone will know that the test is on Friday, and so it won't be a surprise. The test cannot be on Thursday. Having established that the test cannot be on Friday, if no test has been given by end-of-class on Wednesday, it will be obvious that the test is on Thursday, so no surprise. The test cannot be on Wednesday, nor on Tuesday, nor on Monday for similar reasons. So no surprise test can be given.
Similarly, it is impossible to threaten someone thusly: "I'll get you when you least expect it!"
This is a paradox because revenge and surprise tests are, of course, perfectly possible. What's going on here?
Mixed Nash equilibrium is going on here :P If you made this into a little game, with the students paying a cost to cram but getting a benefit if they crammed the night before the test, and the teacher wanting the minimum number of students to cram, you could figure out what the actual ideal strategies would be, and the teacher would indeed have a mixed strategy.
What the non-probabilistic (that is, deductive) reasoning really shows is that there is no way to always surprise a student. If you make things probabilistic, that means you're only claiming to surprise your students the most you can. This problem is weird because it demonstrates how in order to be surprising overall, sometimes you actually do have to choose bad options (at least when you're playing against perfect reasoners :D )! It's only by sometimes actually choosing Friday that the teacher can ever get students to not cram before class Thursday - the times the quiz is on Friday are sacrificed in order to be surprising the other times.
That's a great reply, thanks!