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.
It’s an amusing paradox and I don’t know what the standard solution to it is, but I resolve it easily in my mind by tabooing the word ‘surprise’, and replacing with “suddenly obtaining certain knowledge of its date”. Then it becomes more of a silly game of words.
Assuming the test is certain to take place (it’s the law!), the students can’t be suprised on Friday, but on Thursday they will either be “surprised” to locate the test on Thursday or “surprised” to locate the test on Friday. The Thursday (or Wednesday or Tuesday or Monday) surprise will therefore be genuine, though no student can be suprised on Friday. They can be “surprised” on Thursday about either a Thursday or a Friday test, though.
The day-by-day progression is a distraction btw. The paradox can be replaced by having five cups, a ball under one of them, lifting them one by one in whatever order. If you’ve lifted three empty cups, you’ll then either be “surprised” to locate the ball under the cup you lift next, or “surprised” to locate it definitively under the cup you haven’t lifted. Both of these will be a surprise, a surprise occurring at the next-to-last cup.
It’s an amusing paradox and I don’t know what the standard solution to it is
The ‘no friday, but any other day is fine’ thing is the closest we have to a standard solution. The taboo game is a good idea, but it could also easily be misleading. The taboo solution works or doesn’t work depending on what you replace ‘surprise’ with, so you have to argue for your replacement. (EDIT: this strikes me now as a general and serious problem with the taboo game)
Here’s an argument against your replacement. If I told you that there would be a test next year, on this day, at exactly 2:00, you would hardly call this a surprise even though you’d just gained ‘sudden and certain knowledge of its date’. It would be impossible to not give a surprise test. This can’t be what ‘surprise’ is supposed to mean, and even if it is, the paradox still makes impossible another kind of surprise test which teachers often take themselves to be able to announce.
A surprise test should probably be understood as a test given in such a way that you do not know, the night before (when you would study) that the test will be the next day. This is what the paradox makes problematic. The students can’t be surprised, on Thursday, to locate the test on either Friday or Thursday, because they know the test won’t be on Friday (since then they’d know about it the night before). So they know the test must be on Thursday (which they would have guessed last night, so no surprise there either).
The paradox can be replaced by having five cups, a ball under one of them, lifting them one by one in whatever order.
You can reproduce the paradox with five cups, yes. But the conditions have to be narrower than you say: the cups have to be lifted in a specific order known to the lifter before hand. The lifter has to be told that he will not know, before he at any stage lifts the cup to find the ball, wether or not the ball will be under that cup. So the player will know that the ball cannot be in the last cup (since then he will know before he lifts it that it is there), and given that, it cannot be under the second to last cup either, and so on.
Furthermore, if you take the “information content” approach to surprise, then you would be more surprised by a test on Monday than by a test on Thursday; but this is made up for by the fact that on Monday, Tuesday, and Wednesday, you were very slightly surprised that there wasn’t a test. Total surprise is conserved.
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!
It’s an amusing paradox and I don’t know what the standard solution to it is, but I resolve it easily in my mind by tabooing the word ‘surprise’, and replacing with “suddenly obtaining certain knowledge of its date”. Then it becomes more of a silly game of words.
Assuming the test is certain to take place (it’s the law!), the students can’t be suprised on Friday, but on Thursday they will either be “surprised” to locate the test on Thursday or “surprised” to locate the test on Friday. The Thursday (or Wednesday or Tuesday or Monday) surprise will therefore be genuine, though no student can be suprised on Friday. They can be “surprised” on Thursday about either a Thursday or a Friday test, though.
The day-by-day progression is a distraction btw. The paradox can be replaced by having five cups, a ball under one of them, lifting them one by one in whatever order. If you’ve lifted three empty cups, you’ll then either be “surprised” to locate the ball under the cup you lift next, or “surprised” to locate it definitively under the cup you haven’t lifted. Both of these will be a surprise, a surprise occurring at the next-to-last cup.
The ‘no friday, but any other day is fine’ thing is the closest we have to a standard solution. The taboo game is a good idea, but it could also easily be misleading. The taboo solution works or doesn’t work depending on what you replace ‘surprise’ with, so you have to argue for your replacement. (EDIT: this strikes me now as a general and serious problem with the taboo game)
Here’s an argument against your replacement. If I told you that there would be a test next year, on this day, at exactly 2:00, you would hardly call this a surprise even though you’d just gained ‘sudden and certain knowledge of its date’. It would be impossible to not give a surprise test. This can’t be what ‘surprise’ is supposed to mean, and even if it is, the paradox still makes impossible another kind of surprise test which teachers often take themselves to be able to announce.
A surprise test should probably be understood as a test given in such a way that you do not know, the night before (when you would study) that the test will be the next day. This is what the paradox makes problematic. The students can’t be surprised, on Thursday, to locate the test on either Friday or Thursday, because they know the test won’t be on Friday (since then they’d know about it the night before). So they know the test must be on Thursday (which they would have guessed last night, so no surprise there either).
You can reproduce the paradox with five cups, yes. But the conditions have to be narrower than you say: the cups have to be lifted in a specific order known to the lifter before hand. The lifter has to be told that he will not know, before he at any stage lifts the cup to find the ball, wether or not the ball will be under that cup. So the player will know that the ball cannot be in the last cup (since then he will know before he lifts it that it is there), and given that, it cannot be under the second to last cup either, and so on.
Furthermore, if you take the “information content” approach to surprise, then you would be more surprised by a test on Monday than by a test on Thursday; but this is made up for by the fact that on Monday, Tuesday, and Wednesday, you were very slightly surprised that there wasn’t a test. Total surprise is conserved.