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.
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!