The unexpected hanging

The Unexpected Hanging is a famous paradox where a prisoner is sentenced to hang. The sentence specifies the hanging will take place at noon one day next week, Monday through Friday. The sentence also specifies the day will be a surprise: the prisoner will not know the date until noon on that day.

The prisoner reasons that the hanging cannot take place on Friday, because he would know on Thursday night. So it must be Monday through Thursday. He then eliminates Thursday because he would know on Wednesday night. He then eliminates Wednesday, Tuesday, and Monday, and relaxes because the sentence cannot be properly carried out on any day. When the executioner comes on Wednesday, he is surprised and complains that the sentence was inconsistent. The executioner points out that each part of the prisoner's sentence turned out to be true.

Some authors have attempted to resolve the paradox by taking 'surprise' to means he cannot be absolutely sure of the date, in which case the prisoner's reasoning does not work. But this makes the puzzle trivial. The sentence could just as well have been "you will be hanged tomorrow, but you can't be sure of that."

The paradox does in fact rely on ambiguity in the term 'surprise'. If  'surprise' means the date of the hanging cannot be deduced using the terms of sentence as axioms, then there is indeed a contradiction. This does not mean the prisoner won't be hanged, he just can't be hanged consistently with the sentence. Thus the paradox is resolved.

There is a version of the paradox where a professor assigns a pop quiz sometime next week. She tells the students they will be surprised. Our intuition tells us that students will indeed be surprised by such a pop quiz if it is early in the week.

I wondered if the paradox could be 'reactivated' if there was a way to define 'surprise' to be consistent with students' intuition. So let's let the students register their surprise by wagering when the quiz will take place. Each night students will be allowed to bet that the quiz will be the next day. Once a student bets he cannot bet again that week. If he is right, the professor pays him $1. If he is wrong he pays the professor $1. If he does not bet by the date of the quiz, his payout is 0. For simplicity let's assume the scenario is repeated every week indefinitely. If we interpret an event to be 'surprising' if the odds of it happening are <50%, then the professors statement is that at least one day exists where the students should not place any bets.

The optimal strategy for such a game is a Nash equilibrium. That means the students will choose their bets according to a probability distribution, and the professor will choose the date of the quiz according to another probability distribution. Furthermore, neither can get an additional advantage by changing their strategy.

I'll save you the trouble of calculus in 10 variables. The students' optimal strategy is to bet with probabilities 1/31, 2/31, 4/31, 8/31, 16/31. The professor's optimal strategy is to hold the quiz with probabilities 16/31, 8/31, 4/31, 2/31, 1/31. If there is a cohort of 31 students who play for 31 weeks, and all the options are exercised proportionally, then each student will gain $1. For example, the 2 students who bet on Tuesdays will each gain $1 8 times and lose $1 7 times (when the quiz is on Wednesday, Thursday, or Friday). Also the professor will lose $1 per week. If she holds the quiz on a Tuesday, she will gain $1 from the Monday student and lose $1 each to the 2 Tuesday students.

Now we can see that the students are never surprised. For example, on Wednesday night, the odds of the test being on Thursday are 4/7. Thus the professor's statement is false and the paradox is resolved.

Unfortunately, our intuition that pop quizzes are surprising is likely a result of the poor reasoning skills of students.