Backward Induction - A Paradox of Backward Induction

A Paradox of Backward Induction

The unexpected hanging paradox is a paradox related to backward induction. Suppose a prisoner is told that she will be hanged sometime between Monday and Friday of next week. However, the exact day will be a surprise (i.e. she will not know the night before that she will be executed the next day). The prisoner, interested in outsmarting her executioner, attempts to determine which day the execution will occur.

She reasons that it cannot occur on Friday, since if it had not occurred by the end of Thursday, she would know the execution would be on Friday. Therefore she can eliminate Friday as a possibility. With Friday eliminated, she decides that it cannot occur on Thursday, since if it had not occurred on Wednesday, she would know that it had to be on Thursday. Therefore she can eliminate Thursday. This reasoning proceeds until she has eliminated all possibilities. She concludes that she will not be hanged next week.

To her surprise, she is hanged on Wednesday.

Here the prisoner reasons by backward induction, but seems to come to a false conclusion. Note, however, that the description of the problem assumes it is possible to surprise someone who is performing backward induction. The mathematical theory of backward induction does not make this assumption, so the paradox does not call into question the results of this theory. Nonetheless, this paradox has received some substantial discussion by philosophers. This paradox is similar to a shorter one: "guess heads or tails but not what you think". While it is easy to come with a flip fulfilling the statement, described reasoning leads to conclusion that there is no way to do that.