On a game show, you are given the choice of three doors leading to three rooms. You know that in one room is $100,000, and the other two are empty. The host asks you to pick a door, and you pick door #1. Then the host opens door #2, revealing an empty room. Do you want to switch to door #3, or stick with door #1?
The answer depends on the host’s algorithm. If the host always opens a door and always picks a door leading to an empty room, then you should switch to door #3. If the host always opens door #2 regardless of what is behind it, #1 and #3 both have 50% probabilities of containing the money. If the host only opens a door, at all, if you initially pick the door with the money, then you should definitely stick with #1.
Which means that (when Monty’s algorithm isn’t given or when there’s uncertainty about how accurate the problem statement is) people who don’t switch are making a very defensible choice by the laws of decision theory. For what plausible reason would he (open a door and) offer to let you switch unless he stood to gain if you did? (Answer: To trick you on the meta level, of course.)
Which means that (when Monty’s algorithm isn’t given or when there’s uncertainty about how accurate the problem statement is) people who don’t switch are making a very defensible choice by the laws of decision theory. For what plausible reason would he (open a door and) offer to let you switch unless he stood to gain if you did? (Answer: To trick you on the meta level, of course.)