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PaperclipPerfector  1mo ago (text post)   25682 thread views

Culture War Roundup for the week of November 10, 2025

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The answer for 95% of those decision theory puzzles/paradoxes is that the puzzle as stated is underspecified.

I disagree. In many cases the intended interpretation is clear, and people who give the wrong answer got the interpretation right but simply did the math wrong.

Specifically for the Monty Hall problem, most people who dispute the correct answer (switching doubles your winning chance) do not claim that the problem is underspecified, but give an answer (switching does not change your winning probability) that is not consistent with any reasonable interpretation of the problem.

Here's an article that covers everything I wanted to say about the topic. Behind Monty Hall's Doors: Puzzle, Debate and Answer?

"Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the other doors, opens another door, say No. 3, which has a goat. He then says to you, 'Do you want to pick door No. 2?' Is it to your advantage to take the switch?"

This is the original formulation of the problem. It's true that it is ambiguous in that it doesn't specifically state whether the host would reveal a goat regardless of whether you picked right or wrong, as pointed out by Martin Gardner (whom I hold in high regard):

"The problem is not well-formed," Mr. Gardner said, "unless it makes clear that the host must always open an empty door and offer the switch. Otherwise, if the host is malevolent, he may open another door only when it's to his advantage to let the player switch, and the probability of being right by switching could be as low as zero." Mr. Gardner said the ambiguity could be eliminated if the host promised ahead of time to open another door and then offer a switch.

But that's not the criticism of most people who dispute the official answer. Those people usually say the answer is exactly 50/50:

Robert Sachs, a professor of mathematics at George Mason University in Fairfax, Va., expressed the prevailing view that there was no reason to switch doors.

"You blew it!" he wrote. "Let me explain: If one door is shown to be a loser, that information changes the probability of either remaining choice -- neither of which has any reason to be more likely -- to 1/2. As a professional mathematician, I'm very concerned with the general public's lack of mathematical skills. Please help by confessing your error and, in the future, being more careful."

I have a similar objection to your interpreation:

if he actually only picks a door when you have picked the winning door: don't

Okay, but if Monty only opens a door if you picked the winner, then obviously you shouldn't switch: your chance of winning would be 0% after switching, not 50%. That doesn't support the 50% answer at all!

You could at least somewhat reasonably assume an adverserial scenario where Monty may decide to reveal a goat or not, with the goal to maximally confuse you and minimize your chances of winning. But in that case, his optimal strategy isn't to reveal a goat only when you're about to win (which only confirms your choice was correct) but to never reveal a goat, regardless of your initial pick, in which case you cannot do better than sticking with your initial guess for an 1/3 chance of winning.

In short, there is no sensible interpretation of the problem where the correct answer is that switching or not doesn't matter. You can only reach the conclusion by getting the math wrong, not by finding a reasonable but unintended interpretation of the problem as stated.

(edit: removed a bit that I need to rethink)

If Monty knows what's behind each door but still opens one at random, it's 50/50 (given you're in a world where he didn't open the car door). I think people are often answering that slightly separate problem, without necessarily realising the ambiguity. Of course Mr Gardner, puzzle-master extraordinaire, would notice, but I think it's perfectly possible to read the (frequently, including in the initial statement) ambiguously-stated problem as the 50/50 one and not realise what you're doing.

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