Why the Monty Hall Problem Drives People Crazy
Why the Monty Hall Problem Drives People Crazy
by Oran Looney
May 13, 2026
Math
Philosophy
Game Theory
This essay isn’t to explain the solution to the Monty Hall<br>problem—you can look that up anywhere—but to ask a related<br>question: why does it seem to drive some people crazy? Why do they get so<br>attached to their wrong answers, and so upset by the correct answer? That’s<br>weird, right? People don’t usually care enough about math problems to get<br>worked up over them, but there’s something about this particular problem that<br>really pushes people’s buttons.
There is an answer to this question, and it’s not just “it’s a bit tricky and<br>people don’t like being wrong.” It’s analogous to what Kahneman called<br>attribution substitution, but instead of substituting an easier version<br>of a question, people substitute an adversarial version of a game, so<br>let’s call it “adversarial substitution.”
This is going to take some work to unpack, but I think it’ll be worth it<br>because it explains a lot about how people understand (and misunderstand) the<br>world around them.
The Monty Hall Problem
The Monty Hall Problem was best described on this episode of Brooklyn 99:
Prior to that, it was discussed by Marilyn vos Savant, to a famously negative<br>public reaction. After giving the correct answer in a newspaper column,<br>she was bombarded with hate mail vehemently insisting she was wrong. But why?
The glib answer is sexism, but this is at most only part of the reason. She<br>wrote on many other topics which did not elicit the same reaction despite<br>having the same author. No, there must be something about the problem itself<br>that triggers an unusually strong reaction.
The Adversarial Variant
The original Monty Hall Problem assumes that Monty always shows you a goat<br>and always gives you the option to switch. The sequence of play looks<br>like this:
We’re now going to introduce an adversarial variation of the game. In this<br>version, Monty has a choice: he can either open a door and give you the option<br>to switch, just like in the original game, or he can immediately give you what’s<br>behind the door you picked. He still has full information about what’s behind<br>the door, of course. Let’s also say this is a zero-sum game: Monty doesn’t<br>want you to win, and will make his choice based on whatever is worse for you.
From Monty’s point of view, the optimal policy is obvious. He knows if the door<br>you chose has a goat or a car behind it. If it’s a goat, he has zero incentive<br>to offer you a chance to switch as that might result in you switching to the<br>car. If it’s a car, then he has every reason to offer you the choice in the<br>hope that you’ll switch.
That means from the player’s point of view, the game now looks like this:
It’s worth comparing the outcomes in these two different variants:
Car Location<br>Original<br>Adversarial
Stay<br>Switch<br>Stay<br>Switch
Initial Door<br>Win<br>Lose<br>Win<br>Lose
Other Door #1<br>Lose<br>Win<br>Lose<br>Lose
Other Door #2<br>Lose<br>Win<br>Lose<br>Lose
The proportion of green in each column tells you the probability of winning<br>under each strategy in each variant. In the original game, switching nets<br>you a 2⁄3 chance of winning, vs. only 1⁄3 if you stay, so your best move is<br>to switch. So much we already knew.
But look at outcomes for the adversarial game on the right. They’re completely<br>different. Now if you switch, you really shoot yourself in the foot: there’s no<br>chance of winning at all. That’s because Monty only gives you the opportunity<br>to switch if he knows you’ve already chosen the car. It’s a trap: he’s giving<br>you a second chance to make a mistake. Your only real option is to choose to<br>stay if given a choice. Of course, you often won’t be given a choice: you’ll<br>pick a goat door, he’ll immediately reveal it, and you’ll think, “ah, bad<br>luck.” You never even know that you were denied an opportunity to switch unless<br>you’d seen the game played before.
Card Forcing
This structure is reminiscent of the classic magician’s “force.” Suppose a<br>magician wants you to end up with a particular card. He of course knows where<br>it is, but he wants to make it seem as if you picked it. He asks you to cut the<br>deck into two piles and point to one. If the force card is in the pile you<br>indicate, he says, “great, we’ll use this one.” But if the force card is in the<br>other pile, he reframes your gesture as eliminating the pile you pointed to:<br>“fine, we’ll get rid of this one.” Either way, the pile containing the force<br>card survives. From your perspective, it feels as though you made a free<br>choice, but the magician was really using hidden information to reinterpret the<br>meaning of your gesture to achieve his own ends.
Of course,...