Background and Motivation
I love the Monty Hall problem because it subverts the intuition. Mathematically, the answer is plain and inevitable. Yet intuitively, it scrambles your brain. For this reason, I have revisited the problem once every few months ever since I studied the problem three years ago to ask myself the question “Is the answer intuitive to me now?” And each time the answer has invariably been no, it is not.
I put myself into the category of “mathematics enthusiast,” but I will be the first to admit that I am not very good at math. I imagine there are people out there to whom the Monty Hall problem poses no dilemma whatsoever. But I speak on behalf of the rest of us when I say that the problem continues to befuddle.
The problem at hand is this: Monty Hall is the host of a game show and you are a contestant. Before you are three closed doors, with one door hiding a car (the desired prize) and two doors each hiding a goat (a dud). You are given the opportunity to select a door to open and win whatever lies behind that door—car or goat.
Here is the twist: after you pick your door, Monty walks over to one of the two unchosen doors and opens the door to reveal a goat. Monty then asks if you want to stick to your initial choice, or switch to the other remaining door. Which will it be, keep or switch?
Your gut reaction might be to keep your door because it’s your door damnit and you think Monty is out to trick you. Or perhaps your reaction might be to not care at all, because how could it possibly matter what you choose if there are two doors left and you don’t know which door hides what?
If your thought process coincided with one of these two possibilities, then your intuition has failed you. It turns out that if you always switch, your odds of winning the car are 2/3, and hence it’s optimal to do so. The answer can be reasoned through conditional probability. Separate your world of possibilities into two buckets, (1) one where your initially chosen door hides a car and (2) one where your initially chosen door hides a goat. If you always switch doors, then in the first case you will always lose because both of your unchosen doors hide goats, and in the second case you will always win because one of the unchosen doors hides a car and the other was eliminated by Monty. Therefore, the odds of winning a car are precisely equal to the odds of initially choosing a goat, or 2/3.
The math works out, but the problem still eludes our intuition. We have a hard time shaking off the inkling that since only two doors remain, with one hiding a goat and the other a car, it must be that the odds are now 50/50 and we simply cannot do better. We know that our reasoning is flawed, but it remains awfully appealing.
Thus the immediate task before us is this: How might we reformulate the Monty Hall problem so that the solution to the problem coincides with our intuition? In other words, can we state an isomorphic problem where our intuition leads us to the desired solution, rather than leading us astray?
Alternative, But Equivalent, Formulations
Below, I will introduce isomorphic formulations of the Monty Hall problem. By “isomorphic,” I mean that the problems may have differing details but are structurally equivalent, and therefore can be thought of as the same problem.
- In the beginning of the game, you choose two doors out of three. Among the two chosen doors, Monty reveals a goat, then asks which of the remaining doors you would like to choose.
- Monty chooses two random doors himself. Between the two doors, Monty opens a door to reveal a goat. Monty then asks which of the remaining doors you would like to choose.
Hopefully it’s clear that the first problem is indeed isomorphic to the Monty Hall problem, since picking one door out of three is the same as picking two doors to exclude. The second problem, however, isn’t as straightforward. You are not even given the initial chance to pick a door yourself — Monty does it for you. However, I argue that this is in fact isomorphic to the Monty Hall problem, and that structurally the problem remains exactly the same. It doesn’t matter who chooses the two doors; it matters how the doors are chosen. Monty picks the two doors randomly and so do you. There is no underlying structural difference.
This second problem is of particular interest to us because it reveals an important insight into Monty’s game. It does not matter who chooses the subset of doors, but it does matter that the subset contains precisely two doors, that the subset was chosen randomly and without the use of Monty’s prior knowledge, and that Monty reveals a goat from this subset. A natural follow up to this insight would be to reformulate the Monty Hall problem using this vocabulary, which leads us to:
The Generalized Monty Hall Problem
(Disclaimer: This is my own formulation of the Generalized Monty Hall problem. This can be even further generalized to suppose that there are C number of cars, instead of fixing C = 1 as I do below. Indeed, cursory research suggests other people have come up with different generalizations. My version, however, suits my needs best.)
In the Generalized Monty Hall problem, there are N doors, where N ≥ 3. Accordingly, there are N-1 goats and a single car, one prize laid behind each door. Monty randomly chooses a proper subset of doors of size M, where 2 ≤ M < N. Then, among the doors in the proper subset, Monty randomly reveals G number of goats, where 0 < G < M. Monty then asks: Do you want to pick a door within this proper subset or outside of the subset?
First, notice how the Generalized Monty Hall problem where N = 3, M = 2, and G = 1 is precisely the problem you are familiar with. In this case, you choose a random subset of size M = 2 out of N = 3 doors by picking an initial door (thereby excluding two doors). Then Monty reveals G = 1 number of goats from this proper subset, and asks if you want to pick a door from this subset (switch doors) or outside of this subset (keep your door).
I argue that the optimal strategy is to always pick a car within the subset (“switch”). Unfortunately, Medium does not support LaTeX, so I’ve attached the proof as a separate PDF:
The math works out yet again, but the problem remains no more intuitive. Specifically, it is hard to fathom why choosing the proper subset of size M should matter at all. The end result appears the same—G number of doors have been eliminated, and N-G number of doors remain. So why does choosing the proper subset matter?
Why does choosing the proper subset matter?
This is the last hurdle to understanding the Monty Hall problem intuitively. Unfortunately, it is also the most difficult hurdle.
The crux of the dilemma is that it shouldn’t matter whether Monty reveals a goat from a proper subset of doors or from all of the doors because the result appears the same: you learn that some door X has a goat, and therefore can rule it out when making your next decision. Since the superficial result is the same (door X has a goat), the process shouldn’t matter.
Of course, mathematics shows that process does matter. The information gained from Monty’s reveal cannot merely be “door X has a goat,” because with that information alone, our odds of winning a car should be symmetrically distributed among the unopened doors. The gained information, then, must incorporate the process through which door X was revealed.
To demonstrate how process matters, we take a look at how a specific process preserves specific structures in the Monty Hall problem. Recall that in the problem, we initially choose a subset of size 2 from which Monty reveals a goat. The probability that this subset has the car is 2/3. Monty then reveals a goat by randomly choosing a goat from this subset, such that the subset retains the 2/3 odds of having a car. Had Monty revealed a goat and shuffled the remaining doors, or had Monty revealed a goat randomly among all three doors, then the unopened doors would be symmetrized and the original subset would lose any meaningful structure. But Monty randomly picks a goat from within the subset, which is a process that does not perturb the subset’s integrity. Thus the information we gain from Monty’s reveal is not merely that “door X has a goat,” but that “door X has a goat WHILE the probability of the subset having the car is preserved.” Crucially, this information captures the asymmetry that now exists between the unopened door in the subset and the door not in the subset.
To clarify this point, another example of a process that would preserve subset probabilities is Monty adding doors with goats to the subset, and then shuffling the doors around within the subset. In some sense, this is the opposite of the Monty Hall problem, where the subset still retains the probability of having the car, but that probability has been diluted across the additional doors within the subset. Asymmetry is introduced such that it is now better to pick a door that’s not in the subset than a door in the subset.
So to answer our original question: choosing the proper subset matters because Monty reveals a goat from the subset such that the subset retains its probability of having the car. To be more precise, picking the subset alone isn’t very meaningful, but Monty’s reveal gives that subset meaning by introducing asymmetry. If we don’t consider this asymmetry when making decisions, then we’re discarding important knowledge and weighing probabilities based on incomplete information, leading to suboptimal choices.
How the Monty Hall Problem Tricks You
The Monty Hall problem first tricks you by making you think that your act of choosing an initial door has more value than serving the sole function of picking a random proper subset from which Monty reveals a goat. The truth is that Monty could have performed that step on your behalf. Anyone could have.
The Monty Hall problem tricks you again by asking whether you would like to keep your door or switch. Beneath the veil, the question really asks if you would like to choose a door from the initially chosen proper subset or not. By framing the question as keep-or-switch, i.e., go with your gut or cave, it adds an emotional element to your decision making that fogs your judgment.
Finally, the Monty Hall problem tricks you because it lures you into focusing on the superficial result—that door X has a goat—instead of the process that preserves subset probabilities. It is not immediately obvious that process matters and so that information is discarded, which means suboptimal decisions are made based on incomplete information.
Conclusion
The next time your friend asks why the Monty Hall problem is so damn unintuitive, you can introduce the Generalized Monty Hall problem and explain how the process through which the door is revealed preserves subset probabilities, thereby creating asymmetry. Then you can explain how the Monty Hall problem is a specific case of the Generalized Monty Hall problem, and how the same principles apply. And if this person is still your friend by the end of your crazy rant, then you may just have succeeded in guiding their intuition towards truth. They will never think of the problem the same way again.
Acknowledgements
Thanks to Alexander Wu for giving feedback.
