Indeed. But the hosts machinations can be arbitrarily more complex; maybe he offers the switch to contestants he finds attractive only when they’ve picked a goat, and contestants he finds unattractive when they’ve picked the car.
This works as a logic puzzle. Assuming the host offers different doors depending on contestant attractiveness makes absolutely no sense. It's a bizarre assumption.
Maybe the goats can wander from door to door, or maybe there is no car, or maybe behind all of the doors there are tigers. Which would be absurd and unrelated to this puzzle.
The hosts' strategy is the core of the puzzle. The question is what information he has just conveyed to the player by opening the door and that depends entirely on his mental state. If he was always going to open a door then the player should switch. If he is opening a door only if the player has picked the car then they should not switch. If he has bizarre ad hoc motivations then the correct decision depends on bizarre ad hoc considerations.
And, as CrazyStat has correctly pointed out, as stated in the linked article the hosts' strategy is an unknown. It could be bizarre. Although I'd still rather say vos Savant was correct in her reasoning; since the answer is interesting it seems fairer to blame the person posing the question for getting a detail wrong.
The source material for the article says otherwise:
> So let’s look at it again, remembering that the original answer defines certain conditions, the most significant of which is that the host always opens a losing door on purpose. (There’s no way he can always open a losing door by chance!) Anything else is a different question.
You need to know why the host did that though. The host might have an adversarial strategy where they only open the door when using vos Savant's logic would make the player lose.
Her logic was really interesting, it is easy to see why she gets to write a column. But at the end of the day the problem is technically ill-formed.
> There’s no way he can always open a losing door by chance!
Yes there is, he might have picked a remaining door at random with the plan of saying "You lose!" if he finds the car. An actor with perfect knowledge can still have a probabilistic strategy. That doesn't really change the decision to switch, but it does have a material impact on the analysis logic.
She's correct that is a necessary assumption to get the most interesting form of the problem. But that isn't what the questioner asked.
I doubt the person writing the question had the more nuanced version in mind. When the column was published, nearly everyone who wrote to Marilyn vos Savant had been in agreement that the host always opened a door with a goat as the problem was specified. This idea that the host did not was made after Marilyn vos Savant was shown to be correct and after she had already confirmed that it was part of the consensus.
Adding ad hoc hypotheses about the host's motivations turns this into a family of related problems, but not The Monty Hall problem.
For any given logic puzzle, you can safely assume anything not specified is outside the problem.
Here, what Monty had for lunch, whether he finds the contestant attractive, or some complex algorithm for his behavior is left unspecified and -- since this is a logic puzzle -- this must mean none of this matters!
Imagine if Monty opened a door with a goat only if he had had goat cheese for breakfast. Sounds ridiculous for the logic puzzle, right?
We can safely assume, like Savant, that Monty always picks a door with a goat, turning this into a logic puzzle about probability.
Anything else is going out of your way to find ambiguity.
Well sure, it doesn't appear that vos Savant was asked the Monty Hall problem. She seems to have been asked an ill formed alternative problem and answered that instead. Then the interpretation of the ill formed question with the most interesting assumptions about the host's behaviour became the Monty Hall Problem.
And the linked article (and by extension Mr. Diaconis & CrazyStat) was talking about the question that vos Savant was asked as opposed to the one where the assumptions to come to an answer are enumerated.
> We can safely assume, like Savant, that Monty always picks a door with a goat, turning this into a logic puzzle about probability.
No we can't. Otherwise we can safely assume any random axiom, like "The answer is always the 3rd door". You have to work with the problem as written.
This is like being asked how to solve a (legal) Rubik cube configuration and then considering how close you can come to a solution if given an illegal configuration. It is not relevant since it is not what was presented. You can always make things more complex by considering variations that are not relevant to the original problem.
This was a real life game show where the host did not always open a door and offer a switch. In fact most of the time he did not offer a switch. See [1] (Ctrl-F cheating for the relevant paragraph).
That would be a rather convenient signal to the player.