[Cushman]

It's definitely fascinating! Though I feel like attributing this to the intentions of the host is misleading, because it implies something mystical or quantum might be going on. It's more straightforward to just say it depends on the actions of the host.

Specifically, the host in the first formulation never reveals a car, even if she wants to. The host in the second formulation reveals a car roughly 1/3 of the time, even if she'd rather not. Thus, intuitively, it makes sense that the former leaks information about the place of the car, and the latter doesn't. [0]

Of course that's just a rewording of your point, based on my sense that "knowledge and intentions" are concepts that apply readily to game show hosts, but awkwardly to rule-following systems in general.

[0] To the extent that revealing something can not leak information about it...

Edit: To elucidate my quibble further, as a third formulation: "The host chooses one of the two remaining doors at random, which, by coincidence, is always a goat." I assert we should always switch: The intention of the host is the same as in the second case, but the behavior of the system, magically known to us, is that of the first. We make decisions based on that.

[dllthomas]

Lots of quantum things are surely going on...

But really, it has to do with the information content of the actions of the host.

[Cushman]

Just speaking for me, I'm not confident there's a "really" about this, so I'm not too concerned with finding it. I'm just trying to find the easiest way for our co-commenters to understand why the correct answer is, in fact, correct :)

[dllthomas]

Right. My point was just that the actions of the host, as far as we observe, can be the same in both cases.

[Cushman]

Sure, I see what you're saying. In a technical sense the information in the host's head does "matter". My comment is more along the lines that "information content in a system" is a sufficiently abstracted idea from "knowledge of a human being" that, especially if you equivocate them, it's not a useful metaphor for someone who doesn't already get why the right answer is right.

[dllthomas]

Yeah, it's not even strictly "does Monty know?" It's whether his action embodied that knowledge or not. Obviously (I hope) Monty knowing but still picking based on a coin flip is the same as Monty not knowing.