That presumably quickly gets into a game-theoretic double-triple-etc-bluff affair where you have to assume the host is using his knowledge against you.

If you choose the door with the car, the host will open another door to tempt you, so then you don't switch, unlike in traditional Monty Hall. And if he didn't open a door, well, that means you choose a goat, so you should switch for a 50% chance. Except the host knows you're thinking this. And you know he knows. And he knows you know he knows. And...

Perhaps, iterated infinitely, this eventually resolves into a limiting expected value for switching and not, but I have no idea how to compute such a scenario.

See: https://en.wikipedia.org/wiki/Monty_Hall_problem#Other_host_...

The puzzle is is commonly stated is not how any TV show ever worked. In a real (repeated) TV show the host behavior will be non-deterministic, and will sometimes be benign, sometimes adverse, to increase the suspense and ratings. If the host follows a deterministic set of rules (as is usually implicitly assumed in the puzzle version), the optimal strategy is pretty easy to work out.

> If the host follows a deterministic set of rules (as is usually implicitly assumed in the puzzle version), the optimal strategy is pretty easy to work out.

And yet it is the subject of endless discussions. The optimum strategy is pretty easy to work out, yet not widely believed.

You can browser the countless HN or even Reddit threads and it's always filled immediately with one of the few standard explanations (which are all in the Monty Hall Wikipedia article). None of the "endless discussions" actually disputes the strategy or nobody needs to be convinced. It's all about whoever things they can come up with the most "intuitive" explanation. At this point I'm starting to thing that it's more a myth than reality that it's "not widely believed".