If we consider the Monty Hall problem de novo, it is evident that much depends on the strategy of the host. The wiki article on the problem lists possible host strategies. For example, there is Monty from Hell, and Angelic Monty among others. However, the wiki article does not seem to draw any final conclusions concerning the ultimate game theoretic outcome.
Assuming we have no information about the host's strategy, does that make the problem undecidable?