In an adversarial setup you basically get 1/3 wins like you started with because the optimal host move is to provide no information whether they reveal or not
In an adversarial setup, the optimal host move is to permit you to switch only if you picked the car. You still get 1/3 wins with optimal play (never switching), but it's better for the host because it allows people who think it's the traditional game to get zero wins.
If the host strategy isn't explicitly specified, it's reasonable to assume an adversarial setup, because seeing the player lose after switching would be more entertaining for the audience.
> seeing the player lose after switching would be more entertaining for the audience
Have you ever watched game shows? Because it is far more satisfying and fun for the audience when players win, not lose. If you are out there rooting for people to lose, you're a miserable bastard.
Now that I think about it, I have watched a few episodes of game shows, and I think you're right. They are always set up so the audience identifies with the players and the hosts are the adversaries, so you want the players to win.
