What if the host does *NOT* always give you a chance to switch? What if the host knows the prizes beforehand and decides whether to give you a chance based on his knowledge? What if you suspect he wants you to lose/win?

I was reading it where you think of it in terms of game theory. The Contestant wants to win the car. The Host may: want the Contestant to win, want the Contestant to lose, or not care.

The Host, though, only gets to choose whether or not to offer the Contestant a switch.

If the Host wants the most possible wins, he'll only show a goat and offer a switch whne the player is on a goat.

If the Host wants the most possible losses, he'll only show a goat and offer a switch when the player is on the car.

What about the other cases?