[T-hawk]

There's even a more dastardly procedure, where the host doesn't follow fixed rules but has free rein to outwit and trick the contestant. If Monty thinks the contestant can be induced into switching away from the car, he can try an offer, but doesn't have to. So he basically follows C (already the most disadvantageous for the contestant), but can also throw in a game-theory curveball of occasionally offering the switch even when the contestant is already wrong. Your described rule leaks information (offering the switch is a telltale that the contestant was already right), which the host can discredit by occasionally behaving otherwise. And of course if Monty is playing the psychology of the contestant, there's no rigid mathematical answer at all, like a poker bluff.

And by most accounts, the actual "Let's Make A Deal" show did let Monty do whatever he liked, so Vos Savant's presentation of the problem following rigid rules didn't really have a basis in reality.