I just ran across a strange statement in the solution of the MH problem with Bayes as it is described here for instance

Assuming that I chose door A the conditional probability that the car is behind door A if I am shown door B ( with a goat) depends on the conditional probability that Monty picks door B given that the car is behind A being 1/2 ( that is Monty is making a random choice) . However, if he has a strategy like ALWAYS picking B when there is no car behind it then the conditional probability of having the car behind A if shown B will change ( become 1/2 in fact)

But this has to mean, that the Bayes analysis has a defect somewhere, whether I switch or not can not depend on a hidden strategy choice of Monty, can it?