Howard--
You said, "Someone coming along at the second stage of the game where there are only 2 doors to choose between has no information on which to base his choice. He faces an entirely random choice between two doors. His chance of winning is therefore 50/50."

You are wrong in that assumption. Lighter change the game such that the second person sees two doors, and has no other information. The game is not set up to give away two cars, so if the original contestant did win the car, there will only be goats left for the second person to choose from. In that case, he would have zero chance of winning the car, and that possibility must be accounted for in determining the probability of winning the car. I showed, earlier today, that the new person has only a 1/3 chance of winning the car.

Now you can change the game further by only having a second player if the first contestant fails to win the car. The new person facing two doors with the assurance there is a car available, and no other information, does have a 50/50 chance.

We have come a long way away from the original Monty Hall Problem.