Here is an interesting variation on the Monty Hall problem. For now I will simply present it cold, without indicating the context of where I saw it. Feel free to leave your proposed solutions in the comments. Everything from vague intuitions to hard-core Bayesian analysis is welcome.
Adam and Barney are contestants on a game show hosted by Monty Hall. Each player knows that the other one exists. They are confronted with three identical doors. One of the doors conceals a car, while the other two conceal goats. Both players select one of the doors, but neither player knows which door was chosen by the other player. After making their selections, Monty now opens a door according to the following rule: If Adam and Barney have selected the same door then Monty opens one of the remaining two doors, always opening one he knows to conceal a goat. (If he has more than one door to choose from, he chooses randomly from his available options). If Adam and Barney have selected different doors, then Monty opens the one remaining door, even if by doing so he reveals the car. After Monty opens a door, both players are given the chance to switch. If both players land on the door with the prize, then they both recieve the car.
Place yourself in the role of Adam. You have chosen door one. You now see Monty open door 3 and it turns out to be empty. Should you switch?
To help you parse this, let me mention a few possible scenarios.
The only way Monty will open the door with the car is if Adam and Barney select different doors, and neither of their doors conceals the car. (For simplicity, we will assume that both players win if Monty reveals the car.
Suppose Adam and Barney both choose door one. If the prize is behind door two, then Monty will be forced to open door three. If the prize is behind door one, then Monty will be able to open either door twoor door three, and he will make his choice randomly.
If Adam choose door one and Barney chooses door three, then Monty will open door two regardless of what is behind it.
Keep in mind that you do not know which door Barney chose. So when you see Monty open the empty door three, it might be the case that Barney chose door two, thereby forcing Monty to open door three, whereupon it was discovered by chance that door three was empty. Or it might be that Barney chose door one, and Monty opened door three because he knew that it was empty. There are other scenarios, of course.
Okay! Have at it. I’m going to go grade finals for the next forty-eight hours. I’ll expect answers when I return.