Deviant argumentation regarding Monty Hall problem

As a note, there's no way to do this empirically in the real world, as the problem does not even remotely match a situation that actually occurred on "Let's Make a Deal".

I don't know what monty might have done in the original show to screw with the probability, but the usual analysis is correct and can be verified with as much simulation as you like.

I suspect that players would be unable to make the correct decision given the pressure and psychological manipulations coming from monty. Especially considering that highly intelligent people will argue for days that it's 50/50 up until they do a simulation and actually see the result; it's not an easy problem for untrained people.

Think of the conformity research. Monty could be saying things like "are you sure you want to switch?". Also, I think people may have a bias that causes them to think "man I'd feel stupid if I switched and got it wrong, but it would be ok if I didn't switch and got it wrong".

This game show is pre-internet so everyone would be on their own with regards to coming up with the solution beforehand. And nearly everyone gets it wrong at first.

I wonder how much selection bias there is in the type of people who actually got on the show.

These and related issues are addressed in Jason Rosenhouse's eponymous book on the Monty Hall Problem. I strongly recommend it. (Gratuitous plug: I reviewed it in detail at my blog).

One thing he points out there is that many people even when playing iterated versions of the problem still don't switch. There's a fair number of psych studies using Monty Hall to try to understand how humans think about probability and game theory. One major issue is that switching triggers feelings of loss aversion. Many people say they chose not to switch because they'd then feel bad if they had been right after all. So even if Monty had played the game as it is depicted (which he didn't) it is likely that many people would still not have switched.

Bonus question: suppose that we allow Monty to choose whether or not to open a door and offer a switch. In fact, Monty acts as follows: whenever the guest picks the correct door, Monty always offers a switch; otherwise Monty offers a switch with probability P.

However, the game show has been going on for years, and therefore the guest, who has seen all the past shows, knows what P is. What should the value of P be in order to ensure that, whenever Monty offers a switch, the odds for switching are 1:1?

What makes you think contestants would have figured it out? People appear regularly on Deal Or No Deal and don't seem to have figured out that there is no element of skill, and that the 'deal' offered is always lower than the expected winnings if the player continues...

I don't think you need to take data.

Don't you think we would have remembered if Monty had ever screwed the contestant, and showed the Grand Prize as the first door the contestant didn't win?

"Ok. Let's look at one of the doors you didn't select. ....... You didn't select the Grand Prize! Hurrah! So, do you want to keep your loser door, or take the other loser door?"

Whatever Monty may or may not have know about probability theory, he surely knew that doing that wouldn't make for an entertaining climax to his show. As long as he was committed and able to avoid that scenario, the usual analysis is fine.

Even if Monty was feeling bitter one day, and decided to screw the contestant, or they just made a mistake and opened the wrong door, that really isn't the point of the problem.

Of course your best strategy is different for Screw You Monty than for Let's Have an Exciting Climax to the Show Monty, but that shouldn't surprise any of us. Any optimal strategy has to assume a prior on the behavior of Monty. No Free Lunch.

He claimed that Monty Hall didn't know the answer to his own problem, and quotes a letter he says was written by Monty Hall in 1990. If this is true, I find it impossible to follow the following New York Times story, from 1991 where Monty explains all the subtleties of the problem in much shorter order than all of the complications on wikipedia.

Unless Monty Hall got a reply to his 1990 letter that explained it to him, and so he knew about it in 1991?

It is trivial to simulate, just do it for yourself.

My results, from back when I did it, follow.

As presented, with Monty always opening a door with a goat that the contestant did not pick:

Switch - 2/3 Stick - 1/3

If Monty opens a random door that the contestant did not pick, with the contestant losing if the car is revealed:

Switch - 1/2 Stick - 1/2

Monty should try and predict whether the player is the type to switch or not. If they are not, and a switch would help them, he should offer the switch. In some of the other cases, perhaps randomly, he should refrain from offering the switch.

If he is behaving purely randomly he should not tell people, or it would be like playing rock paper scissors against a random number generator instead of a human - lower drama.