three door monty

Monty Hall has died. He was 96 years old, so he had a good deal. Here's the pilot from his show.

I wasn't a big fan of Let's Make a Deal, so I don't really know what to say about him. But I’ve always wanted to write an essay about the Let’s Make A Deal Problem (LMaDP),  and I’ve never had an opportunity to do so. Well, it's now or never.

I’m not sure how well known LMaDP is outside of actuarial and mathy types of crowds. It was made famous in 1990 in a letter to Marilyn Vos Savant, who answered questions for her “Ask Marilyn” column in Parade Magazine.

The gist of the problem is as follows:

You are on a TV gameshow, and given a choice of three doors. One door hides a prize. The other two doors each hide garbage. You pick a door, and win what’s behind it. But before the host opens the door to reveal what you’ve won, he picks one of the other two doors and reveals that it hides garbage. He then gives you a choice between staying with the door you originally chose and switching to the other as-yet-unrevealed door. Should you switch? Before you answer, let me add some information which is essential, but often elided in the retelling: Going in, the host knows which door hides the prize, and is required by the rules of the game to open a door that you didn’t pick, and which hides garbage behind it. This, presumably, is to build excitement. Now, with that extra bit of information, should you switch or stick.

When Vos Savant was presented with the problem, she (correctly answered that you should switch. Specifically, you have a 2/3 chance of winning if you switch and a 1/3 chance of winning if you stick. The response was angry and vocal. She was roundly condemned as a know-nothing, setting back the cause of education, blah blah blah. I remember some of the senior actuaries at work were similarly incensed. Her response does initially seem counterintuitive; since you have two doors to choose from, and they seem the same. But she stuck to her guns. And she was right to do so.

What I'd like to do now is argue in several different ways. Then, finally, I want to end by answering the question of what makes the doors different.

Arguments

Argument 1: 100 doors

One of the most commonly used arguments presents a situation that is similar, but different in one crucial way. Specifically, it supposes that there are 100 doors, with one hiding a prize. After you pick a door, the host opens 98 others, revealing them all to be losers. he then gives you the chance to switch. Are you better off switching? Presumably the fact that there are so many more doors makes it easier to visualize that it's unlikely that you won with your initial pick.

Argument 2: Under what circumstances do you win by switching/sticking

Consider what circumstances lead you to win by switching and what circumstances lead you to win by sticking. If you initially choose the winning door (something which happens one time in three), you will win by sticking. If you initially chose a losing door (something which happens two times in three), you will win by switching. QED.

Argument 3: Enter the Bayesians

This is actually closely related to argument 2. Assume it doesn't matter whether you stick or switch. That means that you have a 50% chance of winning with your door. But you know that, once you choose your door, the host will open another and give you the choice. In other words, no matter what he does (i.e., whichever other door he reveals), your door will then have a 50% chance of being the winner. That means that, before he does anything your door has a 50% chance of being the winner, since the chance of your door being a winner (before he does anything) is equal to the weighted average of it being the winner after he does something (weighted by the probabilities of his taking each possible action).

So let's get this straight. If the game were different -- you pick a door and then, without anything being revealed, you can keep it or switch to one of the others -- the chances of your door being the winner are one in three. But adding the extra rules of the game as stated above, your initial door goes up from 1/3 to 1/2! That's a pretty good Schroedinger's door, there.

Argument 4: Simulation by spreadsheet
I created a little spreadsheet to simulate the situation. It's here. You can play along at home. Sheet1 has 10,000 simulations. In each, there are three doors -- A, B and C. The following happens:

• In column C, the winning door is determined randomly
• In column E, the door you (as the contestant) is determined randomly
• In column H, the door the host opens is determined randomly, subject to the rule that it not be either the door chosen in column C or the door chosen in column E
• Columns I and K display the doors you have by sticking and by switching
• Columns L and M indicate whether you win by sticking or by switching (with a "1" in the appropriate column).
• Column N is simply a check column. It should always be 1. Cell N1 serves as a check that every entry in column N is a 1. N1 should be zero, indicating that everything else is the same.
• Cells L1 and M1 add up the number of simulations in which you win by sticking and in which you win by switching.

You can rerun the 10,000 simulations by selecting an empty cell (e.g., R2) and hitting the delete key. The relevant cells will be recalculated, and you can see that cells L1 and M1 will be somewhat close to 3,333 and 6,667 (respectively). I have rerun this simulation several times, and each time I get close to that split. Note that the odds of getting that exact split are low. If you flip a fair coin 1000 times, you will likely get close to 500 heads and 500 tails. But you are unlikely to get exactly that.

What Makes the doors different?

As I wrote above, it initially seems counterintuitive to say that one door is more likely to win than the other. There are two possibilities, and on the surface it can seem like they are the same, and therefore each has a 50% chance of winning. If the chances are not the same, there must be some asymmetry between them. So what is that asymmetry?

The asymmetry is in the way the two doors came to be candidates (i.e., why each is not the one that was revealed as not the winner).

• The door you did not choose was not revealed to be a loser for one of two possible reasons:
• Possibly, you chose the winner to start, in which case the host could have opened either remaining door, and he just chose the one he did; or
• Possibly, you chose a loser, in which case one of the remaining doors is a winner, and the host had no choice but to reveal the one he revealed.
• The door you initially chose was not revealed because the host is not allowed to reveal it.

Since the two remaining doors are not the same. the argument that they are equally likely to win because they are the same is false.