The Monty Hall Problem

When I was younger, I would often play a game with myself: He knows.  I know he knows.  He knows I know he knows. I pondered these sentences, turning them over slowly in my mind, thinking that eventually, I would have a feel for them.  Unsurprisingly, for the mind of a little boy, the examples that came to mind were adversial.  He tried to out know me, as I tried to out know him.  So, I could see this as a game of countering strategies.  When I felt I had an intuition built up, I tried for more,

I know he knows I know he knows.

He knows I know he knows I know he knows!

I know he knows I know he knows I know he knows!!

He knows I know he knows I know he knows I know he knows!!!

I know he knows I know he knows I know he knows I know he knows!!!!

There was always a point at which the sentences attained true meaninglessness.  It was at that point, I gave things a rest until next time.

Today, I share this game for reason.  I realized that in two mathematics puzzles, the real break through comes from getting inside the other guy’s head.

There is a problem called the Monty Hall Problem which most people seem to get wrong.  It comes in the form of a simple game, with a simple goal and a simple dilemma.  One is presented with three doors by the game show host: behind one of the doors is a prize and the goal is to pick the door behind which the prize lurks.  One has a first opportunity to pick a door and having picked one, another door, different from the one you picked is opened.  Subsequently, one is given a choice of switching the door to the remaining yet unopened door or sticking with the door that one has already picked.  The central question is this: is it a better strategy to stay with the original choice or to switch doors?

Most people get this wrong.

Most people think it doesn’t matter either way.

This is wrong.

Our intuition does not seem to work in this situation.  SO WHAT’S GOING ON HERE?

It’s a matter of perspective.  Let me explain — with diagrams.  From the perspective of the participant in the game,  this diagram lays out the scenario.   First, one chooses a door.  Second, the gameshow host makes a choice of another door.  Finally, the participant makes a choice between the remaining doors.   (One should note that the diagram only shows what happens when door one is chosen first.   The name of the door doesn’t matter.   For instance, before we play the game, we can always have an assistant quickly rename all the doors so that the first door we choose is always  called door number one.  As long as we don’t move around the labels during the rest of the game, it does not change the reasoning.)

The next two diagrams highlight that in the case of switching and in the case of staying with the same door, there are two situations where one can win and two situations where one can loose.

What’s the catch?  The key insight is looking at things from the perspective of the host and realizing these pictures are not the whole story.   The host made one more move even before the participant makes the first: the host chooses where to place the prize.  As the first two diagrams show, the host is playing a secret game of avoiding the prize.   If  any of the first two situations occurs, the host telegraphs where the prize is by avoiding it.  Therefore, by switching one can take advantage of this weakness.

In the last case, the participant has guessed correctly.

Switching is a losing move here.  In the first two cases, switching is always a good idea.   So there  are two cases where switching is always a win and one case where switching loses the game.   Therefore, switching is the better strategy which wins two out of three times.   This is the Monty Hall Problem.

Although the common failure of intuition in this problem is often cast as a mathematical failure or a failure of knowledge of probability, I think that the issue is not realizing when the game started, which is before the participant ever makes a move.

If you would like to try some of these ideas for yourself, I have provided programs in three forms:

HERE

There is an executable file which should run if you double click it on any windows system. There is a Mathematica Notebook; and there is the C++ source code for the executable.