Let’s See What’s Behind Door …

[ Read this in Dutch ]

I just love game theory.

In the final play of a popular Dutch quiz, the remaining candidate is presented with three doors. Behind one of the doors is the Grand Prize. The candidate picks a door. Something happens. The chosen door opens. The Grand Prize is there, or it is not.

Few quiz games are more brainless than this*. In fact, both the quiz and the quizmaster were discontinued** decennia back.

But the probability calculus mysteries surrounding this straightforward three-way split prompt fierce discussions to this day, sometimes even leading to fights and lifelong feuds. All because of the something that happens between picking the door and opening it.

The something that happens after the candidate picks his door is that the quizmaster always opens one of the two remaining doors and demonstrates that the Grand Prize is not behind that one. After which he always offers the candidate the option of switching to the second remaining door. The probabilistic confusion, discussions, fights, and feuds all revolve around this Earth-shaking question:

Should he switch?

Well yes, in fact, he should. That is, if he wants to double his chance of getting the Grand Prize.

Let’s take a seat in the audience and look at the quiz stage. Here’s our candidate—let’s call him Mat—standing before the door he picked, door A. The Grand Prize might be behind his door! Then again, it might not. Who can tell? Next to him is the quizmaster, who has just opened door B. Behind door B, the audience can clearly see a whole lot of nothing. And then there’s door C, which is as firmly shut as door A, and which obviously may contain the Grand Prize, or it may not. With door B out of the game, it would seem logical to conclude that the Grand Prize is either behind door A, where Mat is standing his ground, or behind door C, the option of switching to which has Mat all confused.

‘Two options means 50/50, right?’ he’s thinking out loud. ‘So it doesn’t matter whether I switch or stay put, right? So why’s that really tall guy in the audience with the laptop gesturing towards door C with his chin?’

Truth be told, the question is not as complicated as a lot of people would have you believe.

With three doors to choose from, and only one Grand Prize, Mat initially had a one-in-three chance of picking the right door, and a complementing two-in-three chance of picking the wrong door***. If he picked the right door, and he switches, he is certain to end up at the wrong door. If he picked the wrong door, and he switches, he is certain to end up with the Grand Prize, because the quizmaster has been helpful enough to give away which of the other two doors is priceless. So: wrong door – switch – Grand Prize; right door – switch – nothing.

With me so far, Mat? If you picked the wrong door initially, you will win the Grand Prize if you switch; if you picked the right door, you will go home empty-handed if you switch.

Now the chance of picking the wrong door initially was two-in-three. So if you switch, the chance of winning the Grand Prize is two-in-three! And if you don’t switch, you win only if you actually picked the right door, the chance of which was one-in-three. In other words, you increase the odds from 1/3 to 2/3 if you switch, Mat!

In confusion and despair, Mat looks to the quizmaster for help, but he looks into the camera and pretends not to understand any probability calculus.

Okay, let’s look at it a different way. What the quizmaster is really offering you is a switch from door A to both other doors. It’s not exactly new information that one of the other doors is empty, now is it?

Mat shakes his head, but more in mounting confusion than in answer to the question.

Okay, never mind logic. (Mat’s eyes brighten in relief.) Let’s just go through the possibilities. Mat might have chosen any one of the three doors, and he could decide to switch or stay put in all three cases, so he has six options. Let’s assume that the Grand Prize is behind door C. If Mat picks door A, he would be wrong—initially. The quizmaster would open door B, showing it empty. Staying put at door A would make Mat a loser; switching to C would yield the Grand Prize. If instead he picks door B, the situation is almost identical: quizmaster shows the emptiness behind A, staying put would send him home without a prize, and switching to C would immortalize Mat as Grand Prize winner. If, however, he picked C initially****, the quizmaster shows him the content of either A or B, which are both empty. In that case, staying put would mean he’s sticking with the Grand Prize, and switching to B or A would mark him for the sad, pathetic loser he really is.

Mat initially picks door

A

B

C

If he stays put nada nothing Grand Prize
If he switches (to C!) Grand Prize Grand Prize zilch

Mat’s options when the Grand Prize is behind door C

In two out of three cases, he gets to switch to the winning door!

While Mat, who stopped listening after the assumption that the Grand Prize is behind door C, opens this door before a breathless audience, the quizmaster counters:

‘But it’s not really three possibilities, is it? If he picks the right door, I could then show him what’s behind either of the other doors, A or B. Picking the right door gives Mat two possibilities to switch to the wrong door! A total of four possibilities, only two of which give him the Grand Prize!’

To paraphrase Douglas Adams, most leading statisticians see this argument as a load of dingo’s kidneys. You can’t just go around adding up options that Mat has (picking door A, B or C) with options that the quizmaster has (opening door A or B). The only independent and thus determining variable is Mat’s choice of door. Mat has three choices, two of which yield him the Grand Prize if he switches.

In fact, the quizmaster’s interjection only serves to further confuse the issue, which brings us to a far more fascinating aspect of the quiz: psychology*****! All this only holds up if the quizmaster always opens an empty door and always gives Mat the option of switching.

But what if the quizmaster, sly and manipulative bastard that he is, does these things only if it serves his purpose******. If he has a choice of opening a door for Mat or not, but Mat doesn’t know this, he would do well to open an empty door only if Mat happens to pick the right door. Knowing his probability calculus, Mat would switch, and be a loser. If Mat picked the wrong door, the quizmaster would of course leave him there and not open any doors or offer any options whatsoever.

Things get even more interesting if Mat knows that the quizmaster has a choice in this, and especially if the quizmaster knows that Mat knows. Mat would then expect the manipulative, cheating behavior described in the previous paragraph, and the quizmaster should anticipate on that and do the exact opposite, again tricking Mat into forfeiting the Grand Prize. But if Mat knows that the quizmaster knows that Mat knows, he would …

Et cetera ad infinitum.

Of course, this only works if Mat actually understands the odds and is smart enough to act on them. And given the fact that he completely failed to understand this explanation, this is a dubious assumption.

In general, it seems that the quizmaster should have profound insight into his candidate’s mental faculties before deciding how to act. I wonder what he should do if his candidate was the author of this post?

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* One that springs to mind is a game called “Nerves”, in which a rapidly increasing dollar amount is shown on a screen. The countup can be stopped by pushing a large red button, and the first candidate to push the button gets the displayed amount.

** Technically, the quiz was discontinued and the quizmaster died, but that’s not the point.

*** Disregarding any good or bad luck that Mat may or may not suffer from.

**** But what are the odds of that, really? Mat picking the right door at once? Well, one-in-three, as a matter of fact.

***** And I don’t mean Mat’s psychology, though I wonder whether psychologically it’s not twice as terrible to switch and lose than to stay put and lose…

****** Which, I assume, is to claim the Grand Prize for himself after the studio closes.