## Tuesday, 11 October 2016

### Four Doors Problem (Not Really Monty)

I'm revisiting Monty because he's so much fun!  That said, the scenario isn't really related to the Monty Hall problem, other than it involves doors (four rather than three) and there is a preferred prize … who wouldn't want the chance to win a goat?

This time, Monty has four doors, each with a little platform in front of it.  Three of the doors have goats behind them and the other one has a large sum of money, a million dollars.  Monty knows which door has the cash behind it.  For ease of reference, the doors are labelled 1, 2, 3 and 4 while the platforms are colour coded: Red, Green, Yellow and Blue.

Rather than selecting a door and being offered the chance to switch, you now have the opportunity to question Monty.  The problem, as it is explained to you, is that Monty can only answer you when standing on one of the four platforms and, depending on which platform he is standing on, he will give you a different sort of answer:

the truth,

a lie,

the truth or a lie - but alternating, so a 50% probability of starting with a lie and then alternating, and

a totally random answer (with a uniform distribution, so 50-50 on a yes-no question, 33-33-33 on an x-y-z question and so on).

Each time you ask Monty a question, the amount of money behind the door is halved.  While you can move him to any platform you like without affecting the prize value, you don't know which colour platform triggers Monty to respond in which way.

So, the metaquestions are these:

What is the optimal number of questions to ask (see below)?

What is the value of your final position (see below)?

What would your questions be? and

How many platforms do you get Monty to stand on?

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By value, I mean that a 25% chance of winning a million would be worth \$250,000.  By optimal, I mean the number of questions that result in highest value.  Note that while they strictly are of equal value in a sense, I'd still put a certainty of winning \$250,000 above the 25% chance of winning a million for psychological reasons - because we tend to prefer certainty over risk.  We are inherently loss averse which means that the 25% probability of winning a million has to be balanced against the 75% probability of having given away the certainty of \$250,000.  If you need to choose between two otherwise equal "values", consider the one with less risk as (closer to) optimal.  (Alternatively, think about the fact that you'll have a goat to deal with if you open the wrong door!  Let's assume that you don't want a goat and would consider winning one as an impost.)

Note that if you could only reach absolute certainty after four questions, then the start position of a 25% chance of taking home a million (along with a 75% chance of winning a goat) would have about four times as much "value" as \$62,500 in the bank.  Hint: you can get there in less than four questions.

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If you like, you can just forget about the goats.  They really aren't important.