The latest Jellymatter poll has been up for a while now, time to discuss what the correct solution is. As well as sounding like a question from a Voight-Kampff test, it is a “double trick question”, based on the Monty Hall problem. It was a little mean of me to post it with my own agenda in mind.
For me, the interesting thing about the Monty Hall problem is vehemency of those who argue for “switch”, option. The argument is nearly always unjustified. Whilst arguing this I will talk about how the problem has been stated in the past: It’s history shows how quickly someones brief, informal argument can change into an unintuitive answer to a ill-posed question and then into a dogmatic belief.
The original article published by Steve Selvin (link) in American Statistician is little more than half a page long, half of which is a dialog between a game show host (Monty Hall) and a contestant. Monty Hall is giving the contestant a choice of three doors, behind two of them is what I now know to be called a “ZONK” (a goat), behind the other is a cash prize. The contestant chooses one door, Monte Hall opens another, demonstrates that it’s empty and states that the contestant has a 50:50 chance of the money being in either of the remaining doors. At this point Selvin objects: “WAIT!!!”.
He is right to object.
… but afterwards he comes up with a strange solution. Let’s have a look at the game tree as I see it. It has more strategies than the game tree that you will find on Wikipedia – This is because the real take home message for this post is that: The knowledge gained from being shown a ZONK is dependent on what strategy you assume the host to be using. This is an vital aspect of the problem that is usually overlooked.
Making the strategy more explicit:
There, are of course far more possible strategies. For example, in a real life version of the back alley pea and cup game given in the poll, I expect the most likely strategy for the ‘host’ is given by:
The game tree representing the host strategy assumed in the usual formulation of the Monty Hall problem is a subtree of big one above:
Here, the chance of you having chosen the correct door given that you have seen that there is a ZONK behind one of the others is 2/3, and the chance of you winning assuming that you play optimally (choosing switch) is also 2/3. I’m not going to explain why here, as an explanation is available here, here, here and in lots other places.
What is the host trying to do?
Is Monty trying to win?
It is hard to explain the behaviour of the host in the accepted solution to the Monty Hall problem: Firstly, he is not trying to stop you from winning, or the optimal strategies of the two ‘players’ would be:
In which case the probability that you have made the correct choice when the ZONK is shown is 1 and the total chance of winning is 1/3. For me, this strategy best describes the poll question – you should probably stick with the cup you chose in the first place because the other person is trying to take your money. Put another way: If someone with whom you share a mutual dislike gave you a heads or tails decision, the result of which benefits them and not you. Then, after looking at the result of the toss they say to you “are you sure you want to stick with your original choice?”, what would you do? I expect you would tell them to get stuffed!
Is Monty trying to loose?
He is not trying to give away the money either, or the strategy would be:
In which case the probability of having made the correct choice given the ZONK shown is 0, the probability of the cash being in the door that you chose is 1/2 and is the chance of winning the whole game is 2/3.
Is Monty just messing with the contestant?
I can only assume then, that he is trying to mess with the contestant as much as possible. Which, admittedly, makes for good television. But, even if this were true, a mixed strategy would be better (but perhaps people could think this unfair). I haven’t been able to find out for sure if Monty Hall always offered a choice. I’ve seen hints that he didn’t always – instead using a mixed strategy (which is different to the usual formulation of the problem).
Aha! So, Monty is messing with the contestant – POORLY!!!!
Monty’s actions being well described as a bad attempt at messing with the contestant seems to explain why the usual result is unintuitive. But, an unintuitive result from adding an unintuitive assumption isn’t “absolutely correct and unintuitive” as I have seen it described; it’s more like “silly”. Although this implies that I think the 2/3 probability is quite silly, the choice of switching is less so. Switching when asked is the correct thing to do in two out of three of the cases where a choice is presented – but still, acting on this depends on a judgement about which strategies are being used i.e. how likely each one is. You need to be braver than me to try and answer that.
The version that began the controversy…
So lets look at the vos Savant formulation, the one that is used by Wikipedia:
Suppose you’re on a game show, and you’re given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1 [but the door is not opened], and the host, who knows what’s behind the doors, opens another door, say No. 3, which has a goat. He then says to you, “Do you want to pick door No. 2?” Is it to your advantage to switch your choice?
This says even less about the hosts strategy than the original. At least in the Selvin formulation the strategy can be determined by empirically observing Monty Halls behavior on Let’s Make a Deal. Here, you have absolutely no idea about what the host is trying to achieve. But, it’s not that vos Savant is unaware of the choice of strategies, but according to her, she makes them explicit only in the answer to the question and says “anything else is a different question” – not exactly – anything else is a different answer. I can’t dislike her argument too much though: it really made so many people aware of interesting probability questions that even (especially) mathematicians were unfamiliar with. So, really I’m quite happy with her, even though the claims that are made are too strong. Anyway, since her first column on the problem, people have been slowly convinced. Her website states, in a rather large typeface:
We’ve received thousands of letters, and of the people who performed the experiment by hand as described, the results are close to unanimous: you win twice as often when you change doors. Nearly 100% of those readers now believe it pays to switch.
So, now everybody (well, possibly 50% according to my poll) thinks a different wrong thing – that assuming a very unnatural behavior is a sensible. Similarly, there is a plethora of websites that just ignore the really interesting part of this problem: that the answer depends on a judgement about how the host behaves. They nearly always assume the “always offer” strategy, which comes from nowhere sensible. I have to admit that, even when it is used for teaching something good, I dislike unspecific, badly posed maths problems. They are always more confusing than they need be, and, solving them always requires not thinking about some important things that will almost surely come back and bite. Then again: controversy works too, and being vague is a good way of generating it.
So, I’m glad that the Monty Hall problem exists, but in my ideal world it would be better posed.