Poll discussion: The Monty Hall Controversy

by Lucas Wilkins

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:

The red boxes are the hosts possible actions, the green boxes are the contestants and the yellow boxes are the possible outcomes at that point in the game

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.

15 Responses to “Poll discussion: The Monty Hall Controversy”

  1. I always thought the deal was that Monty shows an empty door every week, as part of the show’s format[*], so that the “always switch” strategy would be correct, for the reason that’s usually given. But, yeah, it turns out it wasn’t originally stated that way.

    It’s probably possible to use the standard tools of game theory (evolutionarily stable strategies and all that) to work out the optimum behaviour for the host and the contestant, if the host has a choice about his actions. I would guess both are mixed strategies (something like the “flip a coin” answer in the poll). But like all game theoretic results, it would assume perfect reasoning on the part of everyone. I imagined that your hypothetical street trickster would have sussed me as a mathematical type who’s probably heard of the Monty Hall problem, and would thus try to get me to switch when it was against my interests – so I chose to stick with the original one.

    [*] though I’ve never seen the show, I’m lead to believe that Monty Hall did in fact show an empty door every week before revealing where the car was, but he didn’t offer the contestant the ability to change their mind after seeing it.

    • Yeah, as I said, there’s hints about what he did – implying that he did’t always offer a choice. I’m not that keen on watching episode upon episode of Let’s Make a Deal to find out though.

      I’ve been trying to work out how to exploit people who believe in the Monty Hall problem too much. The difficulty is in making them sure that you’re using the “always switch” strategy without telling them so – making you a cheat (yet another reason to think that there is something fishy about it).

      As for the ESS, my intuition is for mixed strategies too.

    • I apologize for answering so belatedly no one will ever care, but I do quite like game shows and the Monty Hall puzzle naturally comes up when mathematics of game shows are discussed. One lovely irony of the puzzle is that this game was neve played on Let’s Make A Deal, neither the Monty Hall original nor the currently-running version.

      It’s a game that could plausibly have been played; it just didn’t, is all. The basic pattern of the many Let’s Make A Deal games is to offer contestants a choice, and then force them to second-guess themselves. But the second-guessing is not to remake the first choice, which is what the three-doors problem is. The more typical Let’s Make A Deal game would give the contestant a choice between curtains 1, 2, or 3, and then reveal the zonk behind one of the doors. Then the host would likely offer to let the contestant keep his curtain, or take the unspecified amount of cash within this envelope. If the contestant picks the envelope, then what was behind the curtain would get revealed. Perhaps the host would then go on to let the contestant keep the envelope or take what’s behind the small box. Typically around this point, the contestant is allowed mercifully to stop making decisions and to keep what’s won or not.

      And, obviously, on these new second-guessings there’s no subtle probability paradoxes at work.

      • I read the original article about this (before the vanity fair one). It makes certain assumptions about the game that aren’t necessarily true, but, the thing he originally criticizing – Monte Hall, on one show once, claiming that the probability is 1/2 – is definitely worthy of comment.

        My experience of probability paradoxes (MH, sleeping beauty, Borel etc) is that they are resolvable by being more explicit about the question. As you say, there’s certainly no paradox when you just don’t know what the host will do.

  2. “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.”

    I think that’s a bit mean Lucas, you’ve assumed that the Monty Hall problem encompasses all rule sets. Whilst the Monty Hall show seems to include other possibilities for the rules, the Monty Hall problem doesn’t relate to all options; it is used as an device (or riddle) to elucidate a miss-calculation of statistics under the one rule set (i.e. you are showed a zonk and you are always allowed to switch). Obviously if the rules are different then the solution tree is different as you’ve shown. The problem is stated for that ONE ruleset because it’s supposed to be educational.

  3. Hahahah! I think in the back alley situation anything goes and if you’re silly enough to play it without settling on a ruleset (if that’s even possible!) then you probably shouldn’t get too attached to the notes in your back pocket! Come to think of it, you probably shouldn’t get attached to your watch, your wallet or your phone either.

    • That’s exactly why it was mean to ask my question, most people wouldn’t be playing in the first place…

      Holden: The tortoise lays on its back, its belly baking in the hot sun, beating its legs trying to turn itself over, but it can’t. Not without your help. But you’re not helping.
      Leon: [angry at the suggestion] What do you mean, I’m not helping?
      Holden: I mean: you’re not helping! Why is that, Leon?
      [Leon has become visibly shaken]

      It’s quite a contrived situation but the opponents intent, thus strategy, is better defined. Perhaps I’ll try and think of a better version.

      • Here’s an interesting variation, where the “opponent”‘s “strategy” is clear, and the answer is definitely different from the standard Monty Hall solution:

        Your friend has gone missing during a freak rainstorm while mountain climbing. There are three identical caves on the mountainside, each suitable for sheltering from the rain, but the storm has caused the entrance of each one to be blocked by a seemingly identical rockfall. The search party has looked everywhere else and found nothing, so you conclude that your friend is trapped in one of the caves. You don’t think you can clear more than one of the entrances before nightfall, and since there’s nothing to choose between them you decide to start with the closest one. However, at that moment, by complete coincidence, the rocks blocking one of the other caves drop away, revealing it to be empty. (The storm water had apparently stacked them in an unstable way.) Should you continue with your plan to start by clearing the entrance to the closest cave, or walk some distance across the mountainside to the other remaining blocked entrance?

      • No wait. I’m wrong. The answer is actually the same as the standard Monty Hall problem. I think. I’m confused.

  4. Actually, I think you can find the ESS (or rather, the single Nash equilibrium) without doing any complicated maths. It turns out to be kind of boring:

    For the contestant there are only two pure strategies: always switch (if given the choice), or never switch (if given the choice). All other possible strategies for the contestant are mixtures of these two, i.e. switch with a given probability.

    Now let’s go through the game tree from the host’s point of view. First the contestant chooses a door. Next there are two possibilities:

    1. The contestant has chosen the correct door. Damn! Now the host can either pay out or offer to let the contestant switch. But there’s no reason not to let the contestant switch – in the worst case you’ll have to pay out, but if you don’t offer the choice you’ll have to pay out anyway. If the contestant is playing the “never switch” strategy then it doesn’t matter whether you pay out now or make the offer, but for any other strategy you’ll be better off letting her switch.

    2. The contestant has chosen the wrong door. Hooray! Now there are again two choices: allow the contestant to switch, or ZONK them. But in this case there’s no reason not to just ZONK them straight away instead of risking the possibility that they might switch and win the car. If the contestant happens to be playing the “never switch” strategy then it doesn’t make any difference, but in all other cases it’s better not to offer the choice.

    So the only sensible strategy for the host, if they’re trying to minimise payouts, is to offer to switch only if the contestant has chosen the door with the car. Therefore, if you’re the contestant, you should never switch if you’re offered the choice, because if you’d chosen the wrong door you’d have been ZONKed already.

    So the best strategy is “never switch”. If all the contestants always play the best strategy then it no longer matters what the host does – but still, if you were the host it would be a bad idea to play anything other than the best strategy, just in case there are any stray “mutant” contestants who do sometimes switch.

    (It turns out that repeatedly using the word “ZONK” is fun. Who knew?)

    • The cave thing isn’t Monty Hall, as there is no reason to think there is correlation between the choice of exposed cave and where your friend is. The optimal thing to do, then, is to not walk a long way.

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