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There's the Monty Hall Problem and then there's the Second Order Monty Hall Problem Problem. That is, did the wording as it was originally printed unambiguousl

by dekhn 3y ago

There's the Monty Hall Problem and then there's the Second Order Monty Hall Problem Problem. That is, did the wording as it was originally printed unambiguously define the problem to be solved, or was it ambiguous enough that some people who got it wrong got it wrong for the right reason? Like many others, I completely missed that when Monty opens a door to show you a goat, he always shows you a goat. I think under close inspection, the problem doesn't actually allow you to interpret otherwise. But, should a problem like this depend on parsing and identifying the ambiguity, or should the problem be defined with explicit details about the door opening? Given how many otherwise extremely smart people have misunderstood the problem, I am still unsure whether this was an intentional choice in framing the question. I got it wrong and thought Vos Savant was wrong until I read one of the replies in her follow-up article (yes I read this when it first came out!) said they wrote a computer program and reproduced her results. The program implemention worked because they correctly modelled Monty's behavior. I think when people end up writing the program out the ambiguity becomes more obvious.

ghaff 3y ago
I think this is the original "Ask Marilyn": 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 #1, and the host, who knows what’s behind the doors, opens another door, say #3, which has a goat. He says to you, "Do you want to pick door #2?" Is it to your advantage to switch your choice of doors? Craig F. Whitaker Columbia, Maryland Yes; you should switch. The first door has a 1/3 chance of winning, but the second door has a 2/3 chance. Here’s a good way to visualize what happened. Suppose there are a million doors, and you pick door #1. Then the host, who knows what’s behind the doors and will always avoid the one with the prize, opens them all except door #777,777. You’d switch to that door pretty fast, wouldn’t you? ----- Yes, the question as posed by a reader was ambiguous but Marilyn made a very reasonable interpretation based on what someone at that time would probably tend to know about game shows. And she even gave an explanation (using more doors) that I find tends to work with folks combined with the explicit assumption that the host will always pick goats.
- dekhn 3y ago
  Yes, I'm literally talking about this sentence: "host, who knows what’s behind the doors, opens another door, say #3, which has a goat." I parsed that as "50% of the time, monty opens another door and it has a car and you win immediately, and 50% of the time, monty opens another door and it has a goat". In retrospect I think my brain just sort of pictured that and proceeded to assume there was no reason to switch, and it wasn't until I read some answers (not her explanation) that I understood I was wrong (and I went through a lot of anger, like many of the commenters, and thought she was very wrong before that). If there had been exactly one more sentence saying "monty always opens a door showing a goat", I'm pretty sure I would have recognized that. BTW I was familiar with the game (I liked Jeopardy better, as Let's Make a Deal was a fairly dull show) and I don't think they had a game that was identical to the problem as described.
  - JackFr 3y ago
    > 50% of the time, monty opens another door and it has a car and you win immediately Why would you think you won? You didn't choose that one with the car and you aren't offered the opportunity to switch to the open one, only the closed one. The only interpretation which makes sense is that he always shows you a goat.
    - beepboopboop 3y ago
      I think they meant you lose immediately.
    - city41 3y ago
      Not to mention Monty always showing a goat is what adds the tension and interest a game show needs.
      - default-kramer 3y ago
        I disagree. If Monty always reveals a goat, there is no tension. You know exactly what is going to happen. Maybe it could be used a way to pad the running time of the show, but it would not add tension. "Coming up next: Monty reveals what's behind one of the other doors!" seems like something a game show would do. Whereas "Coming up next: Monty reveals a goat!" does not.
Scarblac 3y ago
I assume that at the time of writing, most readers would have been familiar with the actual game show and the fact that the host always opens a door with a goat?
- dekhn 3y ago
  The show uses a different game.
- dboreham 3y ago
  I had never heard of the show when I was introduced to the problem so all my knowledge comes from the problem description.
- jcl 3y ago
  As Monty Hall pointed out in interviews, he was not obligated to offer the opportunity to switch and sometimes did not. So in this case knowledge of the actual show may have contributed to confusion. https://en.wikipedia.org/wiki/Monty_Hall#Monty_Hall_Problem https://en.wikipedia.org/wiki/Monty_Hall#Monty_Hall_Problem (But vos Savant stated in later columns that most of the critical responses she received assumed that the host was obligated to offer the switch, so they were genuinely confused by the paradox.)
kristjansson 3y ago
As someone who misunderstood the problem as you describe, I'm much more invested in this version of the problem :) The problem statement is just a mess: > You pick a door, say #1, and the host, who knows what’s behind the doors, opens another door, say #3, which has a goat. even tiny edits like: > You pick a door, say #1, and the host, who knows what’s behind the doors, opens another door which has a goat, say #3. or even > You pick a door, say #1, and the host, who knows what’s behind the doors, opens another door, say #3, which he knows has a goat. make it clear that the last phrase is describing an invariant of the problem, not an artifact of the illustrative hypothetical.
- svantana 3y ago
  That's much better, but still not enough. We also need to add that the host will open a door regardless of which door I initially choose.
  - kristjansson 3y ago
    Ah yeah, good point: > You pick a door and the host, who knows what’s behind the doors, opens exactly one of the two remaining doors according to the following algorithm: if both doors contain goats, he picks a door at random, else he picks the door he knows to contain the goat.
empyrrhicist 3y ago
Your re-framing is actually not quite right either (or at least, not complete). Key point: if the host picks the car, the game essentially re-sets. Imagine the extreme scenario with 999 goats and 1 car. You select the first door, the host opens 998 doors at random, leaving your selected door and one other. Two scenarios are now possible: 1. There was a car in the 998 doors that got opened. Tough luck, you lose the game automatically now (game resets). 2. There was not a car in the 998 doors. In this scenario, you are still advised to switch. Edit - I guess I misremembered a simulation I once did about it.
- Swizec 3y ago
  Also you're picking between Reward and Bigger Reward. Positive EV no matter your choice so eh don't worry about it too much :) A nice goat fetches up to $1000 to according to a quick google.
  - klyrs 3y ago
    A nice goat, you say? How much for a mean goat, jaded from years of being the spoiler prize that nobody wants?
    - bemusedthrow75 3y ago
      Whenever someone talks about mean goats I think of only one thing: Buttermilk the baby goat, being a jerk. https://www.youtube.com/watch?v=AWvefaN8USk https://www.youtube.com/watch?v=AWvefaN8USk Enjoy.
      - deleted 3y ago
        [deleted]
  - bemusedthrow75 3y ago
    I am suddenly reminded of the clever, morally educational joke about the dead horse raffle.
- pxx 3y ago
TotempaaltJ 3y ago
Vos Savant mentions this, and she's kept track of who did and didn't understand the conditions: > And a very small percentage of readers feel convinced that the furor is resulting from people not realizing that the host is opening a losing door on purpose. (But they haven’t read my mail! The great majority of people understand the conditions perfectly.) https://web.archive.org/web/20181118225305/http://marilynvossavant.com/game-show-problem/ https://web.archive.org/web/20181118225305/http://marilynvos...
- topaz0 3y ago
  Nice find! I have wondered this. It's easy to say after you have come around that you had first misunderstood the scenario and done the math right in that context, even if in truth you (implicitly) understood the scenario but just neglected to account for the constraints.
topaz0 3y ago
What astonishes me is not that people interpret the scenario differently, but that they are so quick to conclude that others are doing the math wrong and feel superior or even angry, rather than thinking about whether you might be talking at cross purposes about different scenarios. So I see the second order problem a bit differently: it's not "was the statement unambiguous", it's about what you do with the (possibility of) ambiguity: do you assume you can't be wrong and therefore others must be, or do you try to figure out whether you were solving different problems?
- kristjansson 3y ago
  Humans don't backtrack well :)
- Verdex 3y ago
  It's kind of like the xkcd airplane on a treadmill blogpost: https://blog.xkcd.com/2008/09/09/the-goddamn-airplane-on-the-goddamn-treadmill/ https://blog.xkcd.com/2008/09/09/the-goddamn-airplane-on-the... There appears to be problems that make it easy for people to assume the worst of others. And on a side note, I discovered about 6 months ago that mythbusters actually did this experiment. At the end of the day, few things work better than seeing something happen in real time.
juped 3y ago
No, there's not. There's the Monty Hall problem and then there's the "can I get away with falling back on claiming the question was somehow wrong after embarrassing myself by getting the Monty Hall problem incorrect" problem. No one who claimed it was unclear or ambiguous did so before getting it wrong. It's really tiresome to have to always pretend to assume good faith when it's really clear what's going on.
c2occnw 3y ago
That interpretation requires believing that the host, who knows what’s behind the doors, would sometimes open the door revealing the automobile and ask the contestant if they want to switch to that door. It doesn't really make sense, so that should tip off readers that they've misunderstood the show format. Marilyn even made this explicit in her answer, writing that the host "knows what's behind the doors and will always avoid the one with the prize" [0]. So readers were arguing her answer was wrong even given this understanding of how the show works. [0] https://www.newspapers.com/article/the-missoulian-the-monty-hall-problem/87196585/ https://www.newspapers.com/article/the-missoulian-the-monty-...
- mlyle 3y ago
  > That interpretation requires believing that the host, who knows what’s behind the doors The host has to always open the door, too. Else-- if your strategy is always to switch when shown a goat, the host can choose to only show you a goat if you have already picked the car (which would cause you to lose 100%). Or various mixed strategies. This is one of those problems that is hard to grasp at baseline, but a little harder to grasp because the situation isn't fully specified. As a result it tends to create a lot of controversy, like the one about an airplane on a treadmill.
bhk 3y ago
> the problem doesn't actually allow you to interpret otherwise How so? The problem is typically presented as a single trial, with nothing that clarifies the intent of the host.

by dekhn 3y ago

ghaff 3y ago

I think this is the original "Ask Marilyn": 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 #1, and the host, who knows what’s behind the doors, opens another door, say #3, which has a goat. He says to you, "Do you want to pick door #2?" Is it to your advantage to switch your choice of doors? Craig F. Whitaker Columbia, Maryland Yes; you should switch. The first door has a 1/3 chance of winning, but the second door has a 2/3 chance. Here’s a good way to visualize what happened. Suppose there are a million doors, and you pick door #1. Then the host, who knows what’s behind the doors and will always avoid the one with the prize, opens them all except door #777,777. You’d switch to that door pretty fast, wouldn’t you? ----- Yes, the question as posed by a reader was ambiguous but Marilyn made a very reasonable interpretation based on what someone at that time would probably tend to know about game shows. And she even gave an explanation (using more doors) that I find tends to work with folks combined with the explicit assumption that the host will always pick goats.

dekhn 3y ago
Yes, I'm literally talking about this sentence: "host, who knows what’s behind the doors, opens another door, say #3, which has a goat." I parsed that as "50% of the time, monty opens another door and it has a car and you win immediately, and 50% of the time, monty opens another door and it has a goat". In retrospect I think my brain just sort of pictured that and proceeded to assume there was no reason to switch, and it wasn't until I read some answers (not her explanation) that I understood I was wrong (and I went through a lot of anger, like many of the commenters, and thought she was very wrong before that). If there had been exactly one more sentence saying "monty always opens a door showing a goat", I'm pretty sure I would have recognized that. BTW I was familiar with the game (I liked Jeopardy better, as Let's Make a Deal was a fairly dull show) and I don't think they had a game that was identical to the problem as described.
- JackFr 3y ago
  > 50% of the time, monty opens another door and it has a car and you win immediately Why would you think you won? You didn't choose that one with the car and you aren't offered the opportunity to switch to the open one, only the closed one. The only interpretation which makes sense is that he always shows you a goat.
  - beepboopboop 3y ago
    I think they meant you lose immediately.
  - city41 3y ago
    Not to mention Monty always showing a goat is what adds the tension and interest a game show needs.
    - default-kramer 3y ago
      I disagree. If Monty always reveals a goat, there is no tension. You know exactly what is going to happen. Maybe it could be used a way to pad the running time of the show, but it would not add tension. "Coming up next: Monty reveals what's behind one of the other doors!" seems like something a game show would do. Whereas "Coming up next: Monty reveals a goat!" does not.

Scarblac 3y ago

I assume that at the time of writing, most readers would have been familiar with the actual game show and the fact that the host always opens a door with a goat?

dekhn 3y ago
The show uses a different game.
dboreham 3y ago
I had never heard of the show when I was introduced to the problem so all my knowledge comes from the problem description.
jcl 3y ago
As Monty Hall pointed out in interviews, he was not obligated to offer the opportunity to switch and sometimes did not. So in this case knowledge of the actual show may have contributed to confusion. https://en.wikipedia.org/wiki/Monty_Hall#Monty_Hall_Problem https://en.wikipedia.org/wiki/Monty_Hall#Monty_Hall_Problem (But vos Savant stated in later columns that most of the critical responses she received assumed that the host was obligated to offer the switch, so they were genuinely confused by the paradox.)

kristjansson 3y ago

As someone who misunderstood the problem as you describe, I'm much more invested in this version of the problem :) The problem statement is just a mess: > You pick a door, say #1, and the host, who knows what’s behind the doors, opens another door, say #3, which has a goat. even tiny edits like: > You pick a door, say #1, and the host, who knows what’s behind the doors, opens another door which has a goat, say #3. or even > You pick a door, say #1, and the host, who knows what’s behind the doors, opens another door, say #3, which he knows has a goat. make it clear that the last phrase is describing an invariant of the problem, not an artifact of the illustrative hypothetical.

svantana 3y ago
That's much better, but still not enough. We also need to add that the host will open a door regardless of which door I initially choose.
- kristjansson 3y ago
  Ah yeah, good point: > You pick a door and the host, who knows what’s behind the doors, opens exactly one of the two remaining doors according to the following algorithm: if both doors contain goats, he picks a door at random, else he picks the door he knows to contain the goat.

empyrrhicist 3y ago

Your re-framing is actually not quite right either (or at least, not complete). Key point: if the host picks the car, the game essentially re-sets. Imagine the extreme scenario with 999 goats and 1 car. You select the first door, the host opens 998 doors at random, leaving your selected door and one other. Two scenarios are now possible: 1. There was a car in the 998 doors that got opened. Tough luck, you lose the game automatically now (game resets). 2. There was not a car in the 998 doors. In this scenario, you are still advised to switch. Edit - I guess I misremembered a simulation I once did about it.

Swizec 3y ago
Also you're picking between Reward and Bigger Reward. Positive EV no matter your choice so eh don't worry about it too much :) A nice goat fetches up to $1000 to according to a quick google.
- klyrs 3y ago
  A nice goat, you say? How much for a mean goat, jaded from years of being the spoiler prize that nobody wants?
  - bemusedthrow75 3y ago
    Whenever someone talks about mean goats I think of only one thing: Buttermilk the baby goat, being a jerk. https://www.youtube.com/watch?v=AWvefaN8USk https://www.youtube.com/watch?v=AWvefaN8USk Enjoy.
    - deleted 3y ago
      [deleted]
- bemusedthrow75 3y ago
  I am suddenly reminded of the clever, morally educational joke about the dead horse raffle.
pxx 3y ago

TotempaaltJ 3y ago

Vos Savant mentions this, and she's kept track of who did and didn't understand the conditions: > And a very small percentage of readers feel convinced that the furor is resulting from people not realizing that the host is opening a losing door on purpose. (But they haven’t read my mail! The great majority of people understand the conditions perfectly.) https://web.archive.org/web/20181118225305/http://marilynvossavant.com/game-show-problem/ https://web.archive.org/web/20181118225305/http://marilynvos...

topaz0 3y ago
Nice find! I have wondered this. It's easy to say after you have come around that you had first misunderstood the scenario and done the math right in that context, even if in truth you (implicitly) understood the scenario but just neglected to account for the constraints.

topaz0 3y ago

What astonishes me is not that people interpret the scenario differently, but that they are so quick to conclude that others are doing the math wrong and feel superior or even angry, rather than thinking about whether you might be talking at cross purposes about different scenarios. So I see the second order problem a bit differently: it's not "was the statement unambiguous", it's about what you do with the (possibility of) ambiguity: do you assume you can't be wrong and therefore others must be, or do you try to figure out whether you were solving different problems?

kristjansson 3y ago
Humans don't backtrack well :)
Verdex 3y ago
It's kind of like the xkcd airplane on a treadmill blogpost: https://blog.xkcd.com/2008/09/09/the-goddamn-airplane-on-the-goddamn-treadmill/ https://blog.xkcd.com/2008/09/09/the-goddamn-airplane-on-the... There appears to be problems that make it easy for people to assume the worst of others. And on a side note, I discovered about 6 months ago that mythbusters actually did this experiment. At the end of the day, few things work better than seeing something happen in real time.

juped 3y ago

No, there's not. There's the Monty Hall problem and then there's the "can I get away with falling back on claiming the question was somehow wrong after embarrassing myself by getting the Monty Hall problem incorrect" problem. No one who claimed it was unclear or ambiguous did so before getting it wrong. It's really tiresome to have to always pretend to assume good faith when it's really clear what's going on.

c2occnw 3y ago

That interpretation requires believing that the host, who knows what’s behind the doors, would sometimes open the door revealing the automobile and ask the contestant if they want to switch to that door. It doesn't really make sense, so that should tip off readers that they've misunderstood the show format. Marilyn even made this explicit in her answer, writing that the host "knows what's behind the doors and will always avoid the one with the prize" [0]. So readers were arguing her answer was wrong even given this understanding of how the show works. [0] https://www.newspapers.com/article/the-missoulian-the-monty-hall-problem/87196585/ https://www.newspapers.com/article/the-missoulian-the-monty-...

mlyle 3y ago
> That interpretation requires believing that the host, who knows what’s behind the doors The host has to always open the door, too. Else-- if your strategy is always to switch when shown a goat, the host can choose to only show you a goat if you have already picked the car (which would cause you to lose 100%). Or various mixed strategies. This is one of those problems that is hard to grasp at baseline, but a little harder to grasp because the situation isn't fully specified. As a result it tends to create a lot of controversy, like the one about an airplane on a treadmill.

bhk 3y ago

> the problem doesn't actually allow you to interpret otherwise How so? The problem is typically presented as a single trial, with nothing that clarifies the intent of the host.