I don't think "motivation of the host" is a great way to accurately describe the issue that Diaconis is calling out, it is rather intended to be more intuitive.
In a precise way, the reason the question is underspecified is because it doesn't say if the probability of the host offering you a chance to choose again is dependent on which choice you make. If the host offers the choice more twice as often when your pick right and when you pick wrong, then changing you pick is the incorrect choice.
Now, colloquially, it can makes sense to assume the host always offers the choice, but practically, if we're looking at how to use statistics in a real world situation, that isn't a safe to always assume that probabilities are independent.
The question as stated does not permit such a choice by the host since if it were a choice, it had already been made.
This is like being presented with a nearly completed game of chess, asked if the loser can lose in 1 move and then arguing that the answer is more nuanced because there might have been other moves taken that produced a different end games rather than the ones that produced this particular end game. We do not care about those other end games, since we are only considering this particular one.
> The question as stated does not permit such a choice by the host since if it were a choice, it had already been made.
Whether the choice was already made by the host makes no difference, what matters is what information about the hidden state can be derived from that choice.
Let's say the rules of the game are modified to sat that the host never offers a re-selection when you already have selected a door with a goat. Then if the host has offered you a re-selection you should definitely not take it because you already have the good prize. You know this because the re-selection offer provides information about what is behind the door you selected.
In fact, any time your choice of door has amyy statistical effect on whether a re-selection is offered, then a re-selection offer (or lack) provides a small amount of information that modifies the expected value of choosing a new door.
> This is like being presented with a nearly completed game of chess, asked if the loser can lose in 1 move and then arguing that the answer is more nuanced because there might have been other moves taken that produced a different end games
It is absolutely nothing like that. That is not a question about statistics or probability.
I think this explanation is just cope. Nothing about the problem should lead you to believe that the host is an evil genie purposefully trying to trick you. Attacking the framing device for the problem is the kind of post-hoc rationalization you make after failing at a probability test.
> Nothing about the problem should lead you to believe that the host is an evil genie purposefully trying to trick you.
Is it really unreasonable to assume that the host would like to keep the car? As I see it, that's the economic intuition behind why most people don't switch.
I think we're disagreeing about how much it's reasonable to assume. I'm happier treating it as a self contained problem (in which case I'd say that the form quoted by CrazyStat is underspecified); but if you're familiar with the TV show it's based on, you can reasonably assume that he always opens a door with a goat and gives you a chance to switch.
My objection is to the claim that "most people get it wrong", if most people are being fed the underspecified problem. I think the gut reaction is not to switch, because in most comparable situations across human experience it would be a mistake (imagine a similar situation at a sketchy-looking carnival game rather than a TV show). They then try to justify that formally and make mistakes in their justification, but the initial reaction not to swap is reasonable unless they've been convinced that Monty Hall always opens a door with a goat and gives a chance to switch.
> My objection is to the claim that "most people get it wrong", if most people are being fed the underspecified problem. I think the gut reaction is not to switch, because in most comparable situations across human experience it would be a mistake (imagine a similar situation at a sketchy-looking carnival game rather than a TV show
This may have a role to play. However there is a long history of people who aren't "going off their gut", including statisticians, getting this wrong with a very high level of confidence. It seems pretty clear that there is more than just an "underspecificity" problem. If you properly specify the problem, you will get similar error levels.
I agree, but I believe the reason for the errors is because people intuitively have a pretty good grasp of the game theory for the situation where someone is trying not to give you something they promised (and it's the sort of thing where IRL you shouldn't believe somebody trying to convince you to change your mind, so it's a useful bias to ignore parts of the problem even when it is fully specified). I believe that the statisticians then try to justify that, and end up making incorrect arguments.
> I agree, but I believe the reason for the errors is because people intuitively have a pretty good grasp of the game theory for the situation where someone is trying not to give you something they promised
Unfortunately this doesn't match reality. The vast majority of people who got the problem wrong when it was first published are not confused about the rules and insist that the chances are even (same chance to get a car switching as not switching). This doesn't match a theory that these people think the host is trying to trick the player in some way.
Additionally, You can reframe the problem and will still see significant error rates.
It contains a clarification that the article omitted from the description:
> So let’s look at it again, remembering that the original answer defines certain conditions, the most significant of which is that the host always opens a losing door on purpose. (There’s no way he can always open a losing door by chance!) Anything else is a different question.
Yes, of the host opens a door, it will always be a losing door. Nobody is disputing that.
The part that is underspeified is: does the host always open a door and if not, does the player's choice of a door impact whether the host opens a door?
I think you should take the time to understand why this explanation matters. It reveals some important things about how people can make mistakes with statistics. Not understanding something doesn't make it "cope".
In a precise way, the reason the question is underspecified is because it doesn't say if the probability of the host offering you a chance to choose again is dependent on which choice you make. If the host offers the choice more twice as often when your pick right and when you pick wrong, then changing you pick is the incorrect choice.
Now, colloquially, it can makes sense to assume the host always offers the choice, but practically, if we're looking at how to use statistics in a real world situation, that isn't a safe to always assume that probabilities are independent.