> I don't believe it's a coincidence that the ill-posed problem gets the high profile attention.

In this case even the well-posed problem(s) can be highly controversial as it can be seen from other comments - including someone who claims that probabilities are meaningless here because "it’s not a repeated game."

> something strange is going on.

The New York Times may be part of it then, I got the second quote from https://www.nytimes.com/1991/07/21/us/behind-monty-hall-s-do...

Sadly I can't read the NYT.

> In this case even the well-posed problem(s) can be highly controversial as it can be seen from other comments - including someone who claims that probabilities are meaningless here because "it’s not a repeated game."

I will take a look at such a comment if you link it, but I'm not going to actively search for such comments as I don't plan to play Monty Hall police.

The most intelligent probably just responded on the internet when they first got acquainted with it, and upon realizing the problem is ill-posed just ignore it. So the visible flamewar tends to be between people who don't realize the problem is ill-posed...

I was referring to https://news.ycombinator.com/item?id=36736947 but I didn’t expect you to read all the exchanges or comment on them.

“In Monty Hall, you’re just making a guess and no matter how clever you are about it, it’s still just a guess. There is no strategy because the game doesn’t last long enough for strategy to matter.”

The point is that the discussions are not always related to a misunderstanding about the protocol followed by the host to open a door.

Someone else doesn’t accept that (after the host’s protocol is fully specified) the problem is conditional on the stated run on events - insisting that it only makes sense to consider “the whole game”: https://news.ycombinator.com/item?id=36737326

> It's great we can agree the problem is ill-posed. But that is a binary thing, either it is ill-posed or it is not. It can't be a little bit ill-posed.

Would the problem be well-posed if the original problem statement was amended to fully specify the host’s behavioral protocol?

Yes if the host's behavioral protocol and the rules of the game were specified (deterministically or even probabilistically) the problem would be well-posed. I assume you mean the problem statement proper were to be amended, and not amending just the "solution", i.e. add and move the requisite givens from the solution to the actual problem statement.

Of course, it would then just reduce to a simple rote calculation, instead of bait and switching.

Another issue is that it is impossible to objectively agree on the "right" problem statement, given the ill-posed problem statement. An ill-posed problem statement is really a set of different problems each with their own answers.

If you decide to do this, I highly recommend providing multiple fixed problem statements, and explicitly indicating the distinction from "the" Monty Hall problem, yet the connection to it: make sure each fixed problem statement is compatible with Whitaker's original sequence of described events.

Make sure the reader understands that "the" Monty Hall problem does not in fact exist. Double check you don't use "the Monty Hall problem" anywhere, as even the name reinforces the idea that we are talking about a specific and thus well-posed problem and not a set of distinct and thus ill-posed problems. (I assume I probably did somewhere use the phrase in the thread above... sigh).

If your medium allows comments etc. make sure the reader is immediately aware of this, otherwise the comments will probably fill with readers unintentionally re-mansplaining the "proper" solution of "the" Monty Hall problem "under standard assumptions" set by the Ministry for Standardizing the Interpretation of Certainly Not Ill-posed But Just A Little Understandardized Problems.

The question is whether “Is it to your advantage to switch your choice of doors?” but the objective is not specified. Do I want to get the car? Do I want to get a goat? The latter objective is compatible with the problem statement.

Let’s say that it is amended again to fully specify the objective. Is it well posed now?

We are not told how the contestant picks a door and how the person responsible for putting the car and the goats behind the doors makes the choice.

There are multiple ways that the choices may have been done which are compatible with the stated run of events. It’s not a given than the location of the car and the door selection are independent.

The contestant may have seen the car, or may have heard the goats, and may have used this knowledge to pick a door. Or maybe the contestant doesn’t knew anything and it was the producer who knew that the contestant would pick door #1 because that’s his lucky number or whatever and intentionally put the car there or avoided doing so.

Game theory disregards the true ulterior objective of a participant, the game rules decide if there's a winner or a draw etc...

For example a king may force an unwilling philosopher to play chess together. The philosopher indeed intentionally loses the game. But according to the rules of chess the king still won.

A necessary condition to compute any probabilities is that the game rules are specified (and the ill-posed Monty Hall problem can be viewed as a single player game with ill-posed rules, or as a 2 player game of host vs contestant with ill-posed host protocol).

That doesn't mean it's a sufficient condition. Don't ask me to specify an explicit checklist to verify a natural language problem is well-posed: the burden of proof lies with the poser. That said, even though the burden of proof shouldn't lay with the challenged, those who are challenged are free to illustrate counterexamples exposing the ill-posed nature of the problem, such as example protocol 1 listed above.

If you truly like formally verifiable puzzles you could consider the formalization of Fermat's Last Theorem. Observe the huge contrast with the ill-posed Monty Hall problem:

X ^ N + Y ^ N = Z ^ N

has no solutions in the strictly positive integers for 3 <= N.

Every high school kid knows what exponentiation is: the power X ^ N is a repeated product of the same base X, where the base appears an exponent N number of times.

Its trivial to understand the question. Its trivially provable that the formula is well formed, and thus has a meaning. But yeah, with this type of puzzle it's less obvious how to rile up the less educated against those who practice careful rigorous precision. And then collectively repeat pretend those who protest "don't get it". Every time it passes I see the same things: a bunch of people pretending they nerdsniped whoever questions the ill-posed nature of the problem. Except they didn't even hit. We are expected to collectively pretend what a bunch of dummies the more rigorous and careful are, for not jumping to conclusions or for not playing along with pretending the hidden given was supposedly really given. It's just organised gaslighting intelligent people, who already know bayesian statistics, and lambasting them for not understanding it when rather often they do, but moreover they understand the problem is ill-posed, even if they don't use the mathematical jargon (they might request some more alternative runs of the game in order to make sure they understand the rules of the game).

To pull a Monty Hall is rather easy (please don't actually go out and do this):

1. just take any textbook problem with solution

2. make sure you understand both problem and solution.

3. eliminate a given from the problem statement.

4. fun and attention (in this world probably ad-profit)

Make sure to select an easy enough problem: you don't need to defend your practices if the fanboys will mansplain your solution over and over for you. The easier the problem the larger the group of fanboys in your defense.

Make sure to select a statistical one: whenever people object ambiguity of the problem statement and describe other possibilities they can be derailed as if they are not talking about the existence of other valid interpretations of the ill-posed problem statement, but as if they are talking and incorrectly analyzing probabilities and possibilities within the Holy interpretation of the ill-posed problem (pretending the hidden given was present in the original problem statement).

Make sure to separate authorship of the problem statement and your "solution", this gives plausible deniability when people point out the problem is ill-posed.