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.
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.