Ah. I didn't understand what you're getting at because I'm dense. Sorry.

So what you're really saying is:

1. You've invented a new game, the Winterclaw game. It has some similary to the Monty Hall game, but the rules are different. I'm OK with that.

2. Playing the best strategy in the Winterclaw game results in a different average probability of success than in the Monty Hall game. That's true, too. In the Winterclaw game, the best average probablity of success is indeed 1/2. I'll explore that a bit more in a second.

3. The Monty Hall game is broken. That I disagree with. It's not at all broken.

What is unclear is what you mean by "broken". First of all, the Monty Hall problem is not the same as the Monty Hall game. You said the problem is broken. But the game is not the problem. The game is the premise of the problem, and the problem is to answer what is the best strategy and why.

We're not talking about whether the game is good or bad, apparently, but rather whether the problem is interesting or not. (Side note - a broken game is one in which you always win or you always lose - not very interesting. Broken games automatically produce no interesting problem of choosing a strategy.)

Now we all agree that in the Monty Hall game as well as the Winterclaw game, you win or lose are both possible, so the game is interesting. The question then: is the problem of choosing a strategy interesting?

In order for a problem of this type to be interesting, there has to be at least two strategies, and one has to be better than the other. If they have the same outcome, then it isn't an interesting problem even if the game is interesting, since your decision to follow one or another would not affect the game outcome.

For example, consider the coin toss game within the game of football. You can choose head or tail. The coin toss game is interesting. I care whether I win or not, and I can lose. But is the problem interesing? is there a best strategy, and if so, which is it. There are two strategies - choose heads or choose tails. The value of both strategies is 1/2. Therefore, while the game is interesting, the coin toss problem is dull and not worth discussing. Do whatever, it won't matter.

Now in that sense, then the Monty Hall problem, which is a question about the Monty Hall game, is interesting. Many smart and/or educated people give the wrong answer as to the best strategy. Either they choose "stay", which is not the best strategy because it has a value of 1/3, or they deny that the "switch" strategy has a value of 2/3. This makes it interesting - the choice of strategy matters, and many people perversely get it wrong.

So, I do not think the Monty Hall problem is broken at all. Quite the opposite. It provokes a lot of thought and discussion, and is surprising and interesting to a lot of people.

Now let's consider the Winterclaw game and the Winterclaw problem.

In the Winterclaw game, there are two strategies as well, stay or switch. And as you've discovered, in the Winterclaw game, the value of both strategies is 1/2.

Which means that the Winterclaw problem is broken. It doesn't matter if you analyze it incorrectly. There is no advantage to you knowing the strategies and the probabilities. If two people compete at the Winterclaw game, one playing with an understanding of the strategies, and the other totally clueless and choosing at random, and we measure the outcomes of these two people over a large number of trials, we wil not be able to tell which one is clueless and which one is wise to the game. This meets my definition of an uninteresting problem.

[Edit]

On the other hand after further study, there is a slight wrinkle on the Winterclaw game and problem.

I suggested the Winterclaw problem is dull because a clueless random player will not do worse or better than a wise player, and that watching these guys, I can't tell which is which.

But there is another kind of player. A player who chooses to always stay, even after being shown the third case, where he knows he has the wrong door.

In that case, the probability of winning is only 1/3.  So the stay strategy has value 1/3, and the switch strategy has value 1/2. Strange isn't it?

Given three players:

DumbGuy - always stays
SmartGuy - totally understand the game and always switches
CoinTosser - if his door remains closed, he tosses a coin to decide to stay or swtich. If his door is opened and reveals that staying is stupid, he switches, but tosses a coin to pick one of the other two doors.

DumbGuy wins 1/3 of the time.
SmartGuy wins 1/2 of the time.
CoinTosser wins 1/2 of the time.

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Renderosity forum reply notifications are wonky. If I read a follow-up in a thread, but I don't myself reply, then notifications no longer happen AT ALL on that thread. So if I seem to be ignoring a question, that's why. (Updated September 23, 2019)