If you're playing against an opponent and trying to devise a winning strategy against him you can't just say "given no additional information or context, all we're left with is assuming his strategy is to always do X" and viola: present a strategy Y that beats X.
In this case X is "always opens a door with a goat behind it" and Y is "always switch doors". This is fascinating but simply incorrect from the math standpoint.
> Your "dice rolling" formulation of the puzzle is nonstandard. If you want to go with it, you must make it clear in the presentation of the puzzle. There are infinite such considerations; maybe Monty observes the phase of the Moon, maybe Monty likes the contestant, and so on... it wouldn't work as a puzzle!
The "dice rolling" it's not a problem formulation, it's one of the solutions to that problem i.e. specific values of X and Y that satisfy all the requirements. I present it to prove that more than one solution exist and furthermore not all solutions have Y="always switch", so you can't establish Y independent of X.
They key difference here is that I don't consider it as a "puzzle", whatever that means. I consider it to be a math problem. Problems of this kind are often encountered in both Game Theory and Probability Theory. It's perfectly fine to reason about your opponents strategies and either try to beat them all or find an equilibrium: this is still math and not psychology.
You can argue that it's a puzzle instead and I don't mind. What I do mind however is saying that Diaconis was wrong. He specifically said "the strict argument would be..." meaning that his conclusions hold when you consider it as a math problem, not as a "puzzle". My whole point is to demonstrate that.
