PARADE!

flabdablet

Condition 3 should be: "Given a choice of goat doors, it doesn't matter what method is used to choose between them, so long as the player does not know that method".

This is to ensure that if, for example,

Monty always chooses the lowest numbered goat

Player selects door #3

Monty opens door #2

If the player knows Monty's method he can deduce that the car MUST be behind door number 1.
If he doesn't know the method, he cannot make that deduction.

The question to be answered is not "does switching your choice exactly double your chance of winning?" but "Is it to your advantage to switch your choice?"

In order for it not to be to the player's advantage to switch doors, there would need to be a scenario in which the player can use information about Monty's choice methods to raise the probability of the initial pick being correct to at least 1/2. PalmerEldritch appears to believe that the scenario where Monty always picks Door 3 when available is one such. So let's assume both that this is what Monty will do and that the player knows it and take it case by equally likely case.

Player picks   Car is behind   Monty opens   Car deducible   Winning move
------------   -------------   -----------   -------------   ------------
Door 1         Door 1          Door 3        No              Keep
Door 1         Door 2          Door 3        No              Switch
Door 1         Door 3          Door 2        Yes             Switch
Door 2         Door 1          Door 3        No              Switch
Door 2         Door 2          Door 3        No              Keep
Door 2         Door 3          Door 1        Yes             Switch
Door 3         Door 1          Door 2        No              Switch
Door 3         Door 2          Door 1        No              Switch
Door 3         Door 3          Door 1 or 2   No              Keep

A few things become clear here: firstly, it's to the player's advantage not to pick Door 3, since deliberately depriving Monty of his preferred choice eliminates any chance of deducing the car's position from his failure to exercise it. Secondly, the only cases in which the car's position can be deduced are the switch-wins cases. Thirdly, there are still twice as many switch-wins cases as keep-wins cases, meaning that it is to the player's advantage to switch even if unable to deduce the position of the car.

Unless I'm mistaken, the only fact that the player will ever be able to glean using knowledge of Monty's choice procedure is that the car is behind the door that the player didn't pick and Monty didn't open, leading to a win by switching. This follows from the fact that when Monty does open the door that the choice procedure says he should, there is no way for the player to find out whether or not that choice was forced. It's only the fact of Monty's having made a non-preferred choice that can reveal the car's location.

Therefore, it's to the player's advantage to switch when given the opportunity even if Monty's choice procedure is known. The eViLegion version of condition 3 is superfluous as well.

eViLegion

@flabdablet said:

stuff

Yes, actually you're quite right... you can never be reduced below 1/2 chance by "always switching".

With additional information you MAY get an opportunity to deduce that you mustn't switch, but it cannot reduce the expected win from "ALWAYS switching" below 50%.

flabdablet

@eViLegion said:

With additional information you MAY get an opportunity to deduce that you mustn't switch

Short of actual prior knowledge of the car's location, or being under strict instructions from your beloved to come home with a goat, can you give an example? Because it seems to me that all you could ever conceivably find out is that you really, really ought to switch, not that you mustn't.

PalmerEldritch

Well you're changing your argument now. In response to an earlier post I made you replied as follows:

@flabdablet said:

@PalmerEldritch said:
I'm just saying the problem as posed is ambiguous, and there are 3 conditions that need to be assumed (if they're not stated) to make the probability of winning by switching doors = 2/3.
And I'm saying you're wrong, because as soon as you assume your first two conditions, which between them specify unambiguously that Monty must open a door to reveal a goat before giving the contestant the option of switching, then the probability of winning by switching becomes 2/3 without needing your third condition; that is, your third condition is superfluous.

The highlighted part of your response is incorrect, the probability of winning by switching in the problem as stated only becomes 2/3 if Monty randomly picks a goat door when he has a choice of 2 goat doors

flabdablet

@eViLegion said:

you can never be reduced below 1/2 chance by "always switching".

Can you think of a scenario that includes PalmerEldritch's conditions 1 and 2 (mandatory reveal of a non-winning door, followed by an opportunity to switch) that can reduce your chance of winning by switching below 2/3? I haven't yet managed to, and although PalmerEldritch seems to be claiming that Bayesian Scripture says such a scenario must exist, he hasn't yet given a correct example.

eViLegion

@flabdablet said:

@eViLegion said:
With additional information you MAY get an opportunity to deduce that you mustn't switch
Short of actual prior knowledge of the car's location, or being under strict instructions from your beloved to come home with a goat, can you give an example? Because it seems to me that all you could ever conceivably find out is that you really, really ought to switch, not that you mustn't.

Actually, I have no example...

I was imagining some unspecified hypothetical non-random process, which gives some you additional information.
Regardless of that process, that additional information could only ever be usable in 1/3 of cases, and will not affect the 2/3 of cases which provide the main grounds for always switching.

Further thought suggests that there is no non-random process that doesn't increase the players chances of deducing he should switch, nor is their any way for Monty to reliably communicate (via his choice of reveal, given their choice) that the player should stick (assuming he's trying to help the player). I certainly cannot contrive one.