>>13162887Let's consider that the scenario unfolds in this manner: both contestants silently pick a door and inform nobody of their choice. You have 33% chance of winning and the other contestant also has 33% chance of winning, that much is clear so far. You might have picked the same door, nobody knows. Since the other contestant goes first, you only have 66% chance to play the game which represents the 66% chance that the other contestant fails to win. If this hurdle is cleared you now have 50% chance to win since two doors remain and one of them contains the prize. It doesn't matter if one of the doors that remains is the door you originally picked. It's misleading to think in terms of whether it's an effective strategy to always switch door since it is not guaranteed that you have the option to switch. All that matters is that you had 66% chance to play followed by 50% to win. Your final odds are 33%. These are the same odds you had at the start which is no mistake: the other contestant improved your odds by opening a losing door but this information had an equal cost in giving you a chance to lose.
You could imagine a similar scenario with 3 contestants: the first has 33% chance to win, the second does as well, and if they both lose then you win which still comes down to a 33% chance, shared equally by all participants regardless of order of play. Next, you can imagine that all 3 doors are picked and opened simultaneously for 3 participants and again the final 33% odds remain. The order of opening and the ability to switch does not seem to come into play.
The original problem only works because a gameshow host is supposed to be all-knowing and neutral. They can open a door, which gives valuable information, without giving you a chance to lose. Other contestants cannot do that. Even if you paid a contestant to help you win, without any information they can only act randomly. They couldn't help you even if they wanted.
