So let’s modify the problem somewhat. Instead of each person being given the “decider” or “non-decider” hat, we give the “deciders” rocks. You (an outside observer) make the decision.
Version 1: You get to open a door and see whether the person behind the door has a rock or not.
Winning strategy: After you open a door (say, door A) make a decision. If A has a rock then say “yes”. Expected payoff 0.9 1000 + 0.1 100 = 910 > 700. If A has no rock, say “no”. Expected payoff: 700 > 0.9 100 + 0.1 1000 = 190.
Version 2: The host (we’ll call him Monty) randomly picks a door with a rock behind it.
Winning strategy: Monty has provided no additional information by picking a door: We knew that there was a door with a rock behind it. Even if we predicted door A in advance and Monty verified that A had a rock behind it, it is no more likely that heads was chosen: The probability of Monty picking door A given heads is 0.9 1⁄9 = 0.1 whereas the probability given tails is 0.1 1.
Hence, say “no”. Expected Payoff: 700 > 0.5 100 + 0.5 1000 = 550.
Now let us modify version 2: During your sleep, you are wheeled into a room containing a rock. That room has a label inside, identifying which door it is behind. Clearly, this is no different than version 2 and the original strategy still stands.
From there it’s a small (logical) jump to your consciousness being put into one of the rock-holding bodies behind the door, which is equivalent to our original case. (Modulo the bit about multiple people making decisions, if we want we can clone you consciousness if necessary and put it into all rock possessing bodies. In either case, the fact that you wind up next to a rock provides no additional information.)
This question is actually unnecessarily complex. To make this easier, we could introduce the following game:
We flip a coin where the probability of heads is one in a million. If heads, we give everyone on Earth a rock, if tails we give one person a rock. If the rock holder(s) guesses how the coin landed, Earth wins, otherwise Earth loses. A priori, we very much want everyone to guess tails. A person holding the rock would be very much inclined to say heads, but he’d be wrong. He fails to realize that he is in an equivalence class with everyone else on the planet, and the fact that the person holding the rock is himself carries no information content for the game. (Now, if we could break the equivalence class before the game was played by giving full authority to a specific individual A, and having him say “heads” iff he gets a rock, then we would decrease our chance of losing from 10^-6 to (1-10^-6) * 10^-9.)
So let’s modify the problem somewhat. Instead of each person being given the “decider” or “non-decider” hat, we give the “deciders” rocks. You (an outside observer) make the decision.
Version 1: You get to open a door and see whether the person behind the door has a rock or not. Winning strategy: After you open a door (say, door A) make a decision. If A has a rock then say “yes”. Expected payoff 0.9 1000 + 0.1 100 = 910 > 700. If A has no rock, say “no”. Expected payoff: 700 > 0.9 100 + 0.1 1000 = 190.
Version 2: The host (we’ll call him Monty) randomly picks a door with a rock behind it. Winning strategy: Monty has provided no additional information by picking a door: We knew that there was a door with a rock behind it. Even if we predicted door A in advance and Monty verified that A had a rock behind it, it is no more likely that heads was chosen: The probability of Monty picking door A given heads is 0.9 1⁄9 = 0.1 whereas the probability given tails is 0.1 1. Hence, say “no”. Expected Payoff: 700 > 0.5 100 + 0.5 1000 = 550.
Now let us modify version 2: During your sleep, you are wheeled into a room containing a rock. That room has a label inside, identifying which door it is behind. Clearly, this is no different than version 2 and the original strategy still stands.
From there it’s a small (logical) jump to your consciousness being put into one of the rock-holding bodies behind the door, which is equivalent to our original case. (Modulo the bit about multiple people making decisions, if we want we can clone you consciousness if necessary and put it into all rock possessing bodies. In either case, the fact that you wind up next to a rock provides no additional information.)
This question is actually unnecessarily complex. To make this easier, we could introduce the following game: We flip a coin where the probability of heads is one in a million. If heads, we give everyone on Earth a rock, if tails we give one person a rock. If the rock holder(s) guesses how the coin landed, Earth wins, otherwise Earth loses. A priori, we very much want everyone to guess tails. A person holding the rock would be very much inclined to say heads, but he’d be wrong. He fails to realize that he is in an equivalence class with everyone else on the planet, and the fact that the person holding the rock is himself carries no information content for the game. (Now, if we could break the equivalence class before the game was played by giving full authority to a specific individual A, and having him say “heads” iff he gets a rock, then we would decrease our chance of losing from 10^-6 to (1-10^-6) * 10^-9.)