Oh right, I see where you’re coming from. When I said “you can’t control their vote” I was missing the point, because as far superrational agents are concerned, they do control each other’s votes. And in that case, it sure seems like they’ll go for the $2, earning less money overall.
It occurs to me that if team 4 didn’t exist, but teams 1-3 were still equally likely, then “heads” actually would be the better option. If everyone guesses “heads,” two teams are right, and they take home $4. If everyone guesses “tails,” team 3 takes home $3 and that’s it. On average, this maximizes winnings.
Except this isn’t the same situation at all. With group 4 eliminated from the get go, the remaining teams can do even better than $4 or $3. Teammates in room A2 knows for a fact that the coin landed heads, and they automatically earn $1. Teammates in room A1 are no longer responsible for their teammates’ decisions, so they go for the $3. Thus teams 1 and 2 both take home $1 while team 3 takes home $3, for a total of $5.
Maybe that’s the difference. Even if you know for a fact that you aren’t on team 4, you also aren’t in a world where team 4 was eliminated from the start. The team still needs to factor into your calculations… somehow. Maybe it means your teammate isn’t really making the same decision you are? But it’s perfectly symmetrical information. Maybe you don’t get to eliminate team 4 unless your teammate does? But the proof is right in front of you. Maybe the information isn’t symmetrical because your teammate could be in room B?
I don’t know. I feel like there’s an answer in here somewhere, but I’ve spent several hours on this post and I have other things to do today.
I do want to add—separately—that superrational agents (not sure about EDT) can solve this problem in a roundabout way.
Imagine if some prankster erased the “1” and “2″ from the signs in rooms A1 and A2, leaving just “A” in both cases. Now everyone has less information and makes better decisions. And in the real contest, (super)rational agents could achieve the same effect by keeping their eyes closed. Simply say “tails,” maximize expected value, and leave the room never knowing which one it was.
None of which should be necessary. (Super)rational agents should win even after looking at the sign. They should be able to eliminate a possibility and still guess “tails.” A flaw must exist somewhere in the argument for “heads,” and even if I haven’t found that flaw, a perfect logician would spot it no problem.
Oh right, I see where you’re coming from. When I said “you can’t control their vote” I was missing the point, because as far superrational agents are concerned, they do control each other’s votes. And in that case, it sure seems like they’ll go for the $2, earning less money overall.
It occurs to me that if team 4 didn’t exist, but teams 1-3 were still equally likely, then “heads” actually would be the better option. If everyone guesses “heads,” two teams are right, and they take home $4. If everyone guesses “tails,” team 3 takes home $3 and that’s it. On average, this maximizes winnings.
Except this isn’t the same situation at all. With group 4 eliminated from the get go, the remaining teams can do even better than $4 or $3. Teammates in room A2 knows for a fact that the coin landed heads, and they automatically earn $1. Teammates in room A1 are no longer responsible for their teammates’ decisions, so they go for the $3. Thus teams 1 and 2 both take home $1 while team 3 takes home $3, for a total of $5.
Maybe that’s the difference. Even if you know for a fact that you aren’t on team 4, you also aren’t in a world where team 4 was eliminated from the start. The team still needs to factor into your calculations… somehow. Maybe it means your teammate isn’t really making the same decision you are? But it’s perfectly symmetrical information. Maybe you don’t get to eliminate team 4 unless your teammate does? But the proof is right in front of you. Maybe the information isn’t symmetrical because your teammate could be in room B?
I don’t know. I feel like there’s an answer in here somewhere, but I’ve spent several hours on this post and I have other things to do today.
I do want to add—separately—that superrational agents (not sure about EDT) can solve this problem in a roundabout way.
Imagine if some prankster erased the “1” and “2″ from the signs in rooms A1 and A2, leaving just “A” in both cases. Now everyone has less information and makes better decisions. And in the real contest, (super)rational agents could achieve the same effect by keeping their eyes closed. Simply say “tails,” maximize expected value, and leave the room never knowing which one it was.
None of which should be necessary. (Super)rational agents should win even after looking at the sign. They should be able to eliminate a possibility and still guess “tails.” A flaw must exist somewhere in the argument for “heads,” and even if I haven’t found that flaw, a perfect logician would spot it no problem.