I think this is clearer in the non-anthropic version of the game.
20 players, and we’re all playing for a charity.
A coin is flipped. Heads, 18 green balls and 2 red balls go in the urn, Tails, 2 green balls and 18 red balls go in the urn.
Everyone secretly draws a ball and looks at it. Then everyone casts a secret ballot.
If you have a red ball, your vote doesn’t matter, you just have to cast the ballot so that nobody knows how many green balls there are.
If everyone with a green ball votes Yes on the ballot, then the charity gains $3 for each green ball and loses $1 for each red ball.
In this game, if you draw a green ball, don’t vote Yes! Even if you think everyone else will.
This might be surprising—if I draw a green ball, that’s good evidence that there are more green balls. Since everyone voting Yes is good in the branch of the game where there are 18 green balls, it seems like we should try to coordinate on voting Yes.
But think about it from the perspective of the charity. The charity just sees people play this game and money comes in or goes out. If the players coordinate to vote Yes, the charity will find that it’s actually losing money every time the game gets played.
I think you’re just making a mistake here. Consider the situation where you are virtually certain that everyone else will vote “yes” (for some reason we needn’t consider—it’s a possible situation). What should you do?
You can distinguish four possibilities for what happens:
Heads, and you are one of the 18 people who draw a green ball. Probability is(1/2)(18/20)=9/20.
Heads, and you are one of the 2 people who draw a red ball. Probability is (1/2)(2/20)=1/20.
Tails, and you are one of the 2 people who draw a green ball. Probability is (1/2)(2/20)=1/20.
Tails, and you are one of the 18 people who draw a red ball. Probability is (1/2)(18/20)=9/20.
If you always vote “no”, the expected reward will be 0(9/20)+12(1/20)+0(1/20)+(-52)(9/20)=-22.8. If you always vote “yes”, the expected reward will be 12(9/20)+12(1/20)+(-52)(1/20)+(-52)(9/20)=-20, which isn’t as bad as the −22.8 when you vote “no”. So you should vote “yes”.
Sorry, right after posting I edited my reply to explain the reasoning a bit more. If you’re purely selfish, then I agree with your reasoning, but suppose you’re “playing for charity”—then what does the charity see as people play the game?
The charity sees just what I say above—an average loss of 22.8 if you vote “no”, and an average loss of only 20 if you vote “yes”.
An intuition for this is to recognize that you don’t always get to vote “no” and have it count—sometimes, you get a red ball. So never taking the bet, for zero loss, is not an option available to you. When you vote “no”, your action counts only when you have a green ball, which is nine times more likely when your “no” vote does damage than when it does good.
Ah, thanks for being patient. Yes, you’re right. Once you’ve fixed beliefs about what everyone will do, then if everyone votes yes you should too. Any supposed non zero sum game theory about what to do can only matter when the beliefs about what everyone will do aren’t fixed.
I think this is clearer in the non-anthropic version of the game.
20 players, and we’re all playing for a charity.
A coin is flipped. Heads, 18 green balls and 2 red balls go in the urn, Tails, 2 green balls and 18 red balls go in the urn.
Everyone secretly draws a ball and looks at it. Then everyone casts a secret ballot.
If you have a red ball, your vote doesn’t matter, you just have to cast the ballot so that nobody knows how many green balls there are.
If everyone with a green ball votes Yes on the ballot, then the charity gains $3 for each green ball and loses $1 for each red ball.
In this game, if you draw a green ball, don’t vote Yes! Even if you think everyone else will.
This might be surprising—if I draw a green ball, that’s good evidence that there are more green balls. Since everyone voting Yes is good in the branch of the game where there are 18 green balls, it seems like we should try to coordinate on voting Yes.
But think about it from the perspective of the charity. The charity just sees people play this game and money comes in or goes out. If the players coordinate to vote Yes, the charity will find that it’s actually losing money every time the game gets played.
I think you’re just making a mistake here. Consider the situation where you are virtually certain that everyone else will vote “yes” (for some reason we needn’t consider—it’s a possible situation). What should you do?
You can distinguish four possibilities for what happens:
Heads, and you are one of the 18 people who draw a green ball. Probability is(1/2)(18/20)=9/20.
Heads, and you are one of the 2 people who draw a red ball. Probability is (1/2)(2/20)=1/20.
Tails, and you are one of the 2 people who draw a green ball. Probability is (1/2)(2/20)=1/20.
Tails, and you are one of the 18 people who draw a red ball. Probability is (1/2)(18/20)=9/20.
If you always vote “no”, the expected reward will be 0(9/20)+12(1/20)+0(1/20)+(-52)(9/20)=-22.8. If you always vote “yes”, the expected reward will be 12(9/20)+12(1/20)+(-52)(1/20)+(-52)(9/20)=-20, which isn’t as bad as the −22.8 when you vote “no”. So you should vote “yes”.
Sorry, right after posting I edited my reply to explain the reasoning a bit more. If you’re purely selfish, then I agree with your reasoning, but suppose you’re “playing for charity”—then what does the charity see as people play the game?
The charity sees just what I say above—an average loss of 22.8 if you vote “no”, and an average loss of only 20 if you vote “yes”.
An intuition for this is to recognize that you don’t always get to vote “no” and have it count—sometimes, you get a red ball. So never taking the bet, for zero loss, is not an option available to you. When you vote “no”, your action counts only when you have a green ball, which is nine times more likely when your “no” vote does damage than when it does good.
Ah, thanks for being patient. Yes, you’re right. Once you’ve fixed beliefs about what everyone will do, then if everyone votes yes you should too. Any supposed non zero sum game theory about what to do can only matter when the beliefs about what everyone will do aren’t fixed.