First let’s work out what joint strategy you should coordinate on beforehand. If everyone pledges to answer “yea” in case they end up as deciders, you get 0.5*1000 + 0.5*100 = 550 expected donation. Pledging to say “nay” gives 700 for sure, so it’s the better strategy.
This suggests to me that you actually can coordinate with everyone else, but the problem isn’t clear on whether or not that’s the case.
Thank you for that comment. I think I understand the question now. Let me restate it somewhat differently to make it, I think, clearer.
All 10 of us are sitting around trying to pre-decide a strategy for optimizing contributions.
The first situation we consider is labeled “First” and quoted in your comment. If deciders always say yay, we get $1000 for heads and $100 for tails which gives us an expected $550 payout. If we all say nay, we get $700 for either heads or tails. So we should predecide to say “nay.”
But then Charlie says “I think we should predecide to say “yay.” 90% of the time I am informed I am a decider, there are 8 other deciders besides me, and if we all say “yay” we get $1000. 10% of the time I am informed I am a decider, we will only get $100. But when I am informed I am a decider, the expected payout is $910 if we all say yay, and only $700 if we all say nay.”
Now Charlie is wrong, but its no good just asserting it. I have to explain why.
It is because Charlie has failed to consider the cases when he is NOT chosen as a decider. He is mistakenly thinking that since in those cases he is not chosen as a decider his decision doesn’t matter, but that is not true because we are talking about a predetermined group decision. So Charlie must recognize that in the 50% of cases where he is NOT chosen as a decider, 90% of THOSE cases are TAILS and his participation in directing the group to vote “yay” in those cases nets an expected $190 each time he is not chosen, while his participation in directing the group to vote “nay” nets $700 in each of those times when he is not chosen.
So the nub of it is that since this is a group decision, Charlie is actually a decider whether he is chosen as a decider or not, his decision, the group decision, affects the outcome of EVERY branch, not just the branches in which Charlie is an “official” decider. The nub of it is that we have equivocation on the word “decider,” when it is a group decision, when we count on every individual reaching the same decision, than every individual analyzing the situation must evaluate payout in ALL branches, and not just the ones in which the setter of the problem has chosen to label him a “decider.”
When Charlie is NOT chosen as a decider, he can conclude that tails was thrown with 90% probability and heads with only 10 probability.
This suggests to me that you actually can coordinate with everyone else, but the problem isn’t clear on whether or not that’s the case.
Thank you for that comment. I think I understand the question now. Let me restate it somewhat differently to make it, I think, clearer.
All 10 of us are sitting around trying to pre-decide a strategy for optimizing contributions.
The first situation we consider is labeled “First” and quoted in your comment. If deciders always say yay, we get $1000 for heads and $100 for tails which gives us an expected $550 payout. If we all say nay, we get $700 for either heads or tails. So we should predecide to say “nay.”
But then Charlie says “I think we should predecide to say “yay.” 90% of the time I am informed I am a decider, there are 8 other deciders besides me, and if we all say “yay” we get $1000. 10% of the time I am informed I am a decider, we will only get $100. But when I am informed I am a decider, the expected payout is $910 if we all say yay, and only $700 if we all say nay.”
Now Charlie is wrong, but its no good just asserting it. I have to explain why.
It is because Charlie has failed to consider the cases when he is NOT chosen as a decider. He is mistakenly thinking that since in those cases he is not chosen as a decider his decision doesn’t matter, but that is not true because we are talking about a predetermined group decision. So Charlie must recognize that in the 50% of cases where he is NOT chosen as a decider, 90% of THOSE cases are TAILS and his participation in directing the group to vote “yay” in those cases nets an expected $190 each time he is not chosen, while his participation in directing the group to vote “nay” nets $700 in each of those times when he is not chosen.
So the nub of it is that since this is a group decision, Charlie is actually a decider whether he is chosen as a decider or not, his decision, the group decision, affects the outcome of EVERY branch, not just the branches in which Charlie is an “official” decider. The nub of it is that we have equivocation on the word “decider,” when it is a group decision, when we count on every individual reaching the same decision, than every individual analyzing the situation must evaluate payout in ALL branches, and not just the ones in which the setter of the problem has chosen to label him a “decider.” When Charlie is NOT chosen as a decider, he can conclude that tails was thrown with 90% probability and heads with only 10 probability.
Very good explanation; voted up. I would even go so far as to call it a solution.