Precommitting to "Yea" is the correct decision.
The error: the expected donation for an individual agent deciding to precommit to "nay" is not 700 dollars. It's pr(selected as decider) * 700 dollars. Which is 350 dollars.
Why is this the case? Right here:
Next, I will tell everyone their status, without telling the status of others ... Each decider will be asked to say "yea" or "nay".
In all the worlds where you get told you are not a decider (50% of them - equal probability of 9:1 chance or a 1:9 chance) your precommitment is irrelevant. Therefore, the case where everyone precommits to yea is logically equivalent to the case where everyone precommits to nay and then change to yea upon being told they are a decider.
So we can talk about two kinds of precommitments: "I precommit to answering yea/nay" and "I precommit to answer yea/nay given that I am informed I am a decider". The expected donation in the first precomittment is yea: 455 dollars and nay: 350 dollars; the expected donation in the second precomittment is yea: 910 dollars and nay: 700 dollars.¹
Yes, the first precommittment is half that of the second: because the first goes through a 50% filter, the second starts on the other side of that filter. Of course if you start yea before the filter and nay afterwards, you're going to get the wrong result.
¹: Summing the expected values of yea. pr(heads) pr(selected as decider) value of 'yea' for heads = 0.5 0.1 100 = 5 dollars, pr(tails) pr(selected as decider) value of 'yea' for tails = 0.5 0.9 1000 = 450 dollars. Sum = 455 dollars.
In all the worlds where you get told you are not a decider (50% of them - equal probability of 9:1 chance or a 1:9 chance) your precommitment is irrelevant.
How can that be, when other people don't know whether or not you're a decider?
Imagine the ten sitting in a room, and two people stand up and say "If I am selected as a decider, I will respond with 'yea'." This now forces everyone else to vote 'yea' always, since in only 5% of all outcomes (and thus 10% of the outcomes they directly control) does voting 'nay' increase the total donation (by ...
The source is here. I'll restate the problem in simpler terms:
You are one of a group of 10 people who care about saving African kids. You will all be put in separate rooms, then I will flip a coin. If the coin comes up heads, a random one of you will be designated as the "decider". If it comes up tails, nine of you will be designated as "deciders". Next, I will tell everyone their status, without telling the status of others. Each decider will be asked to say "yea" or "nay". If the coin came up tails and all nine deciders say "yea", I donate $1000 to VillageReach. If the coin came up heads and the sole decider says "yea", I donate only $100. If all deciders say "nay", I donate $700 regardless of the result of the coin toss. If the deciders disagree, I don't donate anything.
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.
But consider what happens when you're already in your room, and I tell you that you're a decider, and you don't know how many other deciders there are. This gives you new information you didn't know before - no anthropic funny business, just your regular kind of information - so you should do a Bayesian update: the coin is 90% likely to have come up tails. So saying "yea" gives 0.9*1000 + 0.1*100 = 910 expected donation. This looks more attractive than the 700 for "nay", so you decide to go with "yea" after all.
Only one answer can be correct. Which is it and why?
(No points for saying that UDT or reflective consistency forces the first solution. If that's your answer, you must also find the error in the second one.)