Stream of conciousness style answer. Not looking at other comments so I can see afterwards if my thinking is the same as anyone else's.
The argument for saying yea once one is in the room seems to assume that everyone else will make the same decision as me, whatever my decision is. I'm still unsure whether this kind of thinking is allowed in general, but in this case it seems to be the source of the problem.
If we take the opposite assumption, that the other decisions are fixed, then the problem depends on those decisions. If we assume that all the others (if there are any) will say yea then U(yea) = 0.9$1000 + 0.1$100 = $910 = 0.91 lives while U(nay) = 0.90 + 0.1$700 = $70 = 0.07 lives, so clearly I say yea. If we assume that all the others (if there are any) will say nay then U(yea) = 0.90 + 0.1$100 = 0.01 lives while U(nay) = 1*$700 = 0.7 lives.
In other words, if we expect others to say yea, then we say yea, and if we expect others t say nay, then we say nay (if people say yea with some probability p then our response is either yea or nay depending on whether p is above some threshold). It appears we have a simple game theory problem with two nash equilibria. I'm not sure of a rule for deciding which equilibrium to pick, so I'll try some toy problems.
If we have 10 people, each in a room with two buttons, one red and one blue, and they are told that if they all press the same colour then $1000 will be donated to Village Reach, but if there is any disagreement no money will be donated, then they have quite a difficult dilemma, two nash equilibria, but no way to single out one of them, which makes them unlikely to end up in either.
If we change the problem, so that now only $500 is donated when they all press blue, but $1000 is still given on red, then the decision becomes easy. I'm sure everyone will agree with me that you should definitely press red in this dilemma. It seems like in general a good rule for game-theory problems is "if faced with multiple nash equilibria, pick the one you would have agreed to in advance". This might not give an answer in games where the various utility functions are opposed to each other, but it works fine in this problem.
So I say nay. Both before and after hearing that I am a decider. Problem solved.
Thinking about this a bit more, it seems like the problem came from Timeless Decision Theory, and was solved by Causal Decision Theory. A rather disturbing state of events.
I like how you apply game theory to the problem, but I don't understand why it supports the answer "nay". The calculations at the beginning of your comment seem to indicate that the "yea" equilibrium gives a higher expected payoff than the "nay" equilibrium, no?
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.)