I got some results I didn't quite expect while thinking about this.
I assumed that if it was good to press the button once, it would be good to consider the effects of pressing the button a lot, and so I tried considering what came up with a scenario that was equivalent on a larger scale:
You're being given an opportunity to break a regulatory tie to approve a massive new mosquito net factory in China.
The mosquito net factory is equal to pressing the button 100,000 times a year. Assume you personally couldn't do that, without getting carpal tunnel.
So it will stop 500,000 Malaria deaths a year, (There are actually more than that many: http://www.who.int/mediacentre/factsheets/fs094/en/) and not only that, but if you approve it, You'll receive a lucrative stock gift, which will pay you 600 MILLION dollars a year. You'll have a golden parachute for the rest of your life! For the purposes of this scenario, that's legal, according to new stock regulations.
Okay, yes, it is VERY polluting. But China doesn't have any air quality standards for these pollutants, so again, it's all legal, and the projected deaths are only 100,000 people a year, mostly in China.
But still, we're saving 400,000 people's lives every year! And we know it's going to be at least one year for the circumstances to change.
And hey, if you want to use your money to invest in better air pollution scrubbers for the plant, go right ahead! I don't have the cost benefit ratios on that right now, since I just had all of my analytics people generate the numbers for the factory.
It seems like the right answer to the scaled up problem is "Shouldn't I either run or have run for me the cost benefits ratio on the air pollution scrubbers BEFORE making a decision which costs at least 100,000 deaths?" It would also seem likely that the result would be that I could install scrubbers at some price below 600,000,000 dollars a year, and then I go home happy with no deaths and the remaining money.
But if I attempt to run the cost benefits on the magic box, it occurs to me that the default assumption is that there IS no cost benefit ratio which applies to saving those people. It feels like the implied result is that they're just magically executed with no protection. Even if they're a billionaire who has invested in cryonics, is in perfect health, and is waiting in a hospital with doctors and cryonicists, they're just permanently dead and can't be saved.
However, that assumption isn't necessarily TRUE. There's no evidence that those people are unsavable in the small scale, just that they likely are in an isomorphic large scale where the deaths actually have a non magical cause.
So if the first thing I would probably have to do is attempt to figure out "What is the cause of the mystery deaths from the box, is it mitigatable, and at what price other than not pressing the button?
Assuming the likely answer "The world is inconvenient, Omega will execute those people and Omega can't be stopped." Then I have a feeling I would end up paralyzed by "But isn't there a way to save everyone?" Except, I'm not paralyzed by that, because if I was, then I would already be paralyzed by that, because I'm faced with those decisions, and I don't feel paralyzed by that. But then again, I don't usually face Omega either.
So in the large scale, build the mosquito net factory, build the air scrubbers with my personal money, save everyone, make money. In the small scale, "Building the scrubbers" is essentially "Kill Omega, and use his box without him going around executing people." But killing Omega is assumed impossible (A potential further complication, Omega may need to be alive for the box to work.)
This makes it seem like my actual answer is "Shove your box Omega, I'm going to make money off of an environmentally safe for profit mosquito net factory using your technology and save everyone." I'm not sure if I should change that answer or not, or if it even makes sense. But it appears to be my current answer.
I'll try to think about this and see if I come up with a better answer.
I was discussing utilitarianism and charitable giving and similar ideas with someone today, and I came up with this hybrid version of the trolley problem, particularly the fat man variation, and the article by Scott Alexander/Yvain about using dead children as a unit of currency. It's not extremely original, and I'd be surprised if no-one on LW had thought of it before.
You are offered a magical box. If you press the button on the box, one person somewhere in the world will die, you get $6000, and $4,000 is donated to one of the top rated charities on GiveWell.org. According to the $800 per life saved figure, this charity gift would save five lives, which is a net gain of four lives and $6,000 to you. Is it moral to press the button?
All of the usual responses to the trolley problem apply. To wit: It's good to have heuristics like "don't kill." There's arguments about establishing Schelling points with regards to not killing people. (This Schelling point argument doesn't work as well in a case like this, with anonymity and privacy and randomization of the person who gets killed.) Eliezer argued that for a human, being in the trolley problem is extraordinarily unlikely, and he would be willing to acknowledge that killing the fat man would be appropriate for an AI in the situation to do, but not a human.
There's also lots of arguments against giving to charity, too. See here for some discussion of this on LessWrong.
I feel that the advantage of my dilemma is that in the original extreme altruism faces a whole lot of motivated cognition against it, because it implies that you should be giving much of your income to charity. In this dilemma, you want the $6,000, and so are inclined to be less skeptical of the charity's effectiveness.
Possible use: Present this first, then argue for extreme altruism. This would annoy people, but as far as I can tell, pretty much everyone gets defensive and comes up with a rationalization for their selfishness when you bring up altruism anyway.
What would you people do?
EDIT: This $800 figure is probably out of date. $2000 is probably more accurate. However, it's easy to simply increase the amount of money at stake in the thought experiment.
Edit 2: I fixed some swapped-around values, as kindly pointed out by Vaniver.