Firstly, I'd take volunteers. If people are suicidal, terminally ill, or just don't feel like living in a world where someone has to actually go through and eliminate billions of people, then who am I to deny them the chance to die for a (edit: potentially) beneficial cause?

Secondly, lottery. It's a tough break for those who lose, I get it, but it's the fairest way to handle such a situation.

That being said, I'm not convinced that fairness should be the primary determinant here, but that is my first instinct. It's hard for me to fight other thoughts though, such as killing the elderly, the violent felons, the mentally disabled. These are all potentially very controversial, but, then again, any solution to this sort of problem would be.

Oh, man, this is a tough one. I'll have to think more about it, but right now, I think I'm still gonna go with volunteers and then a lottery.