I'd pull the lever to kill one person immediately. Assuming the decision maker at each stage is a different person with different opinions on moral, ethical, religious, and logical questions, then it's a near certainty that someone is going to pull the lever to kill the people at their stage. If you're lucky, it's the very next guy. If you're not, it's the guy killing a million people a couple of iterations later. If I'm the first guy, I'll take the moral hit to save the larger number of people.
Step in front of the train: Tell your manager this whole project is dumb, provide a list of reasons why it's a bad idea and explain you are prepared to resign rather than enable its further development.
I’ve been thinking about this. I estimate a few people per 1000 would do an atrocity for no reason if they were guaranteed no consequences, and the deaths if the switch is pulled are 2^(n-1) for the nth switch. The expected deaths will cross 1 somewhere in the high single-digits, then (since it’s outcome*chance), so the death minimising strategy is actually to pull yours if the chain is at least that long.
Edit: This assumes the length of the chain is variable but finite, and the trolley stops afterwards. If it's infinite obviously you pull the switch.
Half-pull the lever so that the points get stuck midway between the two tracks. That should derail the trolley. Someone could conceivably still get hurt, but it improves everyone's chances.
(What? You mean it isn't a literal trolley that has to obey the laws of physics? Damn.)
If we keep doubling, will I eventually be a person on the tracks? There are a finite number of people, so eventually I would be, right? So, passing the buck would be equivalent to handing my fate to a stranger.
OTOH, if there are an infinite number of people, then this thought experiment is creating people out of thin air. Do these imaginary people's rhetorical lives even matter?
Either way, it seems better to kill 1 person at the start.
I don't have to be a soldier on anyone's ethical recursion war, so since the default position is set to kill 1 person, that gets done by the problem itself and the whole thing is solved without me having to do anything.
As a further bonus, now the lever people on the next branches are free to get out of the levels and go release the other prisoners.