This could be another argument for completeness wrt lives: Super-Strong Pareto ("better for someone + worse for no one → better overall") + Acyclicity → completeness!

Probably-minor point: Taken literally (if this is *enough* to explain one lottery being better than another), this reasoning leads to cycles of betterness (for lotteries, if not for outcomes):
(a, a)> (a−, b) > (b, b−) > (b−, a+) > (a,a). (See fn 4 on p 3 of our paper.)

...though as I mentioned in my comments to Brian, I also have my own weird/idiosyncratic reasons for liking Negative Dominance, that go beyond what we say in the paper. :-)

I basically agree with this -- something we mention in the paper but maybe should have emphasized more is the insistence on grounding the evaluation of lotteries in *particular* outcomes ("ways things might actually turn out") rather than aggregates or summary statistics.

Thanks for taking the time to write this up! Just got a chance to read it, and I think you do identify a real leap in our case for Negative Dominance, though I'm not sure if it's fatal to that case. I had more thoughts than I could comfortably fit in a thread, so just sent some comments by email.

If you can assign probabilities to your future choices, then your example would be subsumed by our argument against incompleteness for lotteries (sec. 6), but if not then not. One thing we didn't consider (but is worth considering!) is imprecise/absent probabilities as a source of incompleteness.

Good question! The argument as we present it in the paper depends on each final outcome corresponding to a definite assignment of "lives" (individual outcomes, welfare states) to individuals.

It's the successor to a very old paper, "Exceeding Expectations", that I've been working on since 2017 -- so I'm very happy that it's finally found a good home!