I prefer "maximise happiness" if only because the Repugnant Conclusion is more interesting than the VHEMT.
https://en.wikipedia.org/wiki/Mere_addition_paradox https://en.wikipedia.org/wiki/Voluntary_Human_Extinction_Mov...
Maybe mean/median/minimum happiness? They don't intuitively appeal to me, but I'd like to hear an argument for them from someone who believes in them (as well as arguments for systems I haven't considered.)
Of course there are plenty of non-utilitarian systems to choose from, but I'm sure few their proponents would like to describe as maximising happiness :-)
Most commonly, maximin is extended to leximin i. e. Looking at the next least welloff. There are traditions of critique both from communitarianism (Sandel et al.) and libertarianism (Nozick et al.), both worth looking into.
In terms of happiness, it makes sure that no one is left behind, and tends to produce a more homogeneous society than optimizing for the average.
Also, [1].
Well, aside from the imperative to euthanise the least happy regardless of their absolute happiness, I guess :-).
And making most individual actions normally called moral or immoral difficult to justify or condemn except that they might indirectly affect the least happy person.
And then there's the weird idea that my local actions may have moral weight depending on the arbitrary existence of an unhappy person in a distant country.
All seems pretty bizarre to me.