Set of utilitarian weights
One is Relative Utilitarianism (RU). Under the axioms below, society's preference can be represented by the simple sum of individuals' vNM utilities (each normalized to between $0$ and $1$). That is, every individual is given equal weight.


Pareto Axiom. (This is just the usual: If everyone prefers lottery $p$ to $q$, then so too does society. And if everyone strictly prefers lottery $p$ to $q$, then so too does society.)
Separability.
Invariance.
Anonymity. (Again, this is just the usual, though a small technical adjustment may be necessary.)


The original papers on RU are Dhillon (1998) and Dhillon &amp; Mertens (1999). However, the first contains fatal mistakes and the second is indecipherable. See Börgers &amp; Choo (2017) for a simple and correct exposition/proof of RU.

(Of course, the literature on utilitarianism and what you ask is vast. This is just one small example.)