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I am wondering if there are literature which studies specific properties of the set of utilitarian weights?

To be more specific, when we have some axioms that characterize the preference of a social planner (and gives a utility representation), we can usually say the set of utilitarian weights is convex and closed. But are there previous papers studying more properties about the set?

I am grateful if some one can point me out some related paper regarding properties of set of utilitarian weights.

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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.

  1. 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.)
  2. Separability.
  3. Invariance.
  4. 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 & Mertens (1999). However, the first contains fatal mistakes and the second is indecipherable. See Börgers & 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.)

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