Under certain conditions, social welfare sums individual utilities.
Harsanyi's 1955 result: if individual and social preferences both satisfy expected-utility axioms, social welfare must be a weighted sum of individual utilities. The weights are what the theorem does not supply.
In practice
- A regulator asked to compare a small certain benefit for many against a large uncertain one for few is doing exactly this arithmetic and cannot state the weights.
- A charity allocating between two programmes must fix an exchange rate between beneficiaries, and no data supplies it.