# Is a social choice aggregation rule defined for a set of weightings over the set of voters (N)?

In Christian List's Stanford Encyclopaedia entry Social Choice Theory (2013, https://plato.stanford.edu/entries/social-choice) he says that:

"an aggregation rule is defined for a fixed set of individuals N and a fixed decision problem, so that majority rule in a group of two individuals is a different mathematical object from majority rule in a group of three."

I was wondering if, in a similar way, social choice aggregation rules are defined for a set of weightings over N? In other words, does a disagreement regarding how to weight the voters entail a disagreement regarding aggregation rule?

E.g. let's say that N = {voter-1, voter-2}. Furthermore, you and I both favour a lexicographic dictatorship. The only difference is that you favour a lexicographic dictatorship which ranks voter-1 over voter-2, and I favour a lexicographic dictatorship which ranks voter-2 over voter-1. Strictly speaking, do we agree on aggregation rule, or do we disagree?

Any help would be greatly appreciated! :)

• If the aggregation rules potentially produce different results then I would say they were different, even if they are in the same family of rules. So you disagree. Civil wars and coups have happened because of this disagreement – Henry Jul 31 '18 at 0:27

Let us say that two social choice rules $F$ and $G$ are different if they sometimes produce different outputs when presented with identical input (e.g. when presented with identical preference profiles). In that case, it is definitely the case that in general, different voter-weights will yield different social choice rules.
For example, let's consider "weighted majority voting" with three voters, $A$, $B$ and $C$. One possibility is that each voter receives equal weight (say, 1/3) ---this is effectively the "unweighted" version, which satisfies anonymity. Another possibility is that voter $A$ receives weight 5, while voters $B$ and $C$ each receive weight 1. It is easily verified that this "weighted majority rule" is actually a dictatorship for $A$.