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

Question

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! :)

Answer 1

First it is important to define our terms. The concept of "weighted voters" does not necessarily make sense for an arbitrary social choice rule. So presumably we are interested in social choice rules where, at some point in the aggregation process, we add up a collection of terms, where each term contains information about the preferences or opinion of some voter. Typically, such rules are "anonymous" in the sense that we can permute the opinions of the voters without changing the answer. We can easily introduce a "weighted" version of such a rule by computing a weighted sum instead of an unweighted sum. (Typically, a weighted version of the rule fails to be anonymous.)

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 $A$.

Of course, we might say that all the different weighted majority rules belong to the same "family" of voting rules, and therefore they all have certain properties in common. For example, there are many theorems which are proved for the family of "all weighted majority rules" (or "for all weighted scoring rules", or whatever). But that does not mean that they are the "same" rule.

When you ask, "does a disagreement regarding how to weight the voters entail a disagreement regarding aggregation rule?", the answer is "yes and no". Clearly, we might agree on the type of voting rule to use (say, we agree to use a weighted majority rule because it satisfies certain desirable properties and we all agree these properties are important), while still disagreeing on the exact weights to use in the rule.