I know many papers about showing the existence (or the impossibility) of a social choice rule satisfying a set of properties.
However, I was wondering whether there were some references trying to answer the following question: given a set of desiderata, what is the cardinal of the set of aggregation functions?
The desiderata in question could be anything:
- A certain structure of preferences at the individual or the social level
- The usual universal domain, Pareto property, independence of irrelevant alternative...
I know in the literature, there are many distinctions between different properties of the aggregation function (SWF, SWFL, Collective choice rule...). I am only interested in any result that concern an aggregation function of any type, i.e. something that takes a profile of the voters preferences and return a choice (or a rationalization of the choice, as a social preference).
All I am looking for is any result that says something like the following:
- Given some restriction on the aggregation rule (like saying it is a social welfare function) the number of SWF satisfying some set of properties is ...
- There is strictly more CCR satisfying ... than SWF. (It is trivial when there is an impossibility for SWF and not for CCR, but maybe this is true whenever there is a possibility for SWF too?)