Possibilities in social choice: number of possible collective decision function?

Question

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?)

Answer 1

I can't think of any theorems which specifically focus on the number of social aggregation rules (of some type) satisfying axioms X, Y, and Z. However, aside from impossibility theorems, most of the theorems in social choice theory fall into one of two categories:

1. An axiomatic characterization of a single rule, e.g. "Rule F is the unique rule satisfying axioms X, Y, and Z."
2. An axiomatic characterization of a well-defined family of rules, e.g. "Rules of class C are the only rules satisfying axioms X, Y, and Z."

In the first case, the "number" of rules is obviously one, which is not very interesting. In the second case, the class C is typically defined by one or more "parameters", so the number of rules in C is just the number of valid parameter values. Since these parameters are usually real numbers, this means that the number of rules in C is usually infinite. Indeed asking about the "cardinality" of C is generally not the right question. The right question is the dimension of the parameter space.

For example: suppose C is the set of scoring rules on a set of N alternatives (e.g. the Borda rule, (anti)plurality rule, etc.). Any scoring rule is defined by N numbers: the "scores" $s_1\leq s_2\leq \cdots \leq s_N$ assigned to the N possible ranks in any voter's preference order. Without loss of generality, we can assume $s_1=0$ and $s_N=1$. Furthermore, since the scores are an increasing sequence, they are completely described by the score differences $\delta_1=s_2-s_1=s_2$, $\ \delta_2=s_3-s_2$, $\ldots$, $\delta_{N-1}=1-s_{N-1}$. Observe that $\delta_1+\cdots+\delta_{N-1}=1$. In other words, the vector $(\delta_1,\ldots,\delta_{N-1})$ is an element of the probability simplex $\Delta^{N-1}$, which is a convex set of dimension $N-2$.

From this parameterization, we see that the cardinality of the set of scoring rules is $\beth_1$. But the parameterization is much more informative than the cardinality.