## Thursday, November 20, 2008

### Infinitely many voters

As we have discussed before, the famous Arrow's Theorem is equivalent to this basic theorem on ultrafilters:

Every ultrafilter on a finite set is principal.

(See for instance Joel W. Robbin's paper for a proof of this equivalence.) Basically, the set on which the ultrafilter is built represents the voters, and the member sets of an ultrafilter can be equated with forcing coalitions, groups of voters which, if unanimous on a given preference, determine the overall result with respect to that preference. The theorem above says that every ultrafilter on a finite set of voters has a forcing coalition of size 1, i.e. a dictator.

If we have an inifite set, however, the Axiom of Choice implies that there are non-principal ultrafilters: so if there were infinitely many voters we could devise (alas, nonconstructively) a fair voting system free of dictatorships. It is interesting to try to take advantage of this result to somehow circumvent the limitations for the finite case, and see what fails (something must fail after all):

Let our voters be represented by the set V={v1,...,vN}: we "extend" them by assigning to each vi an enumerable set of fictitious voters vi1, vi2,... Thus the set V' of fictitious voters is infinite and we can have a non-principal ultrafilter F on it. We design the following voting system: for any given preference the vote of each vi is obtained and assigned to the associated fictitious voters vij; if the overall resulting set of fictitious voters supporting the preference is a forcing coalition according to F, the preference is approved, otherwise rejected.

We already know that this voting system cannot be fair, and in fact it is easy to uncover the underlying flaw: the range of our mapping f : VV' is not the entire powerset of V', but only a finite subset of it (with cardinality 2N); although F is non-principal, it becomes principal (streching somewhat the meaning of the concept) when restricted to range(f): F necessarily contains then a pseudodictatorial forcing coalition comprising exactly all the fictitious voters associated to a single vi, who is the resulting dictator for the so devised voting system.