I’m fascinated by but completely failing to grasp your first comment. Specifically:
Suppose we:
Take a finite set FS of N voters
Define an infinite set IS of hypothetical voters, fully indexed by the positive integers, such that hypothetical vote n+1 is the same as real vote (n mod N)+1
Use a “non-principal ultrafilter” to resolve the result
Which of Arrow’s criteria is violated when considering this to be a result of the votes in FS but not violated when considering this to be a result of the votes in IS?
Good question! It’s dictatorship. In such a situation, any non-principal ultrafilter picks out one of the congruence classes and only listens to that one.
More generally, given any partition of an infinite set of voters into a finite disjoint union of sets, a non-principal ultrafilter picks out one member of the partition and only listens to that one. In other words, a non-principal ultrafilter disenfranchises arbitrarily large portions of the population. This is another reason it’s not very useful for actually conducting elections!
I’m fascinated by but completely failing to grasp your first comment. Specifically:
Suppose we:
Take a finite set FS of N voters
Define an infinite set IS of hypothetical voters, fully indexed by the positive integers, such that hypothetical vote n+1 is the same as real vote (n mod N)+1
Use a “non-principal ultrafilter” to resolve the result
Which of Arrow’s criteria is violated when considering this to be a result of the votes in FS but not violated when considering this to be a result of the votes in IS?
Good question! It’s dictatorship. In such a situation, any non-principal ultrafilter picks out one of the congruence classes and only listens to that one.
More generally, given any partition of an infinite set of voters into a finite disjoint union of sets, a non-principal ultrafilter picks out one member of the partition and only listens to that one. In other words, a non-principal ultrafilter disenfranchises arbitrarily large portions of the population. This is another reason it’s not very useful for actually conducting elections!