Herzberg, Frederik
Frederik
Herzberg
Eckert, Daniel
Daniel
Eckert
Impossibility results for infinite-electorate abstract aggregation rules
Center for Mathematical Economics
2010
2017-03-21T08:39:21Z
2018-07-24T12:59:26Z
working_paper
https://pub.uni-bielefeld.de/record/2909316
https://pub.uni-bielefeld.de/record/2909316.json
0931-6558
urn:nbn:de:0070-pub-29093162
332002 bytes
application/x-download
It is well known that the literature on judgment aggregation
inherits the impossibility results from the aggregation of preferences
that it generalises. This is due to the fact that the typical judgment
aggregation problem induces an ultrafilter on the the set of individuals,
as was shown in a model theoretic framework by Herzberg and
Eckert (2009), generalising the Kirman-Sondermann correspondence and
extending the methodology of Lauwers and Van Liedekerke (1995). In the
finite case, dictatorship then immediately follows from the principality
of an ultrafilter on a finite set. This is not the case for an infinite set
of individuals, where there exist free ultrafilters, as Fishburn already
stressed in 1970. The main problem associated with free ultrafilters in the
literature on aggregation problems is however, the arbitrariness of their
selection combined with the limited anonymity they guarantee (which
already led Kirman and Sondermann (1972) to speak about invisible
dictators). Following another line of Lauwers and Van Liedekerke's (1995)
seminal paper, this note explores another source of impossibility results
for free ultrafilters: The domain of an ultraproduct over a free ultrafilter
extends the individual factor domains, such that the preservation of the
truth value of some sentences by the aggregate model - if this is as
usual to be restricted to the original domain - may again require the
exclusion of free ultrafilters, leading to dictatorship once again.