Impossibility results for infinite-electorate abstract aggregation rules

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.Arrow-type preference aggregation, judgment aggregation, model theory, first-order predicate logic, filter, ultrafilter, reduced product, ultraproduct, existential quantifier

Impossibility results for infinite-electorate abstract aggregation rules

Herzberg F, Eckert D. Impossibility results for infinite-electorate abstract aggregation rules. Working Papers. Institute of Mathematical Economics. Vol 427. Bielefeld: Center for Mathematical Economics; 2010.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

Impossibility Results for Infinite-Electorate Abstract Aggregation Rules

Herzberg F, Eckert D. Impossibility Results for Infinite-Electorate Abstract Aggregation Rules. Journal of Philosophical Logic. 2012;41(1):273-286

Aggregating infinitely many probability measures

Herzberg F. Aggregating infinitely many probability measures. Center for Mathematical Economics Working Papers. Vol 499. Bielefeld: Center for Mathematical Economics; 2014.The problem of how to rationally aggregate probability measures occurs in particular (i) when a group of agents, each holding probabilistic beliefs, needs to rationalise a collective decision on the basis of a single ‘aggregate belief system’ and (ii) when an individual whose belief system is compatible with several (possibly infinitely many) probability measures wishes to evaluate her options on the basis of a single aggregate prior via classical expected utility theory (a psychologically plausible account of individual decisions). We investigate this problem by first recalling some negative results from preference and judgment aggregation theory which show that the aggregate of several probability measures should not be conceived as the probability measure induced by the aggregate of the corresponding expected-utility preferences. We describe how McConway’s (Journal of the American Statistical Association, vol. 76, no. 374, pp. 410– 414, 1981) theory of probabilistic opinion pooling can be generalised to cover the case of the aggregation of infinite profiles of finitely-additive probability measures, too; we prove the existence of aggregation functionals satisfying responsiveness axioms à la McConway plus additional desiderata even for infinite electorates. On the basis of the theory of propositional-attitude aggregation, we argue that this is the most natural aggregation theory for probability measures. Our aggregation functionals for the case of infinite electorates are neither oligarchic nor integral-based and satisfy (at least) a weak anonymity condition. The delicate set-theoretic status of integral-based aggregation functionals for infinite electorates is discussed