Judgment aggregation functions and ultraproducts∗
Frederik Herzberg† ‡§
The relationship between propositional model theory and social decision making
via premise-based procedures is explored. A one-to-one correspondence between
ultrafilters on the population set and weakly universal, unanimity-respecting, sys-
tematic judgment aggregation functions is established. The proof constructs an
ultraproduct of profiles, viewed as propositional structures, with respect to the ul-
trafilter of decisive coalitions. This representation theorem can be used to prove
other properties of such judgment aggregation functions, in particular sovereignty
and monotonicity, as well as an impossibility theorem for judgment aggregation in
finite populations. As a corollary, Lauwers and Van Liedekerke’s (1995) represen-
tation theorem for preference aggregation functions is derived.
Key words: Judgment aggregation function; ultraproduct; ultrafilter.
Journal of Economic Literature classification: D71.
AMS Mathematics Subject Classification: 91B14; 03C20.
Ultrafilters have an almost four-decades long history of successful application in the
theory of preference aggregation. Initiated by Fishburn (1970) and Hansson (1971,
Postscript 1976), a seminal contribution was made by Kirman and Sondermann (1972)
whose results motivated numerous other papers in this area; see Monjardet (1983) for
A quarter-century later, Lauwers and Van Liedekerke (1995) provided an axiomatic
foundation for the ultrafilter method in the theory of preference aggregation: They con-
structed a one-to-one correspondence between preference aggregation functions satis-
fying Arrovian axioms and ultrafilters on the population set. The bijection is given by
therestriction oftheultraproduct1—withrespecttotheultrafilter ofdecisivecoalitions
— of the family of individual preference orderings to the original set of alternatives.
∗I would like to thank Dr. Daniel Eckert and Dr. Franz Dietrich for helpful comments on a previous
version of this paper.
†Department of Mathematics, University of California, Berkeley, CA 94720-3840, United States of
‡Institut für Mathematische Wirtschaftsforschung, Universität Bielefeld, Universitätsstraße 25, D-33615
§This work was supported by a research grant from the German Academic Exchange Service (DAAD).
1Ultraproducts were first studied extensively by Ło´ s (1955). Expositions of ultraproducts can be found in
model-theoretic monographs such as Chang and Keisler (1973). The existence of non-principal ultrafilters
(i.e. non-dictatorial large coalitions) on infinite sets was established by Ulam (1929) and Tarski (1930), under
the assumption of the Axiom of Choice. (Strictly speaking, the Ultrafilter Existence Theorem only needs the
Boolean Prime Ideal Theorem, which is a weaker set-theoretic axiom than the Axiom of Choice, as Halpern
and Le! vy (1971) have shown. See also Halpern (1964) and Banaschewski (1983) for related results.)
Ultrafilters have now also entered the theory of judgment aggregation, through pa-
pers by Gärdenfors (2006), Daniëls (2006), Dietrich and Mongin (2007), Eckert and
Klamler (2008),2and some of the results in this article have already been discovered,
via different methods, by Dietrich and Mongin (2007) (see Remark 12 below). It is
not surprising that the ultrafilter method is an appealing tool in judgment aggregation
theory, since ultrafilters are a means of relating the propositional logical structure of
the electorate’s a! genda with the algebraic structure of the voting coalitions.
In this paper, we use the methodology of Lauwers and Van Liedekerke (1995) to
find an axiomatisation of the relation between propositional model theory and social
decision making via premise-based procedures in general, and between judgment ag-
gregation functions and ultrafilters in particular. We prove that a given judgment ag-
gregation function maps profiles to ultraproducts of profiles — with respect to the ul-
trafilter of decisive coalitions — if and only if it satisfies the axioms of weak univer-
sality, respect for unanimous decisions as well as systematicity. This correspondence
between ultrafilters and certain judgment aggregation functions will be used to prove
several other properties of these aggregation functions, as well as an impossibility the-
orem for judgment aggregation in finite populations. We show that the main theorem
of Lauwers and Van Liedekerke (1995) is contained in our results.
In fact, even the converse is true: Many of our results could also be obtained in
a lengthy indirect argument via suitable corollaries of the findings of Lauwers and
Van Liedekerke (1995) through replacing the binary preference predicate on alterna-
tives by a unary truth predicate on propositions. However, the technical translation
effort required would be substantial, and the resulting arguments would therefore ulti-
mately not be significantly shorter than the direct derivation in this paper; furthermore,
it would conceal the simplicity of the ultraproduct construction for propositional struc-
aggregation functions (ultraproducts) and families of decisive coalitions (ultrafilters).
The paper is self-contained and only assumes basic familiarity with propositional
logic on the part of the reader.
2 Judgment sets
Where possible, we follow the terminology of List and Puppe (2007) and the notation
of Eckert and Klamler (2008).
Let N be a finite or infinite set, the population set. Its elements are also called
individuals. Consider a set Y of at least two propositional variables, and let Z be the set
of all propositions, in the sense of propositional calculus, with propositional variables
from Y . Z is called agenda with base Y and should be conceived as the agenda of a
premise-based procedure of social decision making. Let T be a consistent subset of Z
(i.e. T ?? ⊥), called the population’s (unanimous) theory. The set X =?
A fully rational judgment set in X given T is a subset A ⊂ X which is complete in
X (i.e. for all p ∈ Y either p ∈ A or ¬p ∈ A) and consistent with T (i.e. A ∪ T ?? ⊥).
The set of fully rational judgment sets in X given T shall be denoted by D.
is called the atomic agenda.
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