Computational Intelligence, Volume 34, Number 1, 2018
A HIERARCHICAL APPROACH TO JUDGMENT AGGREGATION
AIRG, Western Sydney University, Penrith, Australia
IRIT, Université Toulouse Capitole, Toulouse, France
Judgment aggregation deals with the problem of how collective judgments on logically connected proposi-
tions can be formed based on individual judgments on the same propositions. The existing literature on judgment
aggregation mainly focuses on the anonymity condition requiring that individual judgments be treated equally.
However, in many real-world situations, a group making collective judgments may assign individual members or
subgroups different priorities to determine the collective judgment. Based on this consideration, this article relaxes
the anonymity condition by giving a hierarchy over individuals so as to investigate how the judgment from each
individual affects the group judgment in such a hierarchical environment. Moreover, we assume that an individ-
ual can abstain from voting on a proposition and the collective judgment on a proposition can be undetermined,
which means that we do not require completeness at both individual and collective levels. In this new setting, we
ﬁrst identify an impossibility result and explore a set of plausible conditions in terms of abstentions. Secondly, we
develop an aggregation rule based on the hierarchy of individuals and show that the aggregation rule satisﬁes those
plausible conditions. The computational complexity of this rule is also investigated. Finally, we show that the pro-
posed rule is (weakly) oligarchic over a subset of agenda. This is by no means a negative result. In fact, our result
reveals that with abstentions, oligarchic aggregation is not necessary to be a single-level determination but can be
a multiple-level collective decision making, which partially explains its ubiquity in the real world.
Received 13 November 2015; Revised 30 June 2016; Accepted 6 September 2016
Key words: judgment aggregation, lexicographic aggregation rule, computational social choice.
Judgment aggregation is an interdisciplinary research topic in economics, philoso-
phy, political science, law, and recently in computer science (Wilson 1975; Mongin 1995;
Konieczny and Pérez 2002; Pigozzi 2006; Dietrich and List 2007a; Brandt et al. 2013). It
deals with the problem of how a set of group judgments on certain issues, represented by
logical propositions, can be formed based on individuals’ judgments on the same issues.
Although most of voting rules for social choice, such as majority rules, unanimity rules
or dictatorships, are applicable to judgment aggregation, their behavior can be signiﬁcantly
different because of possible logical links among the propositions on which a collec-
tive decision has to be made. A well-known example is the so-called doctrinal paradox
(Kornhauser and Sage 1993; List 2012), which shows that the majority rule fails to
guarantee consistent group judgments.
Suppose a court consisting of three judges has to reach a verdict in a breach-of-contract
case. There are three propositions on which the court is required to make judgments:
p : The defendant was contractually obliged not to do a particular action.
q : The defendant did that action.
r : The defendant is liable for a breach of contract.
According to the legal doctrine, propositions p and q are jointly necessary and sufﬁcient for
proposition r,thatis,p ^q $ r. Now, the three judgments on the propositions are shown in
Address correspondence to Guifei Jiang, AIRG, and IRIT, Université Toulouse Capitole, France; e-mail:
© 2017 Wiley Periodicals, Inc.