Judgment Aggregation - Gabriella Pigozzi Synthesis Lectures on Artificial Intelligence and Machine Learning

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We conclude our comment of Definition 2.4 by noticing that it builds two key properties into the notion of aggregation function. First, it assumes that the domain of the aggregation consists of all possible judgment profiles or, intuitively, that all profiles of individual opinions are admissible as input for the aggregation. This property is commonly referred to as universal domain. Second, it assumes the aggregation to be resolute, that is, to yield for each profile only one set of formulae. In this book we will work almost exclusively with functions that satisfy universal domain and resoluteness. Aggregation functions that do not satisfy universal domain will be presented later in Chapter 4. Irresolute functions yielding for each profile a non-empty set of sets of formulae will be studied later in Chapter 4 and especially in Chapter 6.

      We now give several examples of aggregation functions as rules for defining the collective set based on a judgment profile. We typically refer to concrete aggregation functions as aggregation rules. The ones that follow in this section will be discussed at several places throughout the book and are the ones most commonly considered in the literature.

       Threshold-based rules

      The rules below determine the collective outcome by checking, for each proposition in the agenda (they are therefore commonly referred to as propositionwise rules), whether the number of individuals accepting that formula exceeds a given threshold. If that is the case, the formula is collectively accepted. Let P ∈ P, we define the following rules.

       Majority rule:

      where, for x ∈ Q, ⌈x⌉ is the smallest integer greater or equal to x. I.e., φ is collectively accepted iff there is a majority of individuals accepting it. We will refer to this rule as the propositionwise majority rule or simply as the majority rule.

       Unanimity rule:

      I.e., φ is collectively accepted iff all individuals accept it. We will refer to this rule as the propositionwise unanimity rule or simply as the unanimity rule.

       Quota rule:

      where

is a tuple of integer thresholds or quotas tφ, one for each formula in the agenda. I.e., φ is collectively accepted iff there are at least tφ individuals that accept it.

      Formula 2.3 defines the class of all propositionwise threshold-based rules. Clearly, the propositionwise majority rule is a particular quota rule whose threshold has been fixed at

for all formulae in the agenda.7 Similarly, the unanimity rule is a quota rule with threshold |N| for all formulae. Quota rules that assign the same threshold to all formulae called uniform.

      It must be noted that the selection of the thresholds has an impact on the ‘rationality’ of the collective set. For instance, it is not difficult to see that the unanimity rule might return incomplete collective sets, and that a uniform quota rule imposing a common threshold lower than

might return collective sets containing both a formula and its negation. In general, one can identify precise constraints on the thresholds, which can enforce a well-behaved output of the aggregation. For instance, for each pair φ and ¬φ, the inequalities

      are necessary and sufficient conditions for the collective set to be complete (i.e., to contain at least one of φ or ¬φ, Formula 2.4) and, respectively, to be such that it never contains both a formula and its negation (i.e., to contain at most one of φ and ¬φ, Formula 2.5). This latter property is usually referred to as weak consistency.

      The class of all quota rules has been studied extensively in [DL07b]. We will come back to the majority rule in much more detail later in Chapter 3, and to quota rules as possible escape routes to some of the impossibility results of judgment aggregation in Chapters 4 and 5.8

       Premise- and conclusion-based rules

      We have already encountered the premise- and conclusion-based rules in Section 1.2. Here we give a more precise formulation of them.

       Premise-based rule:

      where: Prem ⊆ A consists of the subagenda containing the issues that are considered premises in the aggregation, and their negations; Conc ⊆ A consists of the subagenda containing the issues that are considered conclusions in the aggregation, and their negations; Prem and Conc are a partition of A; and PPrem (respectively, PConc) denotes the profile obtained from the restrictions of the judgment sets to the formulae in Prem (respectively, Conc). I.e., φ is collectively accepted iff it is a premise and it has been voted by the majority of the individuals or it is a conclusion entailed by the premises accepted by the majority.

       Conclusion-based rule:

      where PConc is as for the premise-based rule. I.e., φ is collectively accepted iff it is a conclusion and it has been voted by the majority of the individuals.

      Intuitively, premise- and conclusion-based rules apply propositionwise aggregation, via the majority rule, only to specific subsets of the agenda, viz., its premises or its conclusions. They have played a pivotal role in the development of the theory of judgment aggregation, and much literature has been dedicated to their analysis (see, for instance, [NP06, DM10]). We will come back to them at several places in the remaining of the book.

       An example

      It is now time to illustrate the workings of all the above rules side by side. We do that with yet another variant of the doctrinal paradox: