Introduction to Judgment Aggregation

The present notes are an improved version of the notes that served as teaching materials for the course Introduction to Judgment Aggregation given at the 23rd European Summer School on Logic, Language and Information (ESSLLI’11, Ljubljana). The notes are structured as follows: Section 1 introduces the field of judgment aggregation, its relations to preference aggregation and some formal preliminaries. Section 2 shows that the paradox that originated judgment aggregation is not a problem limited to propositionwise majority voting but a more general issue, illustrated by an impossibility theorem of judgment aggregation that is here proven. The relaxation of some conditions used in impossibility results in judgment aggregation may lead to escape routes from the impossibility theorems. These escape routes are explored in Section 3. Section 4 presents the issue of manipulation that arises when voters strategically misrepresent their true vote in order to force a different outcome in the aggregation process. Finally, we conclude by sketching a list of on-going research in the field of judgment aggregation (Section 5). 1 Logic Meets Social Choice Theory Outline. We start by presenting the paradox that originated the whole field of judgment aggregation and by looking at how judgment aggregation relates to the older theory of preference aggregation. Section 1.2 formally introduces the three central notions in the theory of judgment aggregation, namely agendas, judgment sets and aggregation functions. 1.1 A Social View on Logic From the Doctrinal Paradox to the Discursive Dilemma. The idea that groups make better decisions than individuals dates back to 18th century social theorists like Rousseau and Condorcet [13]. However, as we will see in much detail, majority voting—the exemplary democratic aggregation rule—is unable to ensure consistent social positions under all situations—this is the bottom line of the now famous Condorcet paradox, to which we will turn later in this section. Whereas voting theory studies the aggregation of individual preferences, the recent theory of judgment aggregation investigates how individual opinions on N. Bezhanishvili et al. (Eds.): ESSLLI 2010/2011, Lectures, LNCS 7388, pp. 160–209, 2012. c ⃝ Springer-Verlag Berlin Heidelberg 2012 Introduction to Judgment Aggregation 161 logically related propositions can be consistently aggregated into a collective position.1 Judgment aggregation has its roots in jurisprudence, building on the doctrinal paradox that Kornhauser and Sager discovered in the decision making procedure of collegial courts [47,48,46]. It is instructive to recall the original example that Kornhauser and Sager used to illustrate the doctrinal paradox [48]. A three-member court has to reach a verdict in a breach of contract case between a plaintiff and a defendant. According to the contract law, the defendant is liable (the conclusion, here denoted by proposition r) if and only if there was a valid contract and the defendant was in breach of it (the two premises, here denoted by propositions p and q respectively). Suppose that the three judges cast their votes as in Table 1. Table 1. An illustration of the doctrinal paradox Valid contract Breach Defendant liable p q r Judge 1 1 1 1 Judge 2 1 0 0 Judge 3 0 1 0 Majority 1 1 0 The court can rule on the case either directly, by taking the majority vote on the conclusion r (conclusion-based procedure) or indirectly, by taking the judges’ recommendations on the premises and inferring the court’s decision on r via the rule (p ∧ q) ↔ r that formalizes the contract law (premise-based procedure). The problem is that the court’s decision depends on the procedure adopted. Under the conclusion-based procedure, the defendant will be declared not liable, whereas under the premise-based procedure, the defendant would be sentenced liable. As Kornhauser and Sager stated: We have no clear understanding of how a court should proceed in cases where the doctrinal paradox arises. Worse, we have no systematic account of the collective nature of appellate adjudication to turn to in the effort to generate such an understanding. [48, p. 12] The systematic account to the understanding of situations like the one in Table 1 has been provided by judgment aggregation. The first step was made by political philosopher Pettit [67], who recognized that the paradox illustrates a more general problem than a court decision. Pettit introduced the term discursive dilemma to indicate a group decision in which propositionwise majority voting on related propositions may yield an inconsistent collective judgment. 1 Among the existing surveys and tutorial papers on judgment aggregation, we recall [56,53]. 162 D. Grossi and G. Pigozzi Then, List and Pettit [55] reconstructed Kornhauser and Sager’ example as shown in Table 2. By adding the legal doctrine to the set of issues on which the judges have to vote, List and Pettit attained a great generality which provided analytical advantages. The discursive dilemma is characterized by the fact that the group reaches an inconsistent decision, like {p, q, (p∧ q) ↔ r,¬r} in Table 2. Table 2. The discursive dilemma Valid contract Breach Legal doctrine Defendant liable p q (p ∧ q) ↔ r r Judge 1 1 1 1 1 Judge 2 1 0 1 0 Judge 3 0 1 1 0 Majority 1 1 1 0 Is the move from the doctrinal paradox to the discursive dilemma an innocent one? Recently, Mongin and Dietrich [60,59] have investigated such shift and observed that: [T]he discursive dilemma shifts the stress away from the conflict of methods to the logical contradiction within the total set of propositions that the group accepts. [...] Trivial as this shift seems, it has far-reaching consequences, because all propositions are now being treated alike; indeed, the very distinction between premisses and conclusions vanishes. This may be a questionable simplification to make in the legal context, but if one is concerned with developing a general theory, the move has clear analytical advantages. [59, p. 2] Instead of premises and conclusions, List and Pettit focused their attention on judgment sets, the sets of propositions that individuals accept. The theory of judgment aggregation becomes then a formal investigation of the conditions under which consistent individual judgment sets can collapse into an inconsistent judgment set. Their approach combines a logical formalization of the judgment aggregation with an axiomatic approach in the spirit of Arrow’s social choice theory. The first question they can address is how general the judgment aggregation problem is, that is, whether the culprit is majority voting or whether the dilemma arises also with other aggregation rules. They obtained a first general impossibility theorem stating that there exists no aggregation rule that satisfies few desirable conditions and that can ensure a consistent collective outcome. Impossibility results will be the topic of Section 2. Preference Aggregation and Judgment Aggregation. How individual preferences can be aggregated into a collectively preferred alternative is traditionally studied by social choice theory [2,72], whose origins can be traced Introduction to Judgment Aggregation 163 back to the works by Borda and Condorcet [5,13]. In particular, Condorcet aimed at an aggregation procedure that would maximize the probability that a group of people take the right decision. His result, known as the Condorcet Jury Theorem, showed that under certain conditions majority voting was a good truthtracking method. However, he also discovered a disturbing problem of majority voting. Given a set of individual preferences, the method suggested by Condorcet consisted in the comparison of each of the alternatives in pairs. For each pair, we determine the winner by majority voting, and the final collective ordering is obtained by a combination of all partial results. Unfortunately, this method can lead to cycles in the collective ordering: the Condorcet paradox. Let a set of alternatives X . Let P be a binary predicate interpreted on a binary relation over X , that we denote by ≻ (or ≻i when we need to make the agent explicit). xPy means “x is strictly preferable to y”. The desired properties of preference relations viewed as strict linear orders are: (P1) ∀x, y((xPy) → ¬(yPx)) (asymmetry) (P2) ∀x, y(x ̸= y → (xPy ∨ yPx)) (completeness) (P3) ∀x, y, z((xPy ∧ yPz) → xPz) (transitivity) Example 1 (Condorcet paradox). Suppose that there are three possible alternatives x, y and z and three voters V1, V2 and V3. Let ≻i denote agent i’s preference over X (without any index, ≻ denotes the collective preference relation). The three voters’ total preferences are the following: V1 = {x ≻1 y, y ≻1 z}, V2 = {y ≻2 z, z ≻2 x} and V3 = {z ≻3 x, x ≻3 y}. According to Condorcet’s method, a majority of the voters (V1 and V3) prefers x to y, a majority (V1 and V2) prefers y to z, and another majority (V2 and V3) prefers z to x. This leads us to the collective outcome x ≻ y, y ≻ z and z ≻ x, which together with transitivity (P3) violates (P1). Each voter’s preference is transitive, but transitivity fails to be mirrored at the collective level. This is an instance of the so-called Condorcet paradox.2 The profile of preferences here considered is known as ‘latin square’, that is, each alternative is ranked top, second and last in someone’s preference. Condorcet’s 1785 Essai was read and understood by a few until Edward J. Nanson (1882) [61] and Duncan Black (1958) [4]. In 1951 a young economist, the future Nobel prize winner Kenneth Arrow, showed that the Condorcet paradox is not a problem specific of pairwise majority comparison [2]. In his famous impossibility theorem, Arrow proved that, when there are three or more alternatives, the only aggregation procedures that satisfy few desirable properties (like the absence of cycles in the collective preference) are dictatorial ones. That is, the collective preference coincides with the preference of one and the same individual of the group. Thus, when we combine individual choices into a collective one, we may lose something that held at the individual level, like transitivity (in the case of preference aggregation) or logical consistency (in the case of judgment aggregation). 2 We will come back later in Section 2 to another simple formalization of the Condorcet paradox (Example 6). 164 D. Grossi and G. Pigozzi A natural question is how the theory of judgment aggregation is related to the theory of preference aggregation. We can address this question in two ways: we can consider what are the possible interpretations of aggregating judgments and preferences, and we can investigate the formal relations between the two theories. On the first consideration, Kornhauser and Sager see the possibility of being right or wrong as the discriminating factor: When an individual expresses a preference, she is advancing a limited and sovereign claim. The claim is limited in the sense that it speaks only to her own values and advantage. The claim is sovereign in the sense that she is the final and authoritative arbiter of her preferences. The limited and sovereign attributes of a preference combine to make it perfectly possible for two individuals to disagree strongly in their preferences without either of them being wrong. [...] In contrast, when an individual renders a judgment, she is advancing a claim that is neither limited nor sovereign. [...] Two persons may disagree in their judgments, but when they do, each acknowledges that (at least) one of them is wrong. [47, p. 85]3 Regarding the formal relations between judgment and preference aggregation, Dietrich and List [17] (extending earlier work by List and Pettit [55]) show that Arrow’s theorem for strict and complete preferences can be derived from an impossibility result in judgment aggregation. In order to represent preference relations, they consider a first-order language with a binary predicate P representing strict preference for all x, y ∈ X . The inference relation is enriched with the asymmetry, completeness and transitivity axioms of P . The Condorcet paradox can then be represented as a judgment aggregation problem as Table 3 illustrates. Voters of Table 3 are perfectly consistent just like the judges of Table 1. The difference is that in the doctrinal paradox individuals are consistent in terms of propositional logic, while in the Condorcet example consistency corresponds to the transitive and complete conditions imposed on preferences. However, it is worth mentioning that Kornhauser and Sager [46] notice that the doctrinal paradox resembles the Condorcet paradox, but that the two paradoxes are not equivalent. Indeed, as stated also by List and Pettit: [W]hen transcribed into the framework of preferences instances of the discursive dilemma do not always constitute instances of the Condorcet paradox; and equally instances of the Condorcet paradox do not always constitute instances of the discursive dilemma. [55, pp. 216-217] Given the analogy between the Condorcet paradox and the doctrinal paradox, List and Pettit’ first question was whether an analogous of Arrow’s theorem could be found for the judgment aggregation problem. Arrow showed that the 3 Different procedures for judgment aggregation have been assessed with respect to their truth-tracking capabilities, see [6,39]. Introduction to Judgment Aggregation 165 Table 3. The Condorcet paradox as a doctrinal paradox xPy yPz xPz yPx zPy zPx V1 = {x ≻1 y, y ≻1 z} 1 1 1 0 0 0 V2 = {y ≻2 z, z ≻2 x} 0 1 0 1 0 1 V3 = {z ≻3 x, x ≻3 y} 1 0 0 0 1 1 Majority 1 1 0 0 0 1 Condorcet paradox hides a much deeper problem that does not affect only the majority rule. The same question could be posed in judgment aggregation: is the doctrinal paradox only the surface of a more troublesome problem arising when individuals cast judgments on a given set of propositions? The answer to this question is positive and that was the starting point of the new theory of judgment aggregation. We will mention some recent work investigating the formal similarities and differences between preference and judgement aggregation in Section 5. It is now time to introduce some formal definitions. 1.2 Preliminary Notions In this section we introduce the three central notions underlying the formal theory of judgment aggregation: agendas, judgment sets and aggregation functions. Propositional Languages. We will work with the aggregation of judgments formulated in a standard propositional language: φ := p ∈ X | ¬φ | φ ∧ φ | φ ∨ φ | φ → φ where X is a set of atoms. Agendas and Individual Judgments. The following is the first key definition of the framework of judgment aggregation: Definition 1 (Judgment aggregation structure). Let L(X) be a propositional language on a given set of atoms X.4 A judgment aggregation structure for L is a tuple J = ⟨N,A⟩ where: – N is a non-empty set of agents; – A ⊆ L (the agenda) such that A = {φ | φ ∈ I} ∪ {¬φ | φ ∈ I} for some I ⊆ L (the set of issues) which contains only positive (i.e., non-negated) contingent5 formulae. An agenda based on a set of issues I will often be denoted ±I. In other words, the agenda is a set of formulae which is closed under complementation, i.e., ∀φ: φ ∈ A iff ¬φ ∈ A, and where double negations are eliminated so that each formula contains at most one negation. 4 We will often drop the reference to X when clear from the context. 5 I.e., which are neither a tautology nor a contradiction. 166 D. Grossi and G. Pigozzi Definition 2 (Judgment sets and profiles). Let J = ⟨N,A⟩ be a judgment aggregation structure. A judgment set for J is a set of formulae J ⊆ A such that: – J is consistent, i.e., it has a model; – J is maximal (or complete) w.r.t. A, i.e., ∀φ ∈ A, either φ ∈ J or ¬φ ∈ J . The set of all judgment sets is denoted J ⊆ ℘(X). A judgment profile P = ⟨Ji⟩i∈N is an |N |-tuple of judgment sets. We denote with P the set of all judgment profiles. That a formula φ follows from a judgment set J will be denoted J |= φ. For a φ in the agenda, the same notation will be often used interchangeably with φ ∈ J to indicate that φ belongs to J . Slightly abusing notation, we will often indicate that a judgment set Ji belongs to a profile P by writing Ji ∈ P . Remark 1 (Judgment sets as valuation functions). Judgment sets are consistent and maximal subsets of the agenda. As such, they can equivalently be viewed as functions J : A −→ {1, 0} preserving the meaning of propositional connectives. If the agenda A is closed under atoms, i.e., it contains all the atomic variables occurring in their formulae, then each judgment set J corresponds to a propositional valuation. More precisely, let X be the set of atoms occurring in the formulae in A. First of all, note that A ⊆ L(X) and each judgment set J corresponds to a function J : X −→ {1, 0}. Aggregation Functions. The aggregation of individual judgments into a collective one is viewed as a function: Definition 3 (Aggregation function). Let J be a judgment aggregation structure. An aggregation function for J is a function f : P −→ J. Notice that the function takes as domain the set of all possible judgment profiles—the so-called universal domain condition. Often, the collective judgment set f(P ) resulting from the aggregation of a profile P = ⟨Ji⟩i∈N via f is simply referred to as J . To substantiate our presentation, let us now give some concrete examples of ways of aggregating judgment profiles, which are arguably of common use, and which we call here aggregation procedures: Propositionwise Majority φ ∈ fmaj(P ) iff |{Ji ∈ P | φ ∈ Ji}| ≥ q, (1) with q = ⌈(|N |+ 1)/2⌉, where ⌈x⌉ is the smallest integer ≥ x. I.e., φ is collectively accepted iff there is a majority of voters accepting it. Introduction to Judgment Aggregation 167 Propositionwise Unanimity φ ∈ fu(P ) iff ∀i ∈ N,φ ∈ Ji (2) I.e., φ is collectively accepted iff all voters accept it. Premise-Based Procedure: In the premise-based procedure only the individual judgments on the premises are aggregated. The collective judgment on the conclusion is determined by entailment from the group decisions on the premises. Let us denote by Prem ⊆ A the subagenda containing the propositions that are premises and their complements, and by Conc ⊆ A the subagenda containing the conclusions and their complements, such that A = Prem ∪Conc. Following the literature, we assume that the aggregation rule is the above propositionwise majority. fpbp(P ) = fmaj(Prem) ∪ {φ ∈ Conc | fmaj(Prem) |= φ} (3) 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 collectively accepted premises. Conclusion-Based Procedure: In the conclusion-based procedure only the individual judgments on the conclusions are aggregated. This implies that there will be no group position on the premises. Again, we assume the aggregation rule is propositionwise majority. fcbp(P ) = fmaj(Conc) (4) I.e., φ is collectively accepted iff it is a conclusion and it has been voted by the majority of the individuals. Remark 2. Are the above aggregation procedures aggregation functions in the precise sense of Definition 3? A simple inspection of the definitions of the procedures will show that the answer is no. In the light of the doctrinal paradox and the discursive dilemma, it must have already been clear that propositionwise majority cannot be an aggregation function—and this is yet another way of ‘phrasing’ those paradoxes. It is, however, almost an aggregation function: it can either be viewed as a partial function from P to J, or as a function from P to the set of all possibly inconsistent judgment sets. Similar considerations can be made for the other procedures mentioned above. The discrepancy between these procedures and the ‘idealized’ notion of aggregation function can well be viewed as the symptom of some deep difficulty involved with the aggregation of individual opinions. What such a deep difficulty is will be investigated in detail in the next section. We conclude this section with one more variant of the doctrinal paradox: Example 2. Let A = ±{p, p → q, q}. In the literature this agenda is often associated with the following propositions [17]: 168 D. Grossi and G. Pigozzi p: Current CO2 emissions lead to global warming. p → q: If current CO2 emissions lead to global warming, then we should reduce CO2 emissions. q: We should reduce CO2 emissions. The profile consisting of the three judgment sets J1 = {p, p → q, q}, J2 = {p,¬(p → q),¬q} and J3 = {¬p, p → q,¬q}, once aggregated via propositionwise majority, gives rise to an inconsistent collective judgment set J = {p, p → q,¬q}. If we assume that Prem = {p, p → q} and Conc = {q}, we can also see the outcomes of the premise and of the conclusion-based procedures. Summarizing this into a table:6 p p → q q J1 1 1 1 J2 1 0 0 J3 0 1 0 fmaj 1 1 0 fpbp 1 1 1 fcbp 0 Notice that fu, propositionwise unanimity, yields the empty judgment set.