Judgment Aggregation. Gabriella Pigozzi

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СКАЧАТЬ y and z to choose from, and three voters V₁, V₂ and V₃ whose preferences are the same as in Figure 1.2. The three voters’ preferences can then be represented by sets of preferential judgments as follows: V₁ = {x ≻ y, y ≻ z, x ≻ z}, V₂ = {y ≻ z, z ≻ x, y ≻ x} and V₃ = {z ≻ x, x ≻ y, z ≻ 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, z ≻ x}, which together with transitivity (P3) violates (P1) (Figure 1.6). Each voter’s preference is transitive, but transitivity fails to be mirrored at the collective level. This is an instance of the Condorcet paradox casted in the form of a set of judgments over preferences on alternatives.¹¹

      What the Condorcet paradox and the discursive dilemma have in common is that when we combine individual choices into a collective one, we may lose some rationality constraint that was satisfied at the individual level, like transitivity (in the case of preference aggregation) or logical consistency (in the case of judgment aggregation). A natural question is then how the theory of judgment aggregation and the theory of preference aggregation relate to one another. We can address this question in two ways: we can consider what the possible interpretations are 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 between judgments and preferences:

      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. [KS86, p. 85].¹²

      Figure 1.6: The Condorcet paradox as a doctrinal paradox.

      Regarding the formal relations between judgment and preference aggregation, Dietrich and List [DL07a] (extending earlier work by List and Pettit [LP04]) capitalize on the representation of the Condorcet paradox given in Figure 1.6 and show that Arrow’s theorem for strict and complete preferences can be derived from an impossibility result in judgment aggregation.¹³

      Despite these natural connections, and the formal results they support, Kornhauser and Sager [Kor92] notice that the two paradoxes do not perfectly match. 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. [LP04, pp. 216–217]

      Given the analogy between the two paradoxes, List and Pettit’s first question was whether an analogue of Arrow’s theorem could be found for the judgment aggregation problem. Arrow showed that the 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? As we shall see in more detail in Chapter 3, the answer to this question is positive and that can be seen as the starting point of the theory of judgment aggregation.

       How likely are majority cycles?

      Even from our brief survey, the reader may have guessed that large parts of the literature in social choice theory focused on the problem of majority cycles. We may wonder how likely such cycles are in reality. There are two main approaches to this question in the literature. One consists in analytically deriving the probability of a Condorcet paradox in an election, while the other looks at empirical evidence in actual elections. One assumption usually made in the first approach is the so-called impartial culture. According to the impartial culture, each preference ordering is equally possible. It should be noted that, even though it is a useful assumption for the analytic calculations, such an assumption has often been criticized as unrealistic. Niemi and Weisberg [NW68] showed that, under the impartial culture assumption and for a large number of voters, the probability of a majority cycle approaches 1 as the number of alternatives increases. However, they also found out that the probability of the paradox is quite insensitive to the number of voters but depends highly on the number of alternatives.

      Yet, these results are in contrast with the findings of the approach that looks at the actual elections.¹⁴ Mackie [Mac03], for example, claims that majority cycles never actually occurred in real elections. One way to explain such discrepancy is that we do not dispose of all the information needed to verify the occurrence of a majority cycle. For example, we typically do not dispose of the voter’s preference order over all the possible candidates.

1.3 FURTHER TOPICS

      The brief survey on social choice theory provided in this chapter has no pretense to be exhaustive. The aim was to give a background against which to frame the birth and development of judgment aggregation. For a broader but still concise introduction to social choice theory see [Lisce], and [Nur10, Pacds] for an introduction to voting theory. Moreover, the reader is referred to [RVW11] for a survey on preference reasoning from a perspective that brings together social choice theory and artificial intelligence. In particular, Chapter 4 of [RVW11] focuses on preference aggregation. There, several voting rules are defined, and manipulation and computational aspects are discussed. Manipulation in judgment aggregation is the object of a separate chapter in the present book (Chapter 5).

      If the traditional domain of social choice theory has been economics and the political sciences, attention in aggregation problems is witnessing a steady growth within the fields of artificial intelligence and multi-agent systems. Aggregation problems often appear in the design and specification of distributed intelligent systems and the very same idea of voting has been applied to problems like recommender systems [PHG00] and rank aggregation for the Web [DKNS01]. In particular, computational social choice [CELM07, BCE13], of which judgment aggregation can be seen as a contributing field, is the discipline stemmed from the interactions between computer science and social choice theory, and which studies, among other topics, the computational complexity of the application and manipulation of aggregation rules [EGP12],¹⁵ the design of aggregation rules based on knowledge representation techniques like merging [Pig06],¹⁶ or the application of logic to reason, within a formal language, about aggregation problems [AvdHW11].

      ¹Duncan Black (called the founder of social choice [Tul91] for being the first to really understand the work of Condorcet and discovering Dodgson’s papers) gave in [Bla58] an excellent СКАЧАТЬ