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

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href="#ulink_f7ac3d5e-3db5-5e8d-a4fd-97b98661c099">Figure 1.5. The difference with Figure 1.4 is that here the legal doctrine has been added to the set of issues on which the judges have to vote. Now the discursive dilemma is characterized by the fact that the group reaches an inconsistent decision, like {p, q, (p ∧ q) ↔ r, ¬r}. The court would accept the legal doctrine, give a positive judgment on both premises p and q but, at the same time, reach a negative opinion on the conclusion r. Clearly, such a position is untenable, as it would amount to release the defendant while saying, at the same time, that the two conditions for the defendant’s liability applied.

      What are the consequences of the reconstruction given in Figure 1.5? Mongin and Dietrich [MD10, Mon11] have investigated such reformulation 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. [Mon11, p. 2]

      Indeed, instead of premises and conclusions, List and Pettit chose to address the problem in terms of judgment sets, i.e., the sets of propositions accepted by the individual voters. The theory of judgment aggregation becomes then a formal investigation on the conditions under which consistent individual judgment sets may collapse into an inconsistent collective judgment set.

      Exactly like Arrow’s theorem showed the full import of the Condorcet paradox, so showed the result of List and Pettit how far-reaching the doctrinal paradox and the discursive dilemma are. In the next section we will look at how the Condorcet paradox relates to these two paradoxes of the aggregation of judgments.

      Let us start by introducing some formal notation. Let X be a set of alternatives, and ≻ a binary predicate for a binary relation over X, where x ≻ y means “x is strictly preferable to y.” The desired properties of preference relations viewed as strict linear orders are:

      Example 1.1 Condorcet paradox as a doctrinal paradox Suppose there are three possible alternatives x, 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

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