Judgment Aggregation. Gabriella Pigozzi

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      We have seen that, thanks to the Condorcet Jury Theorem, majority rule enjoys an attractive property: some conditions being satisfied, groups make better decisions than individuals. Yet, unfortunately, the Condorcet paradox also showed that this same rule is unable to ensure consistent social positions under all situations.

      Classical social choice theoretic models focus on the aggregation of individual preferences into collective outcomes. Such models focus primarily on collective choices between alternative outcomes such as candidates, policies or actions. However, they do not capture decision problems in which a group has to form collectively endorsed beliefs or judgments on logically interconnected propositions. Such decision problems arise, for example, in expert panels, assemblies and decision-making bodies as well as artificial agents and distributed processes, seeking to aggregate diverse individual beliefs, judgments or viewpoints into a coherent collective opinion. Judgment aggregation fills this gap by extending earlier approaches developed by social choice theory for the aggregation of preferences.⁹

       Doctrinal paradox

      Judgment aggregation has its roots in jurisprudence. The paradox of a group of rational individuals collapsing into collective inconsistency made its first appearance in the legal literature, where constitutional courts are expected to provide reasons for their decisions. The discovery of the paradox was attributed to Kornhauser and Sager’s 1986 paper [KS86]. However, Elster recently pointed out that structurally similar problems have been first indicated by Poisson in 1837 [Els13]. What is now known as the doctrinal paradox [KS93, Kor92, Cha98] was rediscovered in 1921 by the Italian legal theorist Vacca [Vac21] (see [Spe09]), who consequently raised severe criticisms to the possibility of deriving collective judgments from individual opinions. The logical problem of aggregation was also noticed by Guilbaud [Gui52, Mon05], who gave a logical interpretation to preference aggregation.

      Figure 1.4: An illustration of the doctrinal paradox.

      In order to illustrate the doctrinal paradox, we recall the familiar example in the literature by Kornhauser and Sager [KS93]. 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 Figure 1.4.

      The court can rule on the case either directly, by taking the majority vote on the conclusion r regardless of how the judges voted on the premises (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).¹⁰ The problem is that the court’s decision depends on the procedure adopted. In this specific example, 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. [KS93, p. 12]

      Legal theorists have discussed both methods and have taken different positions about them, either by arguing for the superiority of one of the approaches or by questioning both and recommending a third way (see Nash [Nas03] for an overview of the proposed solutions). In particular, Kornhauser and Sager argue against the use of a uniform voting protocol and favor instead a context-sensitive approach, where courts choose the method on a case-by-case basis, by voting on the method to be applied.

       Discursive dilemma

      Judgment aggregation has provided a systematic account of situations like the one arising in Figure 1.4. The first step was made by the political philosopher Pettit [Pet01], who recognized that the paradox illustrates a more general problem than just an impasse in a court decision. Pettit introduced the term discursive dilemma to indicate any group decision in which the aggregation on the individual judgments depends on the chosen aggregation method, like the premise-based and the conclusion-based procedures.

      Figure 1.5: The discursive dilemma.

      Then, List and Pettit [LP04] reconstructed Kornhauser and Sager’s example as shown in 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.

1.2.2 PREFERENCE AGGREGATION AND JUDGMENT AGGREGATION

      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 СКАЧАТЬ