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

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thus appeared that social welfare must be based on just the n-tuple of ordinal interpersonally non-comparable, individual utilities. […] This “informational crisis” is important to bear in mind in understanding the form that the origin of modern social choice theory took. In fact, with the binary relation of preference replacing the utility function as the primitive of consumer theory, it made sense to characterise the exercise as one of deriving a social preference ordering R from the n-tuple of individual orderings {Ri} of social states. [Sen86, p. 1074]

      Figure 1.3: An example of Dodgson’s rule.

      The need for functions of social welfare defined over all the alternative social states was made explicit by Abram Bergson [Ber38, Ber66] and Paul Samuelson [Sam47]. Economists turned to the mathematical approach to elections explored by Condorcet, Borda, and Dodsgon only when—following the informational restriction decreed by Robbins—they searched for methods to aggregate binary relations of preference into a social preference ordering. Thus, social choice theory stemmed from two distinct problems—how to select the winning candidate in an election, and how to define social welfare—and the relations between these problems became clear only in the 1950s.

      Young economist and future Nobel prize winner, Kenneth Arrow defined a social welfare function as a function that maps any n-tuple of individual preference orders to a collective preference order. His axiomatic method outlined the requirements that any desirable social welfare function should satisfy.5 In 1950 he proved what still is the major result of social choice, the “General Possibility Theorem,” now better known as Arrow’s impossibly result6 [Arr50, Arr63]. The theorem shows that there exists no social welfare function that satisfies only just a small number of desirable conditions.

      Let us informally present these conditions: the first is that a social welfare function must have a universal domain, that is, it has to accept as input any combination of individual preference orders. Another commonly accepted requirement is the Pareto condition, which states that, whenever all members of a society rank alternative x above alternative y, then the society must also prefer x to y. The independence of irrelevant alternatives condition states that the social preference over any two alternatives x and y must depend only on the individual preferences over those alternatives x and y (and not on other—irrelevant—alternatives).7 Finally, non-dictatorship requires that there exists no individual in the society such that, for any domain of the social welfare function, the collective preference is the same as that individual’s preference (i.e., the dictator). Arrow’s celebrated result shows that no social welfare function can jointly satisfy these conditions.8

      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

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