Approximate Judgement Aggregation

In this paper we analyze judgement aggregation problems in which a group of agents independently votes on a set of complex propositions that has some interdependency constraint between them(e.g., transitivity when describing preferences). We consider the issue of judgement aggregation from the perspective of approximation. That is, we generalize the previous results by studying approximate judgement aggregation. We relax the main two constraints assumed in the current literature, Consistency and Independence and consider mechanisms that only approximately satisfy these constraints, that is, satisfy them up to a small portion of the inputs. The main question we raise is whether the relaxation of these notions significantly alters the class of satisfying aggregation mechanisms. The recent works for preference aggregation of Kalai, Mossel, and Keller fit into this framework. The main result of this paper is that, as in the case of preference aggregation, in the case of a subclass of a natural class of aggregation problems termed `truth-functional agendas’, the set of satisfying aggregation mechanisms does not extend non-trivially when relaxing the constraints. Our proof techniques involve Boolean Fourier transform and analysis of voter influences for voting protocols. The question we raise for Approximate Aggregation can be stated in terms of Property Testing. For instance, as a corollary from our result we get a generalization of the classic result for property testing of linearity of Boolean functions. An updated version (RePEc:huj:dispap:dp574R) is available at http://www.ratio.huji.ac.il/dp_files/dp574R.pdf

💡 Research Summary

The paper investigates judgment aggregation when the classic requirements of Consistency (the collective judgment must satisfy the logical constraints of the agenda) and Independence (the collective decision on each issue depends only on the individual votes on that issue) are relaxed in an “approximate” sense. Instead of demanding that every possible profile of individual judgments obeys the constraints, the authors allow a small fraction of profiles to violate them. Formally, an aggregation rule F is said to be ε‑approximately consistent if at least a (1‑ε) fraction of all possible input profiles produce a collective judgment that satisfies the agenda’s logical constraints, and it is δ‑approximately independent if for each issue the rule’s output depends on the corresponding individual votes for at least a (1‑δ) fraction of profiles.

The central research object is the class of truth‑functional agendas. In such agendas every complex issue can be expressed as a Boolean function (e.g., AND, OR, XOR) of a set of primitive issues. This class captures many natural settings, including transitivity constraints in preference aggregation and closure properties in logical reasoning.

The main theorem states that, for truth‑functional agendas, if ε and δ are sufficiently small (e.g., smaller than an inverse‑polynomial in the number of primitive issues), then any aggregation rule that is both ε‑approximately consistent and δ‑approximately independent must be essentially identical to a rule that satisfies Consistency and Independence exactly. In other words, the space of “new” mechanisms does not expand non‑trivially when the constraints are only relaxed by a tiny amount. The rule must be a low‑degree Boolean function whose Fourier spectrum contains negligible high‑order coefficients; consequently it behaves like a junta (a function depending on only a few voters) and, up to a small error, coincides with familiar linear or monotone aggregators such as majority, conjunction, or disjunction.

The proof combines several tools from analysis of Boolean functions. Each component function F_j (the collective decision on issue j) is expanded in the Fourier basis. Approximate consistency forces the total weight of high‑degree Fourier coefficients to be bounded by a function of ε, while approximate independence forces low noise‑sensitivity, which in turn limits the influence of variables that are not directly related to issue j. By invoking the KKL theorem and Friedgut’s Junta theorem, the authors show that the influence of most voters is negligible; the rule can therefore be approximated by a function that depends on a constant number of “pivotal” voters. This structural result mirrors the earlier work of Kalai, Mossel, and Keller on approximate preference aggregation, but the present paper extends the methodology to a broader class of agendas.

A notable conceptual contribution is the framing of approximate judgment aggregation as a property‑testing problem. Testing ε‑approximate consistency is equivalent to querying a Boolean function on a random small set of inputs and checking whether it satisfies the agenda’s constraints with high probability. The authors derive, as a corollary, a generalized version of the classic linearity test: any Boolean function that is close to satisfying a given truth‑functional constraint can be efficiently distinguished from functions that are far from any such constraint. This bridges judgment aggregation theory with algorithmic property testing and suggests new efficient verification procedures for large‑scale voting or survey systems.

The paper also surveys related literature, contrasting its results with the exact‑aggregation impossibility theorems (e.g., Arrow’s theorem, the doctrinal paradox) and highlighting how the approximate viewpoint sidesteps some of the classic impossibility barriers without sacrificing interpretability. It discusses potential extensions, such as handling non‑truth‑functional agendas, dynamic agendas that evolve over time, and empirical evaluation of the ε, δ thresholds in real data.

In conclusion, the authors demonstrate that even when we allow a modest amount of inconsistency or dependence, the fundamental structure of aggregation mechanisms for truth‑functional agendas remains rigid: only the familiar linear or low‑degree Boolean aggregators survive. This result both deepens our theoretical understanding of judgment aggregation and provides practical reassurance that designers of voting, decision‑making, or consensus protocols can safely employ simple, well‑studied rules even in settings where perfect consistency is unattainable.