Iterative Judgment Aggregation

The best positive case is a high-multiplicity model of a large deliberative population. Fix an agenda \(\mathcal A\) of logically related binary issues and a consistency constraint \(\Gamma\). A type is a complete rational judgment \(J\in\mathcal J_{\mathcal A,\Gamma}\), not merely a partial opinion: agents of the same type have the same answers, revision options, and costs. The society is a distribution \(\mu\) over these types, where \(\mu_J\) is the fraction holding \(J\). Each individual judgment remains discrete and rational; only the population is continuous.

A plausible regime is a national consultation, large professional association, or online policy deliberation involving thousands or millions of participants but a moderate number of recurring judgment types. The institutional agenda is fixed, and many participants genuinely have identical complete positions. Thus \(N\gg |\operatorname{supp}(\mu)|\), with each type potentially represented by a large cohort. The scenario is not the paper’s small-agent story verbatim, but it is a natural high-multiplicity version of its underlying judgment-aggregation problem.

The action is the same as in the paper: an infinitesimal mass of type \(J\) may move to an adjacent rational judgment \(J'\) in the relevant agenda graph, provided \(J'\) lies in the convex hull of the current support and reduces its aggregate distance to the rest of society. For a graph distance \(d\), define \(D_\mu(K)=\sum_L\mu_L d(K,L)\). The best available moves from \(J\) minimize \(D_\mu(K)\) subject to \(d(J,K)=1\), rationality, and hull membership. A population update is therefore a transport kernel sending each type’s mass either to itself or to one of these improving neighboring types. Consensus means that the resulting distribution is a point mass \(\delta_J\).

My lead mirror is Continuous Model-Graph Move Computation. Its instance consists of \((\mathcal A,\Gamma,\mu,J)\), with rational masses \(\mu_L\) given explicitly for the occupied types. Using the model graph \(G^m_{\mathcal A,\Gamma}\), the question is whether type \(J\) has an admissible move, or, more strongly, to output the complete set of best improving destinations \(J'\). This is precisely the paper’s move-selection question with the profile list replaced by its empirical distribution.

The anchor is Theorem 18, proved in this paper: “For \(G^m_{\mathcal A,\Gamma}\), determining \(\mathit{moves}\) of a single iteration of \(\alg\) is coNP-hard.” The paper explains that adjacency in the model graph requires checking that no rational judgment lies between two given judgments; this nonexistence test is coNP-hard, and it must be performed for candidate moves at every iteration. The continuous formulation preserves exactly that difficulty. Given any discrete profile \(P\), set \(\mu_J=n_J/n\), where \(n_J\) is the number of agents holding \(J\). All distance comparisons and hull computations are then the corresponding weighted comparisons. Multiplying every \(n_J\) by a large common factor even gives an honestly high-multiplicity instance without changing the decision. I would therefore expect this mirror to be Class B: the hardness lives in the agenda’s rational-judgment geometry, not in named individuals or population multiplicity.

This is not a softened version of the authors’ problem. The agenda, logical constraint, model distance, convex hull, local adjacency, and individual improvement criterion are unchanged. Only the profile representation has been continuized. The natural follow-up questions are whether exact move computation remains coNP-hard under restricted constraints, whether approximation of \(D_\mu\) helps, and whether column-generation or separation methods can handle special classes of \(\Gamma\).

A second, genuinely positive anchor is Continuous Hamming-Graph Move Computation. Its input is the same kind of distribution \(\mu\), but moves use the Hamming graph \(G^h_{\mathcal A,\Gamma}\). The task is to compute, for every occupied type \(J\), its set of best improving one-issue moves and hence one valid updated distribution \(\mu'\), under a specified tie-breaking rule. This is a fully specified one-round computational problem; it does not ask us to pretend that a continuum automatically guarantees eventual consensus.

The anchor is Proposition 19, proved in this paper: “For \(G^h_{\mathcal A,\Gamma}\), a single iteration of \(\alg\) runs in deterministic polynomial time.” The proof observes that only at most \(|\mathcal A|\) candidate judgments need be checked for each current judgment. In the continuous version, each weighted distance \(D_\mu(K)\) is computable by summing over the support, and Hamming-hull membership is checkable coordinatewise. Thus the move set for a support of size \(s\) can be computed in time polynomial in \(s\), \(|\mathcal A|\), the encoding of \(\Gamma\), and the bit length of the masses. I would expect this to remain Class A.

This paper therefore supports a useful two-sided mirror: model-graph iteration gives a high-multiplicity problem whose hardness should transfer, while Hamming-graph iteration gives a high-multiplicity problem with a direct polynomial algorithm. I would not claim that Theorem 14 or Theorem 21 automatically transfer: their hypotheses use an odd finite number of agents and probabilistic convergence, whereas a continuum has no parity and may converge only asymptotically. That is the weakest point of the case. But it does not undermine the two stated mirrors, both of which are exact one-iteration computational questions and retain the paper’s central objects.