The Measurable Majority

This theoretical note studies the finite axiomatizability of strict majority reasoning in finite social decision frames. Moss and Pedersen (2026) <doi: 10.48550/arXiv.2606.23853> introduce a coherence criterion that characterizes exactly when qualitative majority judgments are representable by a finitely additive measure. The question addressed here is whether that coherence criterion can be replaced, in the finite setting, by any bounded finite fragment. We prove that it cannot. For every $k\ge 1$, we construct a maximal standard frame whose shortest coherence violation has length exactly $2k+2$. Hence there is no uniform finite bound on the incoherence index of social decision frames, resolving Conjecture 5.7 stated by Moss and Pedersen (2026). The construction is geometric, in the sense that it proceeds via orthogonality and dimension in rational vector spaces, and self-contained: it isolates a symmetric family of half-sized voting blocs and extends it to a maximal frame in which every shorter balanced obstruction is excluded. Along the explicit infinite sequence of universe sizes obtained in the construction, this also establishes the middle-layer family predicted by Conjecture B.25 by Moss and Pedersen (2026). Together with the soundness and completeness theorem for the Moss-Pedersen minimal logic for strict majorities, this establishes that measurable social decision frames are not finitely axiomatizable in that language.

Majority illusion is an undesirable phenomenon in social networks in which agents incorrectly perceive a minority opinion as dominant. This can severely distort collective behavior and decision-making. We study the fundamental question of detecting whether a social network allows for a majority illusion. Formally, in the $q$-Majority Illusion problem, we ask whether there exists a binary labeling of agents in which at least a $q$-fraction of agents have the majority of neighbors with the minority label. We investigate how various structural properties of the underlying social network influence the tractability of this question, and provide a detailed map of its computational complexity.

Majority voting is one of the few black-box interventions that can improve a fixed stochastic predictor: repeated access can be cheaper than changing a high-capability model. Classical fixed-competence theory makes this intervention look monotone -- more votes help above the majority threshold and hurt below it. We show that this picture is fundamentally incomplete. Under the de Finetti representation for exchangeable repeated correctness, voting is governed by a latent distribution of per-example correctness probabilities. Even simple latent mixtures can generate sharply different voting curves, including nonmonotone behavior and, in an explicit construction, infinitely many trend changes. The full latent law determines the curve, but the curve does not determine the law. The exact object recovered by voting is a signed voting signature: at each binomial variance scale, it records excess latent mass above rather than below the majority threshold. Our main theorem proves that the complete odd-budget curve and this signature are equivalent: the curve increments are signed Hausdorff moments, and the full curve recovers the signature uniquely. This viewpoint explains shape phenomena, branch-symmetric nonidentifiability, realizability, variation, and endpoint rates. It also separates estimation regimes: direct per-example success-probability information targets the full signature, whereas fixed-depth grouped labels reveal only a finite prefix.