What Makes Majority Illusion Easy to Detect?

This paper studies strict majority reasoning in finite electorates using so-called $\textit{social decision frames}$: finite sets of voters equipped with distinguished families of coalitions interpreted as those voting blocs evaluated to form a strict majority. A coherence criterion for qualitative majority judgments is identified and shown to give an exact characterization for representability of strict majorities by finitely additive measures. In addition, a minimal natural logic for reasoning about strict majorities is shown to be sound and complete. These developments motivate examination of associated combinatorial questions concerning incoherence in finite families of sets; partial results and a conjecture are given. Finally, the results of this paper are applied to correct a classical representation theorem for weak qualitative probability structures due to Patrick Suppes and to establish a May-type characterization for ordinary strict majority rule for social decision frames.

In this paper, we study the detection of an echo chamber in a social network, i.e., the identification of a set of nodes that agree on a topic, while disagreeing with the rest of nodes. We argue that this problem is different from other social network analysis problems such as community detection, and from other graph problems such as maximum graph cut and maximum clique. To the best of our knowledge, we are the first to formalize the objective function of echo chamber detection, by using the theory of Fourier transforms of set functions (Stobbe and Krause, 2012). We propose scalable semidefinite relaxation, solved via an interior point method and sparse linear algebra. Experimentally, our algorithm recovers the ground truth echo chamber better than competing methods on small synthetic experiments. Our algorithm produces echo chambers with better network properties than competing methods on large real-world datasets. To independently validate our proposed objective function, we show that our algorithm finds echo chambers with more agreements with suspended users than competing methods on a small real-world dataset.

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.