Minimal Effort to Consensus (MEC) polarization measure

2026-06-11

AI Summary

This work proposes a polarization measure grounded in the principle of Minimum Effort Consensus (MEC), defining group opinion polarization as the minimal weighted movement cost required to achieve consensus while simultaneously yielding an optimal endogenous consensus point. The approach employs a parametric family of weighted \(L^\beta\) transport costs—centered on the 1-Wasserstein distance when \(\alpha\) and \(\beta\) take canonical values—and satisfies established axioms of polarization measurement. It naturally gives rise to key interpretive principles, including “distance from consensus intensifies polarization,” the minority principle, and a critical threshold method. An efficient bisection algorithm computes the MEC measure in \(O(n \log(1/\varepsilon))\) time. Benchmarking against 60 expert-annotated cases shows that MEC[2,1.15] achieves a Kendall’s tau of 0.89, substantially outperforming the Van der Eijk and Tastle–Wierman indices and matching the performance of the best-tuned Esteban–Ray parameterization.

Abstract

We introduce the Minimum Effort to Consensus (MEC), a measure that quantifies polarization as resistance to consensus: a population is highly polarized when much effort is needed to bring its members to a common position, and weakly polarized when little is needed. Given an opinion distribution, MEC is the minimum effort required to turn it into a consensus distribution, taken over all consensus points, and it returns both a scalar value and an endogenous optimal consensus point. In the basic case MEC equals the 1-Wasserstein distance (Earth Mover's Distance) to the nearest consensus configuration, so that polarization becomes proximity to maximum disagreement. A two-parameter family with exponents alpha, beta >= 1 writes MEC as a weighted L^beta cost whose weights are the alpha-power of the group masses, recovering mean absolute deviation and variance-like dispersion as special cases and giving alpha and beta natural readings as identification and alienation. We prove a Shifting Away from Consensus principle, by which displacing a whole group's mass away from the optimal consensus point strictly increases polarization, and use it to show that MEC is maximized by the extremal distribution that splits the population equally between the two extremes, establishing that MEC is a polarization measure in the standard sense. We also obtain a Minority Principle and a Tipping Point method, showing that polarization is not monotone in extremism. MEC further satisfies the three axioms of Esteban and Ray with a central-split monotonicity property. Empirically, MEC[2,1.15] attains Kendall's tau near 0.89 against a sixty-expert benchmark, matching the strongest Esteban-Ray parametrization and outperforming the Van der Eijk and Tastle-Wierman measures, and it is computable by bisection in O(n log(1/epsilon)) time.

Problem

• polarization
• consensus
• opinion distribution
• MEC
• Wasserstein distance

Innovation

• Minimum Effort to Consensus
• polarization measure
• Wasserstein distance
• Esteban-Ray axioms
• computational efficiency