Geometric Comparisons of Electoral Rules Under Feedback

2026-04-21

AI Summary

This study investigates how electoral rules influence political polarization in the context of dynamic adaptation by voters and candidates. It introduces two geometric metrics—the "winner radius" and the "supporter centroid radius"—to construct the first geometry-based framework for comparing electoral systems. Through theoretical analysis and large-scale simulations encompassing seven prominent voting rules and a convex combination benchmark, the work demonstrates that proximity of the winner to the median voter does not necessarily reduce polarization. The findings reveal a fundamental trade-off: reducing the winner radius mitigates voter polarization but increases the divergence between candidates and their support bases, and vice versa. This tension underscores that minimizing the winner radius and minimizing voter disagreement entail markedly different long-term dynamic consequences.

Technology Category

• Game Theory and Economic Paradigms: Social Choice / Voting
• Humans and AI: Voting
• Machine Learning: Learning Preferences or Rankings

Application Category

• Social Networks and Social Media: Media and governance, opinion dynamics, filter bubbles, polarization
• Economics, Online Markets and Human Computation: Economics and fairness of platforms and recommendation systems
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

Abstract

We study how electoral rules shape polarization dynamics when voters and candidates both adapt to repeated election outcomes. We introduce two geometric primitives for comparing rules under this feedback: the \emph{winner radius} $R_t = \max_i \|x_i - w^{(t)}\|$, the distance from the winner to the farthest voter, and the \emph{supporter centroid radius} $S_t = \max_j \|c_j - s_j^{(t)}\|$, the largest gap between any candidate and their support base. We show that $R_t$ controls a one-step contraction bound on voter disagreement and $S_t$ plays the analogous role for candidate dispersion, and that these two objectives are in tension. Rules that reduce $R_t$ tend to increase $S_t$, and vice versa. A winner close to the voter median does not resolve the tension, since proximity to the median and proximity to the Chebyshev center are different objectives. We use this framing to organize a simulation study across seven standard electoral rules and one convex-combination benchmark, comprising 1000+ runs across diverse electorate profiles, voter mechanisms, and camp-balance settings. The empirical results confirm the theoretical tradeoff: winner-take-all rules achieve small $S_t$ at the cost of large $R_t$ and weaker voter depolarization, while convex-combination rules reverse this. An oracle comparison further shows that minimizing $R_t$ per step and minimizing voter disagreement per step are distinct objectives with different long-run consequences for both voter and candidate dynamics.

Problem

• electoral rules
• polarization dynamics
• feedback adaptation
• geometric comparison
• candidate-voter alignment

Innovation

• winner radius
• supporter centroid radius
• polarization dynamics
• electoral rules
• geometric primitives