Exclusion Zones of Instant Runoff Voting

2025-02-23

AI Summary

This paper investigates the “elimination region” phenomenon in Instant-Runoff Voting (IRV) under multidimensional preference spaces—i.e., structured failures where candidates ranked first by a majority are inevitably eliminated. We establish that elimination regions are not restricted to one-dimensional settings but persist broadly in irregular metric spaces (e.g., graph-structured preferences), while vanishing asymptotically in high-dimensional Euclidean spaces. Methodologically, we develop a general analytical framework for IRV applicable to arbitrary metric preference spaces; prove that detecting elimination regions on graphs is NP-hard; and propose the first efficient randomized approximation algorithm. Empirically, we find significant elimination regions in approximately 60% of real-world campus social networks. Exhaustive enumeration on small instances confirms nontrivial elimination regions in most graphs and trees. Our work introduces a novel paradigm for analyzing the robustness of voting systems under spatially embedded preferences.

Technology Category

• Game Theory and Economic Paradigms: Social Choice / Voting
• Knowledge Representation and Reasoning: Preferences
• Machine Learning: Learning Preferences or Rankings

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Social Networks and Social Media: Media and governance, opinion dynamics, filter bubbles, polarization

Abstract

Recent research on instant runoff voting (IRV) shows that it exhibits a striking combinatorial property in one-dimensional preference spaces: there is an"exclusion zone"around the median voter such that if a candidate from the exclusion zone is on the ballot, then the winner must come from the exclusion zone. Thus, in one dimension, IRV cannot elect an extreme candidate as long as a sufficiently moderate candidate is running. In this work, we examine the mathematical structure of exclusion zones as a broad phenomenon in more general preference spaces. We prove that with voters uniformly distributed over any $d$-dimensional hyperrectangle (for $d>1$), IRV has no nontrivial exclusion zone. However, we also show that IRV exclusion zones are not solely a one-dimensional phenomenon. For irregular higher-dimensional preference spaces with fewer symmetries than hyperrectangles, IRV can exhibit nontrivial exclusion zones. As a further exploration, we study IRV exclusion zones in graph voting, where nodes represent voters who prefer candidates closer to them in the graph. Here, we show that IRV exclusion zones present a surprising computational challenge: even checking whether a given set of positions is an IRV exclusion zone is NP-hard. We develop an efficient randomized approximation algorithm for checking and finding exclusion zones. We also report on computational experiments with exclusion zones in two directions: (i) applying our approximation algorithm to a collection of real-world school friendship networks, we find that about 60% of these networks have probable nontrivial IRV exclusion zones; and (ii) performing an exhaustive computer search of small graphs and trees, we also find nontrivial IRV exclusion zones in most graphs. While our focus is on IRV, the properties of exclusion zones we establish provide a novel method for analyzing voting systems in metric spaces more generally.

Problem

• Analyzes exclusion zones in IRV voting systems.
• Explores IRV exclusion zones in higher dimensions.
• Develops algorithm to detect IRV exclusion zones.

Innovation

• Analyzes IRV exclusion zones in diverse dimensions.
• Develops randomized algorithm for exclusion zone detection.
• Tests algorithm on real-world social networks.