Exclusion Zones of Instant Runoff Voting
Kiran Tomlinson, Johan Ugander, Jon Kleinberg
TL;DR
This work generalizes the notion of IRV exclusion zones from one-dimensional voter spaces to higher-dimensional metric spaces and graphs. It proves that for voters uniformly distributed over a $d$-dimensional hyperrectangle with $d>1$, IRV has no nontrivial exclusion zones under $L_1$ or $L_2$ metrics, though irregular higher-dimensional shapes can exhibit nontrivial zones. A key methodological contribution is the Condorcet Chain Lemma, which organizes configurations to certify nonexistence of nontrivial zones, and a randomized approximation algorithm for identifying approximate exclusion zones in graphs, supported by fixed-parameter tractability results. The paper also establishes computational hardness results (co-NP-complete for exclusion verification and NP-hard for minimal exclusion-zone computation) and provides empirical evidence from real-world school networks that exclusion zones are common and typically large. Overall, exclusion zones offer a robust lens for analyzing voting systems across diverse metric spaces beyond the classic one-dimensional setting.
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.