Exclusion Zones of Instant Runoff Voting

Abstract

Recent research on instant runoff voting (IRV) shows that it exhibits a striking property over unimodal one-dimensional preferences: there is an exclusion zone around the median voter such that the winner must come from the exclusion zone, unless no such candidate exists. Thus, IRV cannot elect an extreme candidate in this setting as long as a sufficiently moderate candidate runs. 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 such 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 have nontrivial exclusion zones. As a further exploration, we study IRV exclusion zones with graph-based preferences, 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 in graphs. Finally, we report on computational experiments with exclusion zones: (i) performing an exhaustive computer search of small graphs and trees, we find nontrivial IRV exclusion zones in most small graphs; and (ii) applying our approximation algorithm to a collection of larger real-world school friendship networks, we find that about 60% of these networks have probable nontrivial IRV exclusion zones. While our focus is on IRV, the properties of exclusion zones we establish provide new methods for analyzing voting systems in metric spaces more generally.