The Moderating Effect of Instant Runoff Voting

TL;DR

The paper formalizes IRV moderation within a 1-D $1$-Euclidean voter model, introducing probabilistic and combinatorial notions of moderation via exclusion zones. It proves that under uniform voters, IRV has a strict exclusion zone $I=[1/6,5/6]$ where the winner must lie as the candidate pool grows, while plurality loses such a constraint and converges to a Uniform$(0,1)$ winner distribution. The analysis extends to symmetric non-uniform voter distributions, identifying conditions under which IRV maintains moderation up to a polarization threshold, and provides exact finite-$k$ winner densities (notably $f_{P_3}$ and $f_{R_3}$) using a polyhedral framework linked to stick-breaking processes. Overall, the work offers a precise, geometry-based framework showing IRV can constrain extreme winners in many settings, while plurality does not, and it establishes exact asymptotics and finite-$k$ characterizations that enable direct comparison of winner distributions across voting rules. These results have implications for the theoretical understanding of moderation in voting systems and for informing policy discussions about ranked-choice voting in polarized electorates.

Abstract

Instant runoff voting (IRV) has recently gained popularity as an alternative to plurality voting for political elections, with advocates claiming a range of advantages, including that it produces more moderate winners than plurality and could thus help address polarization. However, there is little theoretical backing for this claim, with existing evidence focused on case studies and simulations. In this work, we prove that IRV has a moderating effect relative to plurality voting in a precise sense, developed in a 1-dimensional Euclidean model of voter preferences. We develop a theory of exclusion zones, derived from properties of the voter distribution, which serve to show how moderate and extreme candidates interact during IRV vote tabulation. The theory allows us to prove that if voters are symmetrically distributed and not too concentrated at the extremes, IRV cannot elect an extreme candidate over a moderate. In contrast, we show plurality can and validate our results computationally. Our methods provide new frameworks for the analysis of voting systems, deriving exact winner distributions geometrically and establishing a connection between plurality voting and stick-breaking processes.

Figures (7)

• Figure 1: Three example voter distributions in one dimension (all Betas). Candidates A, B, C, D are positioned at 0.2, 0.3, 0.4, and 0.85. The black line shows the density function of the voter distribution. Regions are colored according to the most preferred candidate of voters in that region and annotated with the approximate vote share of that candidate. As an example, the preference ordering of a voter at 0.5 is C, B, A, D (regardless of the voter distribution). Similarly, a voter at 0.1 has preference ordering A, B, C, D. In the moderate voters example (left), C is both the plurality and IRV winner. In the uniform voters example (center), D is the plurality winner and C is the IRV winner. In the polarized voters example (right), D is the plurality winner and A is the IRV winner.
• Figure 2: Visual depiction of the proof of \ref{['thm:1/6']}. IRV eliminates candidates until a final candidate $x$ remains in the exclusion zone $[1/6, 5/6]$. At this point, $x$ gets more than $1/3$ of the vote share and cannot be eliminated next (regardless of where they are in $[1/6, 5/6]$). Candidates outside of $[1/6, 5/6]$ are thus eliminated until $x$ wins.
• Figure 3: The distributions of the winning position with $k=3, 4, 5,$ and $100$ candidates and continuous 1-Euclidean voters (both uniformly distributed) under plurality and IRV. The histograms are from 1 million simulation trials for $k=3, 4, 5$ and 100000 trials for $k=100$, while the curves plotted for $k=3$ (shown up to 1/2) are the exact density functions given in \ref{['thm:plurality-dsn', 'thm:irv-dsn']}, with pieces separated by color. Note that the IRV winner is only at a position $<1/6$ or $>5/6$ when no candidates fall in $[1/6, 5/6]$ by \ref{['thm:1/6']}; the dashed vertical lines outline this exclusion zone. The probabilistic moderating effect for IRV is already strong at with only $k=5$ candidates.
• Figure 4: IRV (top) and plurality (bottom) winner positions with Beta$(\alpha, \alpha)$-distributed voters and candidates. The violin plots show empirical distributions from 100,000 simulation trials with $k=30$ candidates at each $\alpha$ value, with whiskers marking extrema. The dashed lines show the bounds from \ref{['thm:very-polarized-voters-exclusion', 'thm:polarized-voters-exclusion', 'thm:moderate-voters-exclusion']} in the annotated ranges. As long as voters are not too polarized, IRV prevents extreme candidates from winning. Plurality, on the other hand, allows arbitrarily extreme candidates to win for $\alpha = 1$, when the voter distribution is uniform.
• Figure 5: Plurality vs. IRV winner positions in 100,000 simulation trials for increasing candidate count $k$ (with uniform voters and candidates). Blue points are trials where the IRV winner was more moderate than the plurality winner, while red points are trials where the plurality winner was more moderate. Green points are trials where the winners were identical. Numbers in each quadrant show the proportion of trials falling in that region (the top right number is the proportion of same-winner trials). Notice that cases where the IRV winner is more extreme only appear beginning at $k=5$, in accordance with \ref{['thm:small-k-not-more-extreme']}. Note the probabilistic moderating effect of IRV compared to plurality: IRV does not elect extreme candidates as $k$ grows large, but plurality does.

Theorems & Definitions (26)

• Theorem 1
• proof
• Corollary 1
• Theorem 2
• proof
• Theorem 3
• Lemma 1
• Proposition 1
• Proposition 2
• Theorem 4