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The Moderating Effect of Instant Runoff Voting
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Proves that instant runoff voting (IRV) has a moderating effect relative to plurality in a 1-D Euclidean model, via exclusion zones derived from voter distributions, and derives exact winner distributions for small k.
Problem
Whether IRV produces more moderate winners than plurality, a claim made by advocates but previously supported only by case studies, simulations, and narrow theory, in a 1-D Euclidean voter model.
Approach
Defines exclusion zones (intervals whose candidate presence precludes outside winners) and shows IRV has them under uniform and many symmetric voter distributions while plurality has none; uses a stick-breaking connection to analyze plurality winner distributions and geometric polytope integration for exact winner densities.
Key results
- Under uniform voters and candidates, IRV winner lies outside [1/6, 5/6] with probability going to 0 as k grows, while plurality winner position converges in distribution to Uniform(0,1).
- For k=3 uniform voters, Var(plurality winner)=23/540 and Var(IRV winner)=25/864, so plurality variance is 184/125 times IRV's.
- For k<=4 with symmetric 1-Euclidean voters, the IRV winner cannot be more extreme than the plurality winner; for k>=5 it can be.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use IRV instead of plurality | Uniform voters and candidates · Uniform voters and candidates, k→∞ | winner-position moderation: [1/6, 5/6] versus Uniform(0, 1) | Helped |
| Use IRV instead of plurality | Uniform voters and any candidate configuration · Uniform voters, any candidate distribution | exclusion zone: [1/6, 5/6] versus no exclusion zones | Helped |
| Use IRV instead of plurality | Uniform voters and candidates, k=3 · Uniform voters and candidates, k=3 | winner-position variance: 23/540 versus 25/864 | Helped |
| Use IRV with symmetric voter distributions that are moderate or polarized below the stated threshold | IRV · Symmetric non-uniform voter distributions | exclusion zone: c ≤ F^{-1}(1/6); c ≤ 2(F^{-1}(1/3) - 1/4) | Helped |
| Use IRV with hyper-polarized voters | IRV · Symmetric voters with F(1/4) > 1/3 and candidates at both extremes | winner location: c ≥ 2 F^{-1}(1/3) | Hurt |
| Compare IRV with plurality for small versus larger candidate sets | IRV · Symmetric 1-Euclidean voters, k≤4 versus k≥5 | relative winner extremity: k ≤ 4; k ≥ 5 | Mixed |
Surprising findings
- Plurality has no combinatorial moderating effect for any non-pathological continuous voter distribution, while IRV's exclusion zone flips under hyper-polarized voter distributions.
Limitations
- Analysis assumes fixed candidate positions or candidates drawn from a distribution, with no strategic candidate behavior.
- Restricted to one-dimensional symmetric 1-Euclidean voter model with full rankings.
- Exact winner distributions derived only for k=3 and the tail for general k; general-k derivation is computationally hard.
Experimental setup
Metrics: winner position distribution probability winner lies in exclusion zone variance of winner position
Baselines: plurality voting
Builds on: stick-breaking processes 1-Euclidean (Hotelling) model single-peaked preferences results
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