Instant Runoff Voting and the Reinforcement Paradox

1. Problem Statement

The paper investigates the susceptibility of Instant Runoff Voting (IRV), also known as Ranked-Choice Voting (RCV), to the Reinforcement Paradox (also known as the Consistency Paradox or Multiple Districts Paradox).

• Definition: A voting method suffers from the reinforcement paradox if a candidate $X$ wins in two separate elections (sub-elections $B_1$ and $B_2$) but loses when the ballots from both elections are merged ($B_1 \cup B_2$).
• Context: Unlike monotonicity paradoxes (where changing a ballot to rank a winner higher causes them to lose), the reinforcement paradox has direct administrative implications. IRV requires the transmission of full ballot rankings to a central location to determine a winner, whereas plurality voting only requires vote totals. If IRV is highly susceptible to this paradox, it undermines the logic of aggregating results from distinct precincts or districts.
• Gap in Literature: Previous studies (e.g., Plassmann & Tideman, Courtin et al.) estimated the frequency of this paradox by generating two independent random elections and checking if their merged result differed. This paper adopts a different approach: starting with a single election and determining if its ballot set can be partitioned into two subsets where the same losing candidate wins both subsets.

2. Methodology

The authors employ a three-pronged methodology combining algebraic characterization, theoretical simulation, and empirical analysis.

A. Algebraic Characterization (Three-Candidate Case)

The authors derive necessary and sufficient conditions for a three-candidate IRV election to admit a partition that creates a reinforcement paradox.

• Variables: They define vote counts for complete and partial rankings (e.g., $X_1, X_2$ for voters ranking Candidate A first, then B or C).
• Propositions:
  • Proposition 1 (Complete Ballots): Provides inequalities determining when a Runner-Up (the candidate who loses in the final round) or a Plurality Loser (the candidate eliminated first) can win both sub-elections.
  • Proposition 2 (Partial Ballots): Extends these conditions to elections allowing truncated ballots (common in real-world IRV).
• Key Insight: For a paradox to occur, the order of elimination in the two sub-elections must differ. The authors prove that if the elimination order is identical in both sub-elections, the winner of the merged election must be the same as the sub-election winners.

B. Theoretical Simulation (Monte Carlo)

The authors estimate the probability of these paradoxes occurring under various voter behavior models using 100,000 simulated elections (with $V=1001$ voters).

• Models Used:
  • Impartial Anonymous Culture (IAC): Uniform distribution over the ballot simplex ($\alpha=1$).
  • Dirichlet Models ($\alpha \neq 1$): Varying levels of competition. Low $\alpha$ favors landslide victories; high $\alpha$ favors competitive elections.
  • Spatial Models: One-dimensional spatial models (Normal, Bimodal, and Bimodal Weighted) to simulate political polarization and the "center squeeze" effect.
• Limiting Case: They also calculate exact limiting probabilities as $V \to \infty$ using the software Normaliz for the IAC model.

C. Empirical Analysis

The authors analyze a dataset of 416 real-world single-winner IRV elections from the US (Alaska, Maine, San Francisco, Minneapolis), Scotland, and the American Psychological Association (APA).

• Data Reduction: Elections with $>3$ candidates are "reduced" by running the IRV algorithm until only three candidates remain, then analyzing the resulting profile.
• Geographic Partitions: For elections with precinct/district data, they use Binary Integer Linear Programming (ILP) to search for "natural" partitions (e.g., North vs. South) that produce the paradox, rather than arbitrary mathematical partitions.
• Bootstrapping: They perform non-parametric bootstrapping (sampling with replacement) on real profiles to test the robustness of the paradox against small perturbations in vote counts.

3. Key Contributions

1. Algebraic Conditions: The paper provides the first rigorous necessary and sufficient conditions (Propositions 1 & 2) for a single three-candidate IRV election to be susceptible to a reinforcement paradox.
2. Methodological Shift: It introduces the "single-election partition" approach, contrasting with the "two-election generation" approach of prior literature. This allows for the analysis of specific real-world elections.
3. Threshold Analysis:
   • Proposition 3: Shows that a plurality loser can have an arbitrarily small percentage of first-place votes ($\epsilon$) and still trigger a runner-up paradox.
   • Proposition 4: Establishes that for a runner-up or plurality loser paradox to occur, that candidate must earn at least 25% of the first-place votes (in the limit).
4. Empirical Frequency: The study provides the first large-scale empirical estimates of reinforcement paradox frequency in real-world data.

4. Key Results

Theoretical Probabilities

• High Susceptibility: Under the IAC model (complete ballots), approximately 24.9% of elections are susceptible to some form of reinforcement paradox. When conditioned on the absence of a majority candidate, this rises to 56.8%.
• Runner-Up vs. Plurality: Runner-up paradoxes are significantly more common than plurality loser paradoxes (approx. 22.5% vs. 6.9% unconditioned).
• Spatial Models: In spatial models, paradox frequencies are lower (11–16% unconditioned) but still significant. The "Bimodal Weighted" model (asymmetric polarization) is the only model where plurality loser paradoxes become more likely than runner-up paradoxes.

Empirical Findings (Real-World Data)

• Frequency: In the dataset of 416 reduced elections, 24.8% demonstrated a reinforcement paradox.
• Correlation: In real-world data, every election susceptible to a plurality loser paradox was also susceptible to a runner-up paradox. This contrasts with theoretical models where they can occur independently.
• Robustness: Bootstrapping reveals that while many paradoxes are robust (90%+ of pseudoprofiles also show the paradox), some real-world paradoxes are fragile (e.g., the 2023 Minneapolis Ward 7 election, where only ~33% of pseudoprofiles showed the paradox).
• Geographic Partitions: When restricting the search to meaningful geographic partitions (precincts/districts), the frequency of paradoxes drops significantly (e.g., 3.0% for precinct partitions). This suggests that while the paradox is mathematically common, "natural" geographic splits causing it are rare.
• Case Studies:
  • 2022 Alaska Special House: Demonstrated a plurality loser paradox where Nick Begich (Condorcet winner) could win both sub-elections defined by House districts, despite losing the overall election.
  • 2021 Portland, ME: Showed a runner-up paradox where the runner-up won both the "Northwest" and "Non-Northwest" partitions but lost the overall seat.

5. Significance and Conclusion

• Administrative Implication: The high susceptibility of IRV to the reinforcement paradox confirms that IRV cannot be computed by simply summing precinct-level winners. Full ballot data must be aggregated centrally, increasing administrative complexity and potential for error.
• Theoretical vs. Empirical: Theoretical models (like IAC) often overestimate the frequency of paradoxes compared to real-world data, likely because real elections are less "random" and more structured. However, the empirical data confirms that the paradox is not a rare theoretical curiosity but a frequent occurrence in three-candidate races.
• Severity: The paper argues that the paradox is most "serious" when the partition corresponds to a meaningful demographic or geographic split (e.g., a candidate winning both urban and rural districts but losing the state). While such "natural" partitions are rare (approx. 3% in the dataset), their existence challenges the fairness perception of the IRV system.
• Future Work: The authors suggest future research should focus on the structure of these partitions (how similar $B_1$ and $B_2$ are) and extend the analysis to Condorcet-consistent methods and multi-winner STV systems.

In summary, McCune and Wilson demonstrate that IRV is highly susceptible to the reinforcement paradox in three-candidate elections, both theoretically and empirically, though the occurrence of "natural" geographic splits causing the paradox is less frequent. This susceptibility necessitates central ballot counting and highlights a fundamental inconsistency in the aggregation of voter preferences under IRV.