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arxiv: 2502.05185 · v4 · submitted 2025-01-23 · ⚛️ physics.soc-ph · cs.GT

Instant Runoff Voting and the Reinforcement Paradox

David McCune , Jennifer Wilson This is my paper

Pith reviewed 2026-05-23 05:19 UTC · model grok-4.3

classification ⚛️ physics.soc-ph cs.GT
keywords instant runoff votingreinforcement paradoxranked choice votingvoting paradoxesthree-candidate electionsmonte carlo simulationelection data analysis
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The pith

In three-candidate IRV, a losing candidate can win both halves when ballots are partitioned into two groups.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines the reinforcement paradox in instant runoff voting, where a candidate wins two separate elections but loses when the ballots are merged into one. For three-candidate races it derives exact conditions that determine when a ballot set can be split so a given loser wins each resulting sub-election. Monte Carlo runs then measure how often such splits arise under different voter preference models, and a dataset of actual ranked-choice elections supplies observed rates. If the conditions hold often, IRV results become sensitive to whether votes are counted together or in separate groups.

Core claim

For three-candidate IRV elections we provide necessary and sufficient conditions under which there exists a partition of the ballot set into two sets of ballots such that a given losing candidate wins each of the sub-elections. Applying these conditions, we use Monte Carlo simulations to estimate the frequency with which such partitions exist under various models of voter behavior. We also analyze the frequency with which the paradox occurs in a large dataset of real-world ranked-choice elections to provide empirical probabilities. Our general finding is that IRV is highly susceptible to this paradox in three-candidate elections.

What carries the argument

Necessary and sufficient conditions on ballot sets that allow a losing candidate under full IRV to win each of two partitioned sub-elections.

If this is right

  • IRV is highly susceptible to the reinforcement paradox in three-candidate elections.
  • Such partitions occur at high rates under the tested models of voter behavior.
  • Real-world ranked-choice election data produces occurrence rates consistent with the simulations.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The result implies that election administrators may need to consider how ballots are grouped or aggregated when using IRV.
  • The partition conditions could be applied to test whether other ranked voting rules exhibit similar sensitivities.
  • Future work might check whether the same conditions appear in elections with more than three candidates.

Load-bearing premise

The various models of voter behavior used in the Monte Carlo simulations are representative of real voter preferences in three-candidate elections.

What would settle it

A study of many three-candidate IRV elections that finds no ballot partitions allowing any loser to win both sub-elections would show the claimed susceptibility does not hold.

Figures

Figures reproduced from arXiv: 2502.05185 by David McCune, Jennifer Wilson.

Figure 1
Figure 1. Figure 1: Visualizations of the two bimodal spatial models. model. One of the fruits of these labors is the software package Normaliz [9], which can provide limiting probabilities under IAC in three-candidate elections given a set of linear constraints. In the complete ballot case, Normaliz determines the pro￾portion of integer values of X1, X2, Y1, Y2, Z1 and Z2 that satisfy the inequalities in Proposition 1 given … view at source ↗
Figure 2
Figure 2. Figure 2: Estimated probabilities that an election demonstrates a reinforcement paradox under the model Dir(α). The top figure shows unconditioned probabilities while the bottom figure shows probabilities conditioned on the absence of a majority candidate [PITH_FULL_IMAGE:figures/full_fig_p021_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: (Top) Bootstrapping frequency results for elections that are susceptible to a reinforcement paradox. (Bottom) Boot￾strapping frequency results for elections that are are not susceptible to a reinforcement paradox but generated at least one pseudopro￾file that is. 6.1. Theoretical Versus Empirical Results. As is often the case in social choice, none of the theoretical models give probabilities which closely… view at source ↗
read the original abstract

We analyze the susceptibility of instant runoff voting (IRV) to a lesser-studied paradox known as a \emph{reinforcement paradox}, which occurs when candidate $X$ wins under IRV in two distinct elections but $X$ loses in the combined election formed by merging the ballots from the two elections. For three-candidate IRV elections we provide necessary and sufficient conditions under which there exists a partition of the ballot set into two sets of ballots such that a given losing candidate wins each of the sub-elections. Applying these conditions, we use Monte Carlo simulations to estimate the frequency with which such partitions exist under various models of voter behavior. We also analyze the frequency with which the paradox occurs in a large dataset of real-world ranked-choice elections to provide empirical probabilities. Our general finding is that IRV is highly susceptible to this paradox in three-candidate elections.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript derives necessary and sufficient conditions for the existence of reinforcement-paradox partitions in three-candidate instant-runoff-voting (IRV) elections. It then estimates the frequency of such partitions via Monte Carlo simulation under multiple voter-preference models and via direct enumeration on a large real-world ranked-choice dataset, concluding that IRV is highly susceptible to the paradox in three-candidate elections.

Significance. The combination of exact mathematical conditions, controlled variation across several preference models, and empirical frequencies from real ballots supplies a multi-pronged assessment that does not rest on any single modeling choice. If the derived conditions and reported frequencies are accurate, the work materially advances the quantitative understanding of IRV pathologies and supplies concrete, testable predictions for election data.

minor comments (3)
  1. [Monte Carlo simulations] The Monte Carlo section should explicitly list the simulation parameters (number of voters per election, exact preference-generation procedures for each model, number of trials) so that the reported frequencies can be reproduced.
  2. [Empirical analysis] The real-data analysis should state the size of the dataset, the source election jurisdictions, and the precise criterion used to identify a reinforcement-paradox partition in the observed ballots.
  3. [Theoretical conditions] Notation for ballot types and the partition sets should be introduced once and used consistently; a small table summarizing the notation would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. The referee's description of the manuscript's contributions is accurate.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper first derives necessary and sufficient conditions for reinforcement-paradox partitions in three-candidate IRV elections using standard ballot-counting logic; these conditions are stated as independent mathematical statements rather than being fitted or defined in terms of the target frequencies. It then applies the conditions in Monte Carlo simulations under multiple external voter models and in direct enumeration on a large real-world ranked-choice dataset. No parameter is fitted to a subset and then relabeled as a prediction, no self-citation supplies a load-bearing uniqueness theorem, and no ansatz or renaming reduces the central claim to its own inputs. The derivation chain therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The abstract does not specify any free parameters or new entities; the main assumptions relate to standard modeling practices in the field.

axioms (1)
  • domain assumption Voter preferences in elections can be modeled using standard probabilistic distributions for simulation purposes.
    The paper applies Monte Carlo simulations under various models of voter behavior.

pith-pipeline@v0.9.0 · 5670 in / 1170 out tokens · 63773 ms · 2026-05-23T05:19:57.964906+00:00 · methodology

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Reference graph

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