Instant Runoff Voting and the Reinforcement Paradox

TL;DR

This study investigates the reinforcement paradox in instant runoff voting (IRV) with three candidates, where a candidate can win in two sub-elections formed by partitioning ballots but lose in the merged election. It provides necessary and sufficient algebraic conditions for such paradoxes under complete and partial ballots, and analyzes minimal vote-share thresholds for the sub-election winners. The authors complement theory with extensive Monte Carlo simulations across multiple spatial and Dirichlet ballot models, and validate findings with a large empirical corpus of real-world three-candidate IRV elections, augmented by bootstrapping. The results show IRV is highly susceptible to reinforcement paradoxes in the three-candidate case, with substantial empirical and simulated frequencies, and discuss implications for administrative transparency and future research across geography-based partitions and higher candidate counts.

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.

Figures (3)

• Figure 1: Visualizations of the two bimodal spatial models.
• Figure 2: Estimated probabilities that an election demonstrates a reinforcement paradox under the model Dir$(\alpha)$. The top figure shows unconditioned probabilities while the bottom figure shows probabilities conditioned on the absence of a majority candidate.
• Figure 3: (Top) Bootstrapping frequency results for elections that are susceptible to a reinforcement paradox. (Bottom) Bootstrapping frequency results for elections that are are not susceptible to a reinforcement paradox but generated at least one pseudoprofile that is.

Theorems & Definitions (16)

• Example 1
• Example 2
• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Proposition 1
• Proposition 2
• proof
• Example 3