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Can ranked-choice voting elect the least popular candidate?

David McCune, Jennifer Wilson

Abstract

We analyze how frequently instant runoff voting (IRV) selects the weakest (or least popular) candidate in three-candidate elections. We consider four definitions of ``weakest candidate'': the Borda loser, the Bucklin loser, the candidate with the most last-place votes, and the candidate with minimum social utility. We determine the probability that IRV selects the weakest candidate under the impartial anonymous culture and impartial culture models of voter behavior, and use Monte Carlo simulations to estimate these probabilities under several spatial models. We also examine this question empirically using a large dataset of real elections. Our results show that IRV can select the weakest candidates under each of these definitions, but such outcomes are generally rare. Across most models, the probability that IRV elects a given type of weakest candidate is at most 5\%. Larger probabilities arise only when the electorate is extremely polarized.

Can ranked-choice voting elect the least popular candidate?

Abstract

We analyze how frequently instant runoff voting (IRV) selects the weakest (or least popular) candidate in three-candidate elections. We consider four definitions of ``weakest candidate'': the Borda loser, the Bucklin loser, the candidate with the most last-place votes, and the candidate with minimum social utility. We determine the probability that IRV selects the weakest candidate under the impartial anonymous culture and impartial culture models of voter behavior, and use Monte Carlo simulations to estimate these probabilities under several spatial models. We also examine this question empirically using a large dataset of real elections. Our results show that IRV can select the weakest candidates under each of these definitions, but such outcomes are generally rare. Across most models, the probability that IRV elects a given type of weakest candidate is at most 5\%. Larger probabilities arise only when the electorate is extremely polarized.
Paper Structure (13 sections, 4 theorems, 36 equations, 11 figures, 6 tables)

This paper contains 13 sections, 4 theorems, 36 equations, 11 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Models and Datasets
  4. Description of Models
  5. Analytical Results
  6. Simulation Methodology and Results
  7. Empirical Results
  8. Discussion
  9. How does IRV choose a weakest candidate?
  10. Differences among the spatial models
  11. Differences between Models and Data
  12. Comparison to other voting methods
  13. Conclusion

Key Result

Proposition 1

If voter preferences are single-peaked and voters cast complete ballots then the IRV winner cannot be the Bucklin loser, and cannot receive the most last-place votes.

Figures (11)

  • Figure 1: Examples of one-dimensional spatial voter distributions under the UNI, BIM($\sigma$), and WBI($\sigma$) models, illustrating increasing polarization as $\sigma$ decreases.
  • Figure 2: Examples of four 2D spatial models.
  • Figure 3: Results under the 1D spatial models BIM$(\sigma)$ and WBI$(\sigma)$. The top two figures give results when every voter provides a complete ranking, and the bottom figures give results when a voter provides a complete ranking with probability 0.65.
  • Figure 4: (Left) An election generated under the model BIM(0.5) where $A$ is the IRV winner and the Borda loser. (Right) An election generated under the model BIM(0.5) where the IRV winner $B$ minimizes social utility.
  • Figure 5: (Left) An election generated under the model WBI(0.5) where $A$ is the IRV winner and the Borda loser. (Right) An election generated under the model BIM(0.5) with partial ballots where the IRV winner $A$ is the Borda AVG loser.
  • ...and 6 more figures

Theorems & Definitions (9)

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