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Ahead of the Count: An Algorithm for Probabilistic Prediction of Instant Runoff (IRV) Elections

Nicholas Kapoor, P. Christopher Staecker

TL;DR

The paper tackles probabilistic prediction of Instant Runoff Voting outcomes by modeling each possible ranking with an independent discrete distribution and summing over all elimination orders in $\Gamma(\mathcal{C})$, yielding the winning probabilities $W(\mathcal{E})$. It introduces a recursive, distribution-convolution-based algorithm that uses elimination probabilities $E^\ell_{A_i}$ and updated round distributions $f_R^{\ell A}$ to propagate predictions through all rounds, leveraging discrete convolution and FFT acceleration. Demonstrated applications include real-time winner prediction from partial tallies and probabilistic recount projections, with concrete Alaska examples and a detailed 3-candidate calculation illustrating the method's mechanics. The work offers a practical framework for IRV analytics, highlights computational limits for larger candidate sets, and outlines extensions to incorporate correlations and multi-winner STV scenarios for broader electoral forecasting and post-election analysis.

Abstract

How can we probabilistically predict the winner in a ranked-choice election without all ballots being counted? In this study, we introduce a novel algorithm designed to predict outcomes in Instant Runoff Voting (IRV) elections. The algorithm takes as input a set of discrete probability distributions describing vote totals for each candidate ranking and calculates the probability that each candidate will win the election. In fact, we calculate all possible sequences of eliminations that might occur in the IRV rounds and assign a probability to each. The discrete probability distributions can be arbitrary and, in applications, could be measured empirically from pre-election polling data or from partial vote tallies of an in-progress election. The algorithm is effective for elections with a small number of candidates (five or fewer), with fast execution on typical consumer computers. The run-time is short enough for our method to be used for real-time election night modeling where new predictions are made continuously as more and more vote information becomes available. We demonstrate the algorithm in abstract examples, and also using real data from the 2022 Alaska state elections to simulate election-night predictions and also predictions of election recounts.

Ahead of the Count: An Algorithm for Probabilistic Prediction of Instant Runoff (IRV) Elections

TL;DR

The paper tackles probabilistic prediction of Instant Runoff Voting outcomes by modeling each possible ranking with an independent discrete distribution and summing over all elimination orders in , yielding the winning probabilities . It introduces a recursive, distribution-convolution-based algorithm that uses elimination probabilities and updated round distributions to propagate predictions through all rounds, leveraging discrete convolution and FFT acceleration. Demonstrated applications include real-time winner prediction from partial tallies and probabilistic recount projections, with concrete Alaska examples and a detailed 3-candidate calculation illustrating the method's mechanics. The work offers a practical framework for IRV analytics, highlights computational limits for larger candidate sets, and outlines extensions to incorporate correlations and multi-winner STV scenarios for broader electoral forecasting and post-election analysis.

Abstract

How can we probabilistically predict the winner in a ranked-choice election without all ballots being counted? In this study, we introduce a novel algorithm designed to predict outcomes in Instant Runoff Voting (IRV) elections. The algorithm takes as input a set of discrete probability distributions describing vote totals for each candidate ranking and calculates the probability that each candidate will win the election. In fact, we calculate all possible sequences of eliminations that might occur in the IRV rounds and assign a probability to each. The discrete probability distributions can be arbitrary and, in applications, could be measured empirically from pre-election polling data or from partial vote tallies of an in-progress election. The algorithm is effective for elections with a small number of candidates (five or fewer), with fast execution on typical consumer computers. The run-time is short enough for our method to be used for real-time election night modeling where new predictions are made continuously as more and more vote information becomes available. We demonstrate the algorithm in abstract examples, and also using real data from the 2022 Alaska state elections to simulate election-night predictions and also predictions of election recounts.
Paper Structure (14 sections, 4 theorems, 42 equations, 5 figures)

This paper contains 14 sections, 4 theorems, 42 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Sample applications of our algorithm
  3. Winner prediction based on partial vote counts
  4. Recount predictions
  5. The IRV algorithm
  6. Discrete convolution
  7. The algorithm for computing winning probabilities
  8. Computing distributions for $\mathcal{E}^{\ell A}$, given those for $\mathcal{E}^{\ell}$
  9. Computing elimination probabilities $E^\ell_{A_i}$
  10. A detailed example calculation
  11. Limitations and future work
  12. Computational complexity
  13. The assumption of independence
  14. Future Work

Key Result

Lemma 6

Let the random variables $\{ R \mid R \in \Gamma(\mathcal{C})\}$ be independent, and let $\ell \in \Gamma(\mathcal{C})$. Then the random variables $\{ R^\ell \mid R \in \Gamma(\mathcal{C}\setminus \ell)\}$ are independent.

Figures (5)

  • Figure 1: Win probabilities for Alaska House 18, as votes are tallied
  • Figure 2: Ternary plot of data from Figure \ref{['house18fig']}
  • Figure 3: Win probabilities for Alaska US Senate race, as votes are tallied
  • Figure 4: Alternative runs of Figure \ref{['ussenatefig']}, using different randomized orderings of precinct portions
  • Figure 5: Histogram of historical vote shifts after recounts in US state elections, 2000--2023

Theorems & Definitions (14)

  • Example 1
  • Example 2
  • Example 3
  • Example 4
  • Example 5
  • Lemma 6
  • Lemma 7
  • Lemma 8
  • proof
  • Example 9
  • ...and 4 more