Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

BallotRank: A Condorcet Completion Method for Graphs

Ismar Volic, Jason Douglas Todd

TL;DR

BallotRank addresses ranked-choice aggregation by converting voter profiles into a directed graph and applying a damping-parameter PageRank variant to produce a full candidate ranking. It introduces weights from pairwise margins and self-loops to distinguish cycle members, yielding a stationary distribution BR that satisfies BR = (I − dℓ)^{-1} (1−d)/n 1, with ℓ(a,b) = M(a,b)/TL(b). The method is Condorcet-consistent at d=1 and, for conventional d, identifies Condorcet winners across large empirical datasets, while providing a rich social welfare function rather than a single winner. BallotRank aligns with several standard social choice criteria (anonymity, neutrality, non-dictatorship, Pareto, Smith) but inherently violates IIA, monotonicity, later-no-harm, no-show, and cloning; nonetheless it offers practical advantages by delivering a canonical full ranking. These properties, along with robust empirical performance and clear practical recommendations, make BallotRank a compelling graph-based approach for multiwinner and complex ranking scenarios.

Abstract

We introduce BallotRank, a ranked preference aggregation method derived from a modified PageRank algorithm. It is a Condorcet-consistent method without damping, and empirical examination of nearly 2,000 ranked choice elections and over 20,000 internet polls confirms that BallotRank always identifies the Condorcet winner at conventional values of the damping parameter. We also prove that the method satisfies many of the same social choice criteria as other well-known Condorcet completion methods, but it has the advantage of being a natural social welfare function that provides a full ranking of the candidates.

BallotRank: A Condorcet Completion Method for Graphs

TL;DR

BallotRank addresses ranked-choice aggregation by converting voter profiles into a directed graph and applying a damping-parameter PageRank variant to produce a full candidate ranking. It introduces weights from pairwise margins and self-loops to distinguish cycle members, yielding a stationary distribution BR that satisfies BR = (I − dℓ)^{-1} (1−d)/n 1, with ℓ(a,b) = M(a,b)/TL(b). The method is Condorcet-consistent at d=1 and, for conventional d, identifies Condorcet winners across large empirical datasets, while providing a rich social welfare function rather than a single winner. BallotRank aligns with several standard social choice criteria (anonymity, neutrality, non-dictatorship, Pareto, Smith) but inherently violates IIA, monotonicity, later-no-harm, no-show, and cloning; nonetheless it offers practical advantages by delivering a canonical full ranking. These properties, along with robust empirical performance and clear practical recommendations, make BallotRank a compelling graph-based approach for multiwinner and complex ranking scenarios.

Abstract

We introduce BallotRank, a ranked preference aggregation method derived from a modified PageRank algorithm. It is a Condorcet-consistent method without damping, and empirical examination of nearly 2,000 ranked choice elections and over 20,000 internet polls confirms that BallotRank always identifies the Condorcet winner at conventional values of the damping parameter. We also prove that the method satisfies many of the same social choice criteria as other well-known Condorcet completion methods, but it has the advantage of being a natural social welfare function that provides a full ranking of the candidates.
Paper Structure (20 sections, 4 theorems, 36 equations, 4 figures, 5 tables)

This paper contains 20 sections, 4 theorems, 36 equations, 4 figures, 5 tables.

Table of Contents

  1. Introduction
  2. From PageRank to BallotRank
  3. Social choice under PageRank and BallotRank
  4. Basic network
  5. Adding weights
  6. Adding the damping parameter
  7. Adding self-loops
  8. Formalizing PageRank and BallotRank
  9. The basic algorithm
  10. Adding the damping parameter
  11. Adding weights
  12. Adding self-loops
  13. Linear algebra interpretation of BallotRank
  14. Working through a toy election
  15. Empirical analyses
  16. ...and 5 more sections

Key Result

Proposition 6.2

BallotRank satisfies anonymity, neutrality, and non-dictatorship for any $0<d\leq 1$ and majority for $d=1$.

Figures (4)

  • Figure 1: 18,345 elections with a Condorcet winner exhibit considerable variation in choice set and electorate sizes; BallotRank correctly pegs the Condorcet winner every time. Data are plotted on a log-log scale.
  • Figure 2: Four case studies demonstrate BallotRank's superiority to IRV, its behavior across the range of $d$, its convenient production of a complete ranking, and its ability to rank candidates within cycles.
  • Figure 3: BallotRank graph for the 2021 Minneapolis Ward 2 election.
  • Figure 4: BallotRank graph for the 2022 Oakland election.

Theorems & Definitions (15)

  • Definition 2.1
  • Definition 6.1
  • Proposition 6.2
  • proof
  • Proposition 6.3
  • proof
  • Proposition 6.4
  • proof
  • Proposition 6.5
  • proof
  • ...and 5 more