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Limited Voting for Better Representation?

Maaike Venema-Los, Zoé Christoff, Davide Grossi

TL;DR

This paper analyzes Limited Voting (LV), an approval-based multi-winner rule with fixed ballot size, to determine when it improves employee representation relative to standard approval voting. It develops two core metrics—CC-improvement and PAV-improvement—to quantify diversity and proportionality gains, and shows that LV can outperform AV in highly aligned (party-list-like) elections, especially when ballots are coordinated within parties. However, LV’s guarantees weaken in less aligned profiles, with CC-guarantee equal to zero in general and proportionality axioms often violated; experiments confirm that LV’s benefits are strongest in party-list structures and decay as $\phi$ (deviation from party-list structure) grows. The paper also introduces LV-games to study strategic behavior and demonstrates that, under certain equilibria, LV can deliver near-proportional representation, suggesting LV as a practical, low-effort fix in specific organizational settings, while outlining avenues for future work on relaxing constraints and exploring alternative proportionality measures.

Abstract

Limited Voting (LV) is an approval-based method for multi-winner elections where all ballots are required to have a same fixed size. While it appears to be used as voting method in corporate governance and has some political applications, to the best of our knowledge, no formal analysis of the rule exists to date. We provide such an analysis here, prompted by a request for advice about this voting rule by a health insurance company in the Netherlands, which uses it to elect its work council. We study conditions under which LV would improve representation over standard approval voting and when it would not. We establish the extent of such an improvement, or lack thereof, both in terms of diversity and proportionality notions. These results help us understand if, and how, LV may be used as a low-effort fix of approval voting in order to enhance representation.

Limited Voting for Better Representation?

TL;DR

This paper analyzes Limited Voting (LV), an approval-based multi-winner rule with fixed ballot size, to determine when it improves employee representation relative to standard approval voting. It develops two core metrics—CC-improvement and PAV-improvement—to quantify diversity and proportionality gains, and shows that LV can outperform AV in highly aligned (party-list-like) elections, especially when ballots are coordinated within parties. However, LV’s guarantees weaken in less aligned profiles, with CC-guarantee equal to zero in general and proportionality axioms often violated; experiments confirm that LV’s benefits are strongest in party-list structures and decay as (deviation from party-list structure) grows. The paper also introduces LV-games to study strategic behavior and demonstrates that, under certain equilibria, LV can deliver near-proportional representation, suggesting LV as a practical, low-effort fix in specific organizational settings, while outlining avenues for future work on relaxing constraints and exploring alternative proportionality measures.

Abstract

Limited Voting (LV) is an approval-based method for multi-winner elections where all ballots are required to have a same fixed size. While it appears to be used as voting method in corporate governance and has some political applications, to the best of our knowledge, no formal analysis of the rule exists to date. We provide such an analysis here, prompted by a request for advice about this voting rule by a health insurance company in the Netherlands, which uses it to elect its work council. We study conditions under which LV would improve representation over standard approval voting and when it would not. We establish the extent of such an improvement, or lack thereof, both in terms of diversity and proportionality notions. These results help us understand if, and how, LV may be used as a low-effort fix of approval voting in order to enhance representation.
Paper Structure (39 sections, 16 theorems, 1 equation, 9 figures, 14 tables)

This paper contains 39 sections, 16 theorems, 1 equation, 9 figures, 14 tables.

Table of Contents

  1. Motivation
  2. Contribution
  3. Related work
  4. Preliminaries
  5. Diversity
  6. CC-improvement
  7. CC-guarantee
  8. Proportionality
  9. Proportionality axioms
  10. PAV-improvement
  11. Computational experiments
  12. Experiments design
  13. Generating approval profiles
  14. Generating ballot profiles
  15. Determining winning committees and scores
  16. ...and 24 more sections

Key Result

Theorem 1

Let $E = ( N, C, k, l, A, L)$ be any broadcasted party-list election with parties $P_1, \ldots, P_g$. The CC-improvement is $IMP_{CC}(E) = \frac{\sum_{i=1}^{{\min(\lceil\frac{k}{l}\rceil, g)}}n_i}{\sum_{j=1}^{s}n_j}$ , where $s$ is the largest integer $t$ such that $\sum_{i=1}^{t}|P_i|\leq k$.

Figures (9)

  • Figure 1: Boxplots of $\mathit{IMP}_{CC}$ for different values of $\phi$ (dispersion).
  • Figure 2: Boxplots of the CC-improvement for different values of $\phi$ (dispersion), $k$ (committee size), $g$ (number of parties), and $l$ (ballot limit).
  • Figure 3: Boxplots of the PAV- improvement for different values of $\phi$, $k$, $g$, and $l$.
  • Figure 4: Probability density of CC- improvement for $g=2, \phi = 0$
  • Figure 5: AV-improvement ($\frac{s_{AV}(LV)}{s_{AV}(AV)}$) for different values of $l$ (with $k\in \{8, 12, 16\}$, all values combined, $\phi=0$). Note that the 'improvement' is always at most 1, because LV cannot have a higher AV-score than AV itself.
  • ...and 4 more figures

Theorems & Definitions (37)

  • Definition 1: Limited Voting (LV)
  • Definition 2: Broadcasting order
  • Definition 3: Broadcasted party-list elections
  • Remark 1
  • Definition 4: CC-improvement
  • Theorem 1
  • proof
  • Corollary 2
  • Corollary 3
  • Theorem 4
  • ...and 27 more