Maximizing Index Diversity in Committee Elections

TL;DR

This work addresses the challenge of selecting diverse, high-quality committees in multiwinner elections with labelled candidates. It introduces two models that optimize diversity via ecological indices while respecting either a total score lower bound or per-agent satisfaction, enabling a more nuanced balance between quality and breadth. The authors adapt several ecology-inspired indices (Ri, Si, Sh, Shannon) and introduce a new Lexicographic Counting Index (LC), accompanied by a rigorous axiomatic analysis and complexity results. They show that maximizing diversity without satisfaction is polynomial-time, while incorporating per-agent satisfaction is NP-hard in general and polynomial for certain separable scoring rules. Empirical results indicate that diversity can be meaningfully increased by using indices as tiebreakers or by relaxing score constraints, with some trade-offs and diminishing returns observed as the allowed score relaxation grows.

Abstract

We introduce two models of multiwinner elections with approval preferences and labelled candidates that take the committee's diversity into account. One model aims to find a committee with maximal diversity given a scoring function (e.g. of a scoring-based voting rule) and a lower bound for the score to be respected. The second model seeks to maximize the diversity given a minimal satisfaction for each agent to be respected. To measure the diversity of a committee, we use multiple diversity indices used in ecology and introduce one new index. We define (desirable) properties of diversity indices, test the indices considered against these properties, and characterize the new index. We analyze the computational complexity of computing a committee for both models and scoring functions of well-known voting rules, and investigate the influence of weakening the score or satisfaction constraints on the diversity empirically.

Figures (15)

• Figure 1: The dimensions of the experimental data, where the color of each point represents the average number of agents of all instances with the given number of labels and candidates.
• Figure 2: The proportion of the optimal diversity reached on the experimental data when using the specified diversity index $D$. "$\mathcal{R}$ best" ("$\mathcal{R}$ worst") refers to the rule choosing the committees with the highest (lowest) diversity among the winning committees of $\mathcal{R}$. The red line indicates the median, the green cross the mean.
• Figure 3: The dimensions of the experimental data, where the color of each point represents the average number of agents of all instances with the given number of labels and candidates.
• Figure 4: The proportion of the optimal diversity reached over all experimental data with $k=10$ when using $\mathit{Ri}$ for the different rules and approaches we consider. "$\mathcal{R}$ best" ("$\mathcal{R}$ worst") refers to the rule that chooses the committees with the highest (lowest) diversities among the winning committees of $\mathcal{R}$. If only $\mathcal{R}$ is written, the diversity of the winning committee that abcvoting returns for $\mathcal{R}$ is considered. The red line indicates the median, the green cross the mean.
• Figure 5: The proportion of the optimal diversity reached over all experimental data with $k=8$ when using $\mathit{Ri}$ for the different rules and approaches we consider. "$\mathcal{R}$ best" ("$\mathcal{R}$ worst") refers to the rule that chooses the committees with the highest (lowest) diversities among the winning committees of $\mathcal{R}$. If only $\mathcal{R}$ is written, the diversity of the winning committee that abcvoting returns for $\mathcal{R}$ is considered. The red line indicates the median, the green cross the mean.

Theorems & Definitions (20)

• remark 1
• remark 2
• remark 3
• theorem 1: \ref{['proof:theorem:explain:bounds']}
• theorem 2: \ref{['proof:theorem:charac']}
• definition 1: MAX-$D$-DSAT
• definition 2: MAX-$\left( D, s \right)$-DSCR
• proof
• definition 3
• theorem 3: \ref{['proof:theorem:DssDnf']}