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Abductive and Contrastive Explanations for Scoring Rules in Voting

Clément Contet, Umberto Grandi, Jérôme Mengin

TL;DR

This work proposes to apply techniques from formal explainability, most notably abductive and contrastive explanations, to identify minimal subsets of a preference profile that either imply the current winner or explain why a different candidate was not elected.

Abstract

We view voting rules as classifiers that assign a winner (a class) to a profile of voters' preferences (an instance). We propose to apply techniques from formal explainability, most notably abductive and contrastive explanations, to identify minimal subsets of a preference profile that either imply the current winner or explain why a different candidate was not elected. Formal explanations turn out to have strong connections with classical problems studied in computational social choice such as bribery, possible and necessary winner identification, and preference learning. We design algorithms for computing abductive and contrastive explanations for scoring rules. For the Borda rule, we find a lower bound on the size of the smallest abductive explanations, and we conduct simulations to identify correlations between properties of preference profiles and the size of their smallest abductive explanations.

Abductive and Contrastive Explanations for Scoring Rules in Voting

TL;DR

This work proposes to apply techniques from formal explainability, most notably abductive and contrastive explanations, to identify minimal subsets of a preference profile that either imply the current winner or explain why a different candidate was not elected.

Abstract

We view voting rules as classifiers that assign a winner (a class) to a profile of voters' preferences (an instance). We propose to apply techniques from formal explainability, most notably abductive and contrastive explanations, to identify minimal subsets of a preference profile that either imply the current winner or explain why a different candidate was not elected. Formal explanations turn out to have strong connections with classical problems studied in computational social choice such as bribery, possible and necessary winner identification, and preference learning. We design algorithms for computing abductive and contrastive explanations for scoring rules. For the Borda rule, we find a lower bound on the size of the smallest abductive explanations, and we conduct simulations to identify correlations between properties of preference profiles and the size of their smallest abductive explanations.
Paper Structure (15 sections, 8 theorems, 15 equations, 2 figures, 5 algorithms)

This paper contains 15 sections, 8 theorems, 15 equations, 2 figures, 5 algorithms.

Table of Contents

  1. Our contribution.
  2. Related work
  3. Preliminaries
  4. Formal explanations
  5. Elections and scoring rules
  6. Formal explanations for elections
  7. Necessary winner and explanations
  8. Computing explanations for voting rules
  9. Computing one explanation
  10. Enumeration and smallest explanations
  11. Smallest abductive explanations for Borda
  12. The size of abductive explanations for Borda
  13. A map of elections for SiAXp s for Borda
  14. Data set composition of experiment on SiAXp s
  15. Conclusions and future work

Key Result

Proposition 1

Given a complete rank matrix $\mathcal{R}$, a voting rule $F$ and a winning candidate $w \in F(\mathcal{R})$, $\mathcal{X}$ is an AXp of $\mathcal{R}$ iff $\mathcal{X}$ is a $\subseteq$-minimal partial rank matrix s.t. $\mathcal{X} \subseteq \mathcal{R}$ and $w \in \text{NW}_{F}(\mathcal{X})$. Moreo

Figures (2)

  • Figure 1: Map of elections for 146 preference profiles generated by fourteen different cultures plus six compass profiles (AN, ID, and four for UN). The map on the right shows for each preference profile the size of its SiAXp, normalised as $\frac{|SiAXp|}{nm}$.
  • Figure 2: Normalised SiAXp size on the $x$-axis compared with agreement index (left) and margin of victory (right) for the 146 profiles in our dataset.

Theorems & Definitions (28)

  • Definition 1
  • example 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • example 2
  • example 3
  • example 4
  • Definition 6
  • ...and 18 more