Selecting the Most Conflicting Pair of Candidates

TL;DR

This work introduces a conflict-focused view for multiwinner voting aimed at identifying the most conflicting pair of candidates under ordinal voter preferences. It defines foundational axioms (notably Conflict Consistency and Reverse Stability) and demonstrates an impossibility result relative to standard unanimity/diversity/proportionality objectives, motivating the design of new conflictual rules. The paper formalizes two pairwise-conflict scores, MaxSumConflict and MaxNashConflict, along with MaxSwap, and rigorously analyzes their properties through proofs and counterexamples, including Matching-Domination and antagonization considerations. It also develops a rich interpretation via partitioning ratio $\alpha$, discrepancy $\beta$, discrepancy balance $\gamma$, and group discrepancy imbalance $\phi$, linking these metrics to rule behavior and polarization. Experiments on synthetic and real data (including political, sushi, and figure skating datasets) validate the theoretical distinctions, showing how conflictual rules uncover polarization structures that traditional rules overlook and highlighting practical implications for selecting conflicting options or fostering dialogue.

Abstract

We study committee elections from a perspective of finding the most conflicting candidates, that is, candidates that imply the largest amount of conflict, as per voter preferences. By proposing basic axioms to capture this objective, we show that none of the prominent multiwinner voting rules meet them. Consequently, we design committee voting rules compliant with our desiderata, introducing conflictual voting rules. A subsequent deepened analysis sheds more light on how they operate. Our investigation identifies various aspects of conflict, for which we come up with relevant axioms and quantitative measures, which may be of independent interest. We support our theoretical study with experiments on both real-life and synthetic data.

Figures (11)

• Figure 1: Area where $\{a_2,b_2\}$ is preferred to $\{a_1,b_1\}$ for different rules. We assume $\alpha(a_2,b_2) = 1$ and $\beta(a_1,b_1) = 1$.
• Figure 2: Distribution of the positions of the winning candidates for different rules and distributions of positions. Each pair of colored points correspond to the winners of a single election.
• Figure 3: Metrics values of the selected pairs of candidates for real data. The coordinates of the pair $\{a,b\}$ are $(\alpha(a,b),\beta(a,b))$, its size is $\gamma(a,b)$ and its color is determined by $\phi(a,b)$ with red corresponding to higher values. These data were gathered from 1000 profiles of 100 voters and 10 candidates.
• Figure 4: Metrics of the pairs of candidates with Mallows models. Each column corresponds to a parameter $\psi \in \{0.1,0.3,0.6\}$. The first row is for Mallows using 1 central ranking and the second row Mallows with 2 central rankings. The coordinates of the pair $\{a,b\}$ are $(\alpha(a,b), \beta(a,b)$, its size depends on $\gamma(a,b)$ and its color on $\phi(a,b)$. These data were gathered from 5 profiles for each model, each containing 1000 voters and 10 candidates.
• Figure 5: Mean or maximum values of different metrics for Mallows models with 1 (full line) or 2 (dotted line) central ranking(s). Each dot is averaged over 50 profiles.

Theorems & Definitions (21)

• Definition 1: Conflict Consistency
• Definition 2: Reverse Stability
• Definition 3: Unanimity
• Proposition 1
• proof
• Example 1
• Definition 4: Matching Domination
• Definition 5: Conflict Monotonicity
• Theorem 1
• proof