Discovering Consistent Subelections

TL;DR

This work studies how to uncover hidden, yet meaningful, subelections within ordinal elections by identifying large voter groups and candidate sets that exhibit consistent preferences under Identity, Antagonism, or Clone relations. It provides a spectrum of algorithmic results, including $\mathsf{NP}$-hardness for Hidden-$\text{ID}$ and Hidden-$\text{AN}$, fixed-parameter tractability with respect to the number of voters or candidates, polynomial-time solutions for Hidden-Clones, and ILP formulations to handle larger instances; a graph-based unanimity approach accelerates verifications. The authors validate their methods with synthetic data and real-world datasets (Sushi and Grenoble), showing that hidden consistent subelections reveal substantive structure and can guide efficient elicitation and interpretation of voter preferences. The findings suggest practical benefits for segmentation and market/policy analysis, as large, coherent subpopulations and candidate clusters emerge from complex ordinal profiles. They also outline promising avenues for future work, including near-identity/near-antagonism, structured-domain analyses, and approval-based settings.

Abstract

We show how hidden interesting subelections can be discovered in ordinal elections. An interesting subelection consists of a reasonably large set of voters and a reasonably large set of candidates such that the former have a consistent opinion about the latter. Consistency may take various forms but we focus on three: Identity (all selected voters rank all selected candidates the same way), antagonism (half of the selected voters rank candidates in some order and the other half in the reverse order), and clones (all selected voters rank all selected candidates contiguously in the original election). We first study the computation of such hidden subelections. Second, we analyze synthetic and real-life data, and find that identifying hidden consistent subelections allows us to uncover some relevant concepts.

Figures (4)

• Figure 1: Maps of elections with 10 candidates and 50 voters. Each point represents a single election, and its color represents the maximum number of voters that a) find certain two candidates clones (left), b) agree on certain five candidates being identity (middle), c) are antagonized over certain five candidates. In other words, the darker the point is, the more voters agree on a certain set of candidates being clones (left), identity (middle), or antagonism (right). On each map, ID label marks the identity election, and AN label marks the antagonism election, and dots representing elections coming from the same statistical culture were connected in clusters with names.
• Figure 2: Comparison of Sushi and Grenoble datasets. The black lines denote the results for impartial culture elections.
• Figure 3: Maps of elections with 10 candidates and 50 voters. Each point represents a single election. The darker the point is, the more voters agree on a certain set of candidates being clones. On each map, ID label marks the identity election, and AN label marks the antagonism election.
• Figure 4: Maps of elections with 10 candidates and 50 voters. Each point represents a single election. The darker the point is, the more voters agree on a certain set of candidates being antagonism. On each map, ID label marks the identity election, and AN label marks the antagonism election.

Theorems & Definitions (46)

• Example 1
• Theorem 1
• proof
• Example 2
• Example 3
• Corollary 2
• Theorem 3
• proof : Proof Sketch
• Proposition 4
• proof