Independence of Approximate Clones

TL;DR

This work investigates whether independence of clones in ordinal elections extends to approximate clones by introducing two proximity notions, \(\alpha\)-deletion clones and \(\beta\)-swap clones. It provides a rigorous analysis showing that for elections with \(m \ge 4\) candidates, the clone-independence property fails for IRV, Ranked Pairs, and Schulze under any positive approximation; for \(m=3\) there are positive weak-independence results under certain conditions. An empirical study across local Scottish elections, figure skating judge rankings, and mini-jury deliberations reveals that approximate clones are common in practice and that the closer two candidates are to perfect clones, the more likely removing one does not change the outcome, especially for clone-friendly rules. The results highlight a gap between theoretical robustness of independence of clones and practical behavior under approximation, motivating future work on randomized rules, clone-sets, and cross-format analyses. The study balances theoretical impossibility results with context-rich empirical evidence to guide robust design of voting rules under near-clone scenarios.

Abstract

In an ordinal election, two candidates are said to be perfect clones if every voter ranks them adjacently. The independence of clones axiom then states that removing one of the two clones should not change the election outcome. This axiom has been extensively studied in social choice theory, and several voting rules are known to satisfy it (such as IRV, Ranked Pairs and Schulze). However, perfect clones are unlikely to occur in practice, especially for political elections with many voters. In this work, we study different notions of approximate clones in ordinal elections. Informally, two candidates are approximate clones in a preference profile if they are close to being perfect clones. We discuss two measures to quantify this proximity, and we show under which conditions the voting rules that are known to be independent of clones are also independent of approximate clones. In particular, we show that for elections with at least four candidates, none of these rules are independent of approximate clones in the general case. However, we find a more positive result for the case of three candidates. Finally, we conduct an empirical study of approximate clones and independence of approximate clones based on three real-world datasets: votes in local Scottish elections, votes in mini-jury deliberations, and votes of judges in figure skating competitions. We find that approximate clones are common in some contexts, and that the closest two candidates are to being perfect clones, the less likely their removal is to change the election outcome, especially for voting rules that are independent of perfect clones.

Figures (6)

• Figure 1: Distribution of the number of voters (who ranked all candidates) and of candidates in the Scottish elections dataset.
• Figure 2: Distributions (in number of instances) of the minimum value of $\alpha$ (left) and $\beta$ (right) for which pairs of candidates are respectively $\alpha$-deletion clones and $\beta$-swap clones in the figure skating competitions dataset.
• Figure 3: Values of $\alpha$ and $\beta$ for all pairs of candidates in the Scottish elections dataset. Red dots correspond to pairs of candidates from the same party, and black dots correspond to pairs from different parties.
• Figure 4: Blue bars indicate the average minimum value of $\alpha$ for which two candidates are $\alpha$-deletion clones over all instances in the Scottish elections dataset, depending on the number of candidates $m$. The red line indicates the theoretical maximum $(m-2)/m$, and the black lines show the extreme values over all profiles for a given $m$.
• Figure 5: Minimum value of $\alpha$ (left) and $\beta$ (right) for which two candidates are respectively $\alpha$-deletion and $\beta$-swap clone (over all pairs of candidates) for profiles with $m=10$ candidates and $n=50$ voters, visualized on the map of electionsszufa2020drawing. Each point corresponds to a profile.

Theorems & Definitions (18)

• definition 1
• definition 2
• definition 3
• definition 4
• definition 5
• definition 6
• proposition 1
• proof
• definition 7
• theorem 1