Independence of Approximate Clones

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.

💡 Research Summary

This paper introduces the notion of approximate clones to bridge the gap between the idealized concept of perfect clones (candidates that are adjacent in every voter’s ranking) and the messy reality of real‑world elections where such perfect adjacency is virtually never observed. Two quantitative measures are proposed. The first, α‑deletion clones, asks how many voters (as a fraction α of the electorate) must be removed so that the remaining voters see the two candidates as perfect clones. The second, β‑swap clones, counts the total number of adjacent swaps needed across all rankings to make the two candidates adjacent, normalised by the number of voters. Both measures reduce to zero for perfect clones and coincide when there are exactly three candidates.

The authors then adapt the classic independence of clones axiom to this approximate setting, defining a weak independence of approximate clones: if two candidates are α‑ or β‑approximate clones, removing one of them should not substantially alter the set of winners. They analyse three well‑known single‑winner voting rules that satisfy the original clone‑independence axiom—Instant Runoff Voting (IRV), Ranked Pairs, and Schulze.

Theoretical results are striking. For elections with four or more candidates, no positive value of α or β can guarantee weak independence for any of the three rules; the authors construct explicit counter‑examples where removing a tiny fraction of voters (or performing a few swaps) flips the winner. Conversely, when exactly three candidates are present, the situation improves: if α is sufficiently small (e.g., α < 1/3) IRV and Ranked Pairs retain weak independence, and Schulze does so for β < 0.5. Thus, the clone‑independent rules exhibit a limited robustness to approximate cloning, but only in very small elections.

To assess practical relevance, the paper conducts an empirical study on three real datasets: (1) local Scottish elections, (2) mini‑jury deliberation experiments, and (3) figure‑skating judges’ rankings. For each dataset the authors compute the minimal α and β for every pair of candidates and then test how often the removal of one candidate changes the winner under each rule. Findings include: in the Scottish data, about 12 % of candidate pairs have α ≤ 0.1, and the winner changes in less than 5 % of such cases for IRV and Ranked Pairs; in the mini‑jury data, roughly 18 % of pairs have β ≤ 0.2, and Schulze’s outcome is virtually unchanged; in the figure‑skating data, approximate clones are rare due to many candidates, but when they appear the clone‑independent rules again show minimal impact. Statistical analysis reveals a clear negative correlation between the degree of approximation (smaller α or β) and the probability of a winner change, especially for the clone‑independent rules.

The discussion highlights that while perfect clones are theoretical curiosities, approximate clones are common and can be systematically measured. The paper argues that the robustness of IRV, Ranked Pairs, and Schulze to approximate clones—particularly in three‑candidate elections—offers a practical safeguard against spoiler effects. It also points to future research directions: extending the analysis to multi‑winner settings, exploring alternative distance metrics (e.g., Kendall‑tau), and investigating strategic behavior when candidates can be introduced or withdrawn to manipulate α or β.

In sum, the study enriches the social‑choice literature by quantifying how close real elections come to the clone scenario, by rigorously testing the limits of clone‑independent rules under these approximations, and by providing empirical evidence that these rules indeed behave more stably when candidates are near‑clones. This work offers both theoreticians and practitioners a nuanced tool for evaluating and designing voting systems that are resilient to the subtle, yet pervasive, phenomenon of approximate cloning.