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.

翻译：在序数选举中，若每位投票者都将两位候选人排列在相邻位置，则称其为完美克隆。克隆独立性公理要求移除其中一个克隆不应改变选举结果。该公理在社会选择理论中已被广泛研究，已知有多种投票规则满足此公理（如IRV、Ranked Pairs和Schulze）。然而，完美克隆在实践中不太可能出现，尤其是在选民众多的政治选举中。本研究探讨序数选举中近似克隆的不同概念。非正式地说，若两位候选人在偏好配置中接近完美克隆状态，则称其为近似克隆。我们讨论了两种量化这种接近程度的度量方法，并展示了在何种条件下已知满足克隆独立性的投票规则也能保持对近似克隆的独立性。特别地，我们证明对于至少包含四位候选人的选举，在一般情况下这些规则均不满足近似克隆独立性。但在仅有三名候选人的情况下，我们得到了更积极的结论。最后，基于三个真实数据集——苏格兰地方选举投票、小型陪审团审议投票和花样滑冰比赛裁判评分——我们对近似克隆及其独立性进行了实证研究。研究发现，在某些情境下近似克隆普遍存在，且两位候选人越接近完美克隆状态，其移除改变选举结果的可能性越低，这一现象在满足完美克隆独立性的投票规则中尤为显著。