Approximate Clones in Ordinal Elections

• The paper introduces generalized clones that allow bounded adjacency imperfections, making the concept applicable to noisy real-world data.
• It details rigorous measures like (q, b)-approximate clones, α-deletion, and β-swap metrics to diagnose near-clonality in voter rankings.
• Algorithmic methods, including FPT and dynamic programming techniques, expose computational challenges in recognizing and partitioning approximate clones.

Approximate clones in ordinal elections generalize the classical notion of perfect clones—sets of candidates that every voter ranks consecutively—by allowing bounded imperfection in adjacencies. This relaxation is motivated by the rarity of perfect clones in large or noisy real-world preference data and offers a flexible lens for analyzing candidate similarity, voting-rule robustness, and computational tractability. Multiple rigorous frameworks for approximate cloning have been introduced, each suited to different analytical, computational, or empirical goals. The study of approximate clones also elucidates foundational properties of voting rules such as independence of clones, and highlights algorithmic and complexity boundaries for recognizing or partitioning approximate clones.

1. Formal Definitions and Measures of Approximate Cloning

The classical definition of a perfect clone requires that every voter rank all members of a subset $X\subseteq C$ (the candidate set) in consecutive positions. Formally, in an ordinal election $E=(C,V)$ with $|C|=m$ candidates and $|V|=n$ voters, $X$ is a perfect clone if for every $v\in V$: $$\max_{x\in X} \mathrm{pos}_v(x) - \min_{x\in X} \mathrm{pos}_v(x) + 1 = |X|.$$ Real-world profiles rarely satisfy this, motivating approximate generalizations:

(a) $(q,b)$-Approximate Clones (Faliszewski et al., 14 Sep 2025)

A set $X\subseteq C$ of size $b$ is a $E=(C,V)$0-approximate clone if, for each $E=(C,V)$1,

$E=(C,V)$2

That is, each voter may have up to $E=(C,V)$3 "intruder" candidates intervening between clone members.

A "distance" metric quantifies imperfection: $E=(C,V)$4 and $E=(C,V)$5 is a $E=(C,V)$6-approximate clone if $E=(C,V)$7.

(b) α-Deletion and β-Swap Approximate Clones (Delemazure, 28 Jan 2026)

For candidate pairs $E=(C,V)$8:

• The $E=(C,V)$9-deletion clone metric measures the minimal fraction of voters that must be disregarded to make $|C|=m$0 and $|C|=m$1 a perfect clone pair: $|C|=m$2
• The $|C|=m$3-swap clone metric measures the minimal average number of adjacent swaps over all voters required to render the pair adjacent everywhere: $|C|=m$4 Both measures provide quantitative diagnostics on the proximity to perfect clonality.

(c) Subelection Approximate Clones (Janeczko et al., 2024)

A pair $|C|=m$5 with $|C|=m$6, $|C|=m$7 is an approximate clone block if $|C|=m$8 is a clone set (appears contiguously) for every voter in $|C|=m$9. Approximation is parametrized by $|V|=n$0. The associated decision problem is: $|V|=n$1

2. Computational Complexity and Algorithms

Recognition and partitioning of approximate clones present sharp complexity transitions depending on the imperfection bound and parameterization:

Problem	Complexity	FPT/XP boundary	Parameters
q-Approximate Clone Recognition	NP-complete	FPT in $|V|=n$2; para-NP-hard in $|V|=n$3	$|V|=n$4: imperfection, $|V|=n$5: voters
q-Approximate Clone Partition	NP-complete	FPT in $|V|=n$6, W[1]-hard in $|V|=n$7	$|V|=n$8: imperfection, $|V|=n$9: blocks
Hidden-Clones (subelections)	In P	—	—

• Theorem: Recognizing whether a given set is a $X$0-approximate clone is NP-complete, even for fixed $X$1 (Faliszewski et al., 14 Sep 2025).
• q-Approximate Clone Recognition admits a bounded-search-tree FPT algorithm: runtime $X$2, exploiting extremal positions and recursive removal of intruders (Faliszewski et al., 14 Sep 2025).
• q-Approximate Clone Partition is also NP-complete and W[1]-hard in the number of blocks $X$3, but FPT in $X$4; a dynamic programming approach over ``segment types'' in a reference vote achieves $X$5 time (Faliszewski et al., 14 Sep 2025).
• Hidden-clone discovery via subelections can be addressed in $X$6 time through interval enumeration and hashing (Janeczko et al., 2024).

3. Theoretical Implications for Voting Rules

Approximate clones have direct implications for the robustness and axiomatic behavior of social choice rules:

• Classical independence of clones requires the winner be invariant under removal of any one candidate from a perfect clone set.
• For approximate clones, even small deviations ($X$7) are sufficient to invalidate clone-independence for all standard rules when $X$8. For rules such as IRV, Ranked Pairs, and Schulze, this failure persists even for the weakest (existential) independence form (Delemazure, 28 Jan 2026).
• Exception: With exactly three candidates, IRV is weakly independent of $X$9-deletion clones for $v\in V$0; Ranked Pairs and Schulze are independent for all $v\in V$1 in resolute profiles (Delemazure, 28 Jan 2026).
• Extension: For profiles with Smith sets of size at most three, any Condorcet-consistent, Smith-dominated-alternative-independent rule inherits weak independence from approximate clones (Delemazure, 28 Jan 2026).

4. Empirical and Practical Perspectives

Empirical analysis of diverse datasets reveals several phenomena:

• Perfect clones are rare in large real-world elections. For example, in Scottish council elections (up to $v\in V$2), $v\in V$3 of profiles exhibit perfect clones among all candidates (Delemazure, 28 Jan 2026).
• Approximate clones (low $v\in V$4 or $v\in V$5) are common, notably among same-party or semantically similar alternatives.
• In the sushi preference dataset (10 candidates, 5000 voters), the top pair appears as clones for $v\in V$6 of voters (Janeczko et al., 2024).
• In mini-jury deliberations ($v\in V$7, $v\in V$8), $v\in V$9 of profiles admit perfect clones; $$\max_{x\in X} \mathrm{pos}_v(x) - \min_{x\in X} \mathrm{pos}_v(x) + 1 = |X|.$$0 have $$\max_{x\in X} \mathrm{pos}_v(x) - \min_{x\in X} \mathrm{pos}_v(x) + 1 = |X|.$$1-swap clones (Delemazure, 28 Jan 2026).
• For an imperfection threshold ($$\max_{x\in X} \mathrm{pos}_v(x) - \min_{x\in X} \mathrm{pos}_v(x) + 1 = |X|.$$2), violations of clone-independence are frequent: IRV/Ranked Pairs show $$\max_{x\in X} \mathrm{pos}_v(x) - \min_{x\in X} \mathrm{pos}_v(x) + 1 = |X|.$$3 violations among same-party pairs in Scottish elections and $$\max_{x\in X} \mathrm{pos}_v(x) - \min_{x\in X} \mathrm{pos}_v(x) + 1 = |X|.$$4 in mini-jury experiments (Delemazure, 28 Jan 2026).
• Robustness is graded: the smaller the $$\max_{x\in X} \mathrm{pos}_v(x) - \min_{x\in X} \mathrm{pos}_v(x) + 1 = |X|.$$5, the less likely removal of an approximate clone pair affects the outcome—e.g., $$\max_{x\in X} \mathrm{pos}_v(x) - \min_{x\in X} \mathrm{pos}_v(x) + 1 = |X|.$$6 of cases for $$\max_{x\in X} \mathrm{pos}_v(x) - \min_{x\in X} \mathrm{pos}_v(x) + 1 = |X|.$$7 in mini-jury, but $$\max_{x\in X} \mathrm{pos}_v(x) - \min_{x\in X} \mathrm{pos}_v(x) + 1 = |X|.$$8 when $$\max_{x\in X} \mathrm{pos}_v(x) - \min_{x\in X} \mathrm{pos}_v(x) + 1 = |X|.$$9 (Delemazure, 28 Jan 2026).

5. Algorithmic Methods and Parameterized Analysis

Algorithmic frontiers for approximate clone discovery are tightly aligned with imperfection parameterization:

• For small $(q,b)$0, both recognition and partition admit FPT algorithms via search-tree branching (for recognition) and dynamic programming over precomputed segments (for partitioning) (Faliszewski et al., 14 Sep 2025).
• Partitioning is in XP with respect to $(q,b)$1 (number of blocks), but W[1]-hard, showing that controlling the number of partitions alone is insufficient for tractability (Faliszewski et al., 14 Sep 2025).
• Subelection-based approaches enable scalable discovery in practice via efficient enumeration and hashing (Janeczko et al., 2024).

6. Interpretative and Methodological Insights

Approximate clones offer diagnostic power in empirical election forensics by quantifying degrees of candidate similarity and their effect on electoral outcomes. The choice between measures (positional block span, $(q,b)$2-deletion fraction, $(q,b)$3-swap distance, or voter/candidate subelection thresholds) enables tailored analyses for theoretical, computational, or applied election studies. The pattern that the probability of outcome change strictly decreases with clone closeness (as measured by $(q,b)$4 or $(q,b)$5) provides an empirical foundation for the design of rules with potential $(q,b)$6-clone-robustness properties—an open direction in the literature (Delemazure, 28 Jan 2026).

Approximate clone theory also connects with the analysis of hidden subelections, revealing latent structure such as undifferentiated clusters and price/ideological orderings in real datasets. The distinct parameterizations—imperfection per voter, number of outsiders, or subset selection—reflect different perspectives on what it means for alternatives to be "almost indistinguishable" in collective rank data.

7. Summary Table: Key Definitions

Clone Type	Mathematical Criterion	Main Parameter(s)
Perfect clone	$(q,b)$7 ∀$(q,b)$8	$(q,b)$9: candidate set
(q, b)-approximate clone	$X\subseteq C$0 ∀$X\subseteq C$1	$X\subseteq C$2: imperfection, $X\subseteq C$3: size bound
α-deletion pair	$X\subseteq C$4	$X\subseteq C$5: voter deletion fraction
β-swap pair	$X\subseteq C$6	$X\subseteq C$7: swaps per voter
Subelection clones	$X\subseteq C$8 $X\subseteq C$9 contiguous for all $b$0	$b$1, $b$2: size thresholds

Approximate clones, in their various forms, provide a robust toolkit for both theoretical and empirical electoral analysis, exposing both computational challenges and deeper axiomatic limitations on voting rules while enabling practical discovery of meaningful near-clone patterns in real data (Faliszewski et al., 14 Sep 2025, Janeczko et al., 2024, Delemazure, 28 Jan 2026).