Independence of Approximate Clones

• Independence of Approximate Clones is a framework defining how relaxed cloning operations affect the stability of algebraic and social choice structures.
• The concept utilizes α-deletion and β-swap metrics to analyze the robustness of voting rules, highlighting failures for m ≥ 4 candidates and resilience for m = 3.
• Investigations in Boolean CSPs and category theory reveal that approximate clone structures determine expressibility independence and complexity classifications despite structural relaxations.

Independence of Approximate Clones encompasses a spectrum of concepts in universal algebra, social choice theory, and categorical algebra, which analyze the stability or expressibility of structures when certain "cloning" operations are introduced or modified. In all cases, the notion of "approximate clones" generalizes the exact notion of structural duplication—whether of candidates in elections, operations in Boolean function clones, or morphisms in algebraic categories—by considering perturbations, relaxations, or approximate equivalence, and studies how related axioms or invariants behave under these relaxations.

1. Formal Definitions and Contexts

The mathematical definition of "clone" varies by field but always describes a set of operations or objects closed under key composition principles. In social choice, clones are sets of candidates ranked adjacently by all voters; in Boolean function theory, clones refer to sets of operations closed under composition; in category theory, clones are Lawvere theories—cartesian operads describing finitary algebraic theories.

The independence of clones axiom in the context of voting states that removing one candidate from a pair of perfect clones should not affect the outcome of an election. However, perfect clones (e.g., two candidates ranked adjacently in every ballot) are rare in large-scale applications. Approximate clones generalize this concept, permitting some deviation from perfect adjacency and requiring new measures of proximity and corresponding axioms (Delemazure, 28 Jan 2026).

Two principal formalizations for approximate clones in ordinal elections are:

• α-deletion clones: A pair {x, y} is α-deletion clones in a profile P if the fraction of voters not ranking x and y adjacently is at most α.
• β-swap clones: The average number of adjacent swaps required to make x and y into perfect clones across all voters does not exceed β.

In functional clone theory on the Boolean domain, approximate or "pps" (primitive product–summation) clones are sets of real-valued functions closed under product, summation, limits, and including equality, capturing the expressibility of weighted counting CSPs and their reduction under approximation-preserving reductions (Bulatov et al., 2011). Category theory analyses the independence of clone structures from underlying comonoid properties via endomorphism operads, and investigates when the Lawvere theory axioms are satisfied even in the absence of canonical structural features (such as cocommutativity) (Krähmer et al., 2021).

2. Independence Properties in Ordinal Elections

In ordinal elections, the independence of approximate clones axiom refines traditional clone-independence to allow for the presence of nearly—but not perfectly—identical candidates. A single-winner rule $f$ satisfies independence of α-deletion clones if, for any pair {x, y} deemed α-deletion clones in a profile P, the removal of x (or y) preserves the set of winners, except possibly for the removed candidate. Weak independence requires this for at least one of x, y (Delemazure, 28 Jan 2026).

Key theoretical results establish:

• Non-existence for $m \ge 4$: For $m \ge 4$ candidates, popular rules such as IRV, Ranked Pairs, and Schulze fail (even weak) independence of α-deletion (or β-swap) clones for any $\alpha > 0$ (or $\beta > 0$), due to spoiler effects. Theorems 4.1 and 4.2 demonstrate that even infinitesimal deviation from perfect cloning leads to outcome changes in explicit constructions.
• Positive results for $m = 3$: For three candidates, IRV is weakly independent of α-deletion (and β-swap) clones for $\alpha \le 1/3$, while Ranked Pairs and Schulze are weakly independent of approximate clones for all $\alpha > 0$, provided the profile produces a unique winner. This reflects a collapse of the two distance measures in the three-candidate case, simplifying the combinatorics of adjacency.

Theoretical impossibility for $m \ge 4$ underscores the brittleness of clone independence once exact adjacency is relaxed, highlighting intrinsic limitations of deterministic voting rules.

3. Functional Clones and Approximate Expressibility

In Boolean valued CSPs, pps-functional clones formalize which weighted constraint functions can be approximately expressed (via efficient polynomial-time approximate transformations) from a base set. The set of all nonnegative real-valued functions $p$ contains nested subclones:

• NEQ-clone: Generated by Boolean equality and unary functions.
• LSM-clone: All log-supermodular functions, satisfying $F(x \vee y)F(x \wedge y) \geq F(x)F(y)$ for all $x, y$.
• Total clone: All possible nonnegative functions.

Key results (Bulatov et al., 2011):

• No proper clone exists strictly between the LSM clone and the total clone once all nonnegative unary functions are available (the conservative case).
• Any function outside LSM generates the entire clone of all functions, making the associated counting problem as hard as #P.
• The “implies” clone, generated by the binary function $IMP(x, y) = 1$ for $(x, y) \neq (1, 0)$, coincides with the LSM clone up to arity 3 via Möbius and Fourier analytic arguments. For arity 4, explicit functions show a strict separation.

This structure establishes expressibility independence of approximate clones: adding any sufficiently expressive non-LSM function immediately boosts the expressibility to the maximum—no intermediate complexity is possible.

4. Approximate Clones in Category Theory and Clone Independence

In categorical algebra, comonoids in a braided monoidal category $(\mathcal{C}, \otimes, 1)$ generalize copying and deletion operations, fundamental to logic and computation. The endomorphism operad of a comonoid $X$ collects all comonoid morphisms $X^{\otimes n} \to X$. A central question is when this operad actually forms a Lawvere theory (clone), i.e., when the structure is cartesian.

The classification (Krähmer et al., 2021):

• For the full monoidal subcategory generated by $X$, the endomorphism operad is a clone if and only if $X$ is cocommutative and the braiding on $X \otimes X$ is symmetric.
• However, existence of a clone structure on the endomorphism operad does not in general imply cocommutativity: explicit non-cocommutative comonoids can yield an approximate clone if one restricts attention to ring-homomorphic morphisms arising from specific algebraic constructions.

This demonstrates a categorical version of independence of approximate clones: clone structure does not necessarily reflect stricter structural features (such as cocommutativity), indicating that "approximate" (i.e., limited) closure under operations can arise independently of global algebraic invariants.

5. Empirical and Complexity-Theoretic Observations

Empirical studies of real-world ordinal election datasets show that approximate clones are prevalent, especially in contexts with structurally similar candidates (e.g., multiple candidates from the same party in Scottish local elections, or figure skating competitions with highly similar judges' preferences). Specific findings (Delemazure, 28 Jan 2026):

• Approximate clones occur with high frequency in certain domains (e.g., 46% perfect clones in figure skating, average minimum α ≈ 0.09).
• Even though theoretical results predict universal failure of independence for $m \ge 4$, voting rules such as IRV and Ranked Pairs frequently respect independence for near-clone pairs (for small α or β), with empirical failure rates substantially below 100%.
• Weak independence of approximate clones holds for >90% of all tested pairs/rules/profiles, reflecting that one-sided spoiler effects dominate rather than universal collapse.

In counting CSPs, the functional clone structure establishes a sharp trichotomy: trivial, #BIS-equivalent (LSM), and #P-complete (#P) complexity classes (Bulatov et al., 2011). Escape from the LSM class (i.e., adding a single non-LSM function) produces universality with respect to approximate counting complexity.

6. Limitations, Open Problems, and Future Directions

Current analyses focus predominantly on pairs of approximate clones; extensions to larger clone sets or higher-arity phenomena are largely unexplored (Delemazure, 28 Jan 2026). In Boolean function theory, the longstanding question of finite generability of the log-supermodular clone remains open: evidence suggests no finite arity suffices (Bulatov et al., 2011). A similar open problem concerns the search for intermediate, nontrivial approximate clone structures in the conservative setting, which appears unlikely.

Categorically, a plausible implication is that further research may identify other structural invariants compatible with clone formation beyond cocommutativity, especially under more restrictive classes of morphisms or subcategories (Krähmer et al., 2021). In social choice, the search for practical voting rules with both theoretical and empirical robustness to approximate clones motivates the consideration of randomized, probabilistic, or utility-based frameworks as well as new axiomatic criteria.

A synthesis of empirical and theoretical perspectives suggests that while strict independence of approximate clones is typically unachievable, practical rules often demonstrate substantial robustness, justifying the use of classical axioms as proxies for spoiler-resistance in realistic settings, despite underlying theoretical brittleness.