From Independence of Clones to Composition Consistency: A Hierarchy of Barriers to Strategic Nomination

TL;DR

The paper addresses how voting rules resist manipulation via cloning and composition using two axioms: IoC and the stronger CC. It establishes a strict hierarchy where CC entails IoC, but not vice versa, and investigates whether common IoC rules satisfy CC; most do not, though a variant of Ranked Pairs, RP_i, does. The authors introduce a PQ-tree–based CC transformation that converts any neutral SCF into a CC rule while preserving key properties such as Condorcet/Smith consistency and decisiveness, and they prove fixed-parameter tractability results tied to the PQ-tree’s decomposition degree. They extend the CC framework to social preference functions and analyze how CC interacts with strategic candidacy, showing CC yields obvious strategy-proofness under their transform. The work has practical implications for designing clone-robust decision rules in AI alignment and RLHF contexts, where cloning-like candidate duplications can arise, and offers a tractable pathway to CC-compliant aggregation in complex candidate spaces.

Abstract

We study two axioms for social choice functions that capture the impact of similar candidates: independence of clones (IoC) and composition consistency (CC). We clarify the relationship between these axioms by observing that CC is strictly more demanding than IoC, and investigate whether common voting rules that are known to be independent of clones (such as STV, Ranked Pairs, Schulze, and Split Cycle) are composition-consistent. While for most of these rules the answer is negative, we identify a variant of Ranked Pairs that satisfies CC. Further, we show how to efficiently modify any (neutral) social choice function so that it satisfies CC, while maintaining its other desirable properties. Our transformation relies on the hierarchical representation of clone structures via PQ-trees. We extend our analysis to social preference functions. Finally, we interpret IoC and CC as measures of robustness against strategic manipulation by candidates, with IoC corresponding to strategy-proofness and CC corresponding to obvious strategy-proofness.

Figures (9)

• Figure 1: A preference profile. Columns show rankings, with the bottom row ranked last. The first row shows the number of copies of each ranking (e.g., leftmost column indicates 6 voters rank $b\succ c\succ a\succ d$).
• Figure 2: (Left) Example profile $\boldsymbol{\sigma}$. (Right) $\boldsymbol{\sigma}\setminus \{a_2\}$.
• Figure 3: (Left) $\boldsymbol{\sigma}^\mathcal{K}$, where clone sets from $\boldsymbol{\sigma}$ in \ref{['fig:bg_eg']} are condensed into candidates $K_a$, $K_b$, and $K_c$. (Right) $\boldsymbol{\sigma}|_{K_a}$, where $\boldsymbol{\sigma}$ is limited to members of $K_a$.
• Figure 4: Behavior of SCFs from \ref{['tab:scfs']} w.r.t IoC/CC. $\dagger$ indicates majoritarian SCFs.
• Figure 5: (Left) The PQ-tree representing $\mathcal{C}(\boldsymbol{\sigma})$ from \ref{['ex:pq-tree']} . (Right) The PQ-tree of $\boldsymbol{\sigma}$ from \ref{['fig:bg_eg']}.

Theorems & Definitions (122)

• Definition 1: Tideman87:Independence; Laffond96:Composition
• Definition 2: Zavist89:Complete
• Example 3
• Definition 4
• Definition 5
• Example 6
• Definition 7: Laffond96:Composition
• Proposition 8
• Theorem 1
• proof