Toward Completing the Picture of Control in Schulze and Ranked Pairs Elections

TL;DR

This work analyzes electoral control for Schulze and ranked pairs, focusing on exact multimode control and control by replacing candidates or voters, in both unique- and nonunique-winner settings. It fixes a flaw in a prior reduction for Schulze-CCDC and establishes NP-hardness in both winner models, while also deriving polynomial-time algorithms for certain destructive-control variants via graph-theoretic s-t path-cut concepts. The authors introduce and leverage path-preserving and colored path-preserving vertex-cut problems to connect control actions to graph structure, and provide a general framework showing that multimode control inherits hardness from standard control under insensitivity to bottom-ranked candidates. Collectively, the results map out a broad landscape of resistance and vulnerability: Schulze and ranked pairs exhibit NP-hardness across many multimode and replacing variants, yet several destructive-control cases remain tractable in the nonunique-winner setting, offering practical insights for designing robust voting systems. The work thus advances both theoretical understanding and practical auditing of these Condorcet-consistent voting rules, with implications for the security and reliability of decision-making processes that rely on such elections.

Abstract

Both Schulze and ranked pairs are voting rules that satisfy many natural, desirable axioms. Many standard types of electoral control (with a chair seeking to change the outcome of an election by interfering with the election structure) have already been studied. However, for control by replacing candidates or voters and for (exact) multimode control that combines multiple standard attacks, many questions remain open. We solve a number of these open cases for Schulze and ranked pairs. In addition, we fix a flaw in the reduction of Menton and Singh [IJCAI 2013] showing that Schulze is resistant to constructive control by deleting candidates and re-establish a vulnerability result for destructive control by deleting candidates. In some of our proofs, we study variants of s-t vertex cuts in graphs that are related to our control problems.

Figures (11)

• Figure 1: WMG corresponding to the election constructed from the $\mathrm{3\textsc{SAT}}$ instance from our counterexample with the new reduction. Dashed edges have weight two and drawn edges have weight four.
• Figure 2: The WMG of the election from the proof of Theorem \ref{['theo:votercontrol']}.
• Figure 3: A WMG of the election in Example \ref{['ex:exact-vs-non-exact']}. Candidates from the set of unregistered candidates $D$ are depicted using dashed lines.
• Figure 4: Weighted majority graph in Example \ref{['ex:schulzemethod']}.
• Figure 4: Candidate pairs ordered by difference in pairwise comparison in Example \ref{['ex:rankedpairsmethod']}.

Theorems & Definitions (24)

• Theorem 1
• proof
• Theorem 2
• Theorem 3
• proof
• Theorem 4
• Lemma 1
• proof
• Lemma 2
• Lemma 3