[Paper Review] X THEN X: Manipulation of Same-System Runoff Elections

Zack Fitzsimmons, Edith Hemaspaandra+1 more|arXiv (Cornell University)|Jan 25, 2013

Internet Traffic Analysis and Secure E-voting26 references3 citations

TL;DR

This paper investigates the computational complexity of manipulating runoff elections using the same voting rule in both rounds, comparing standard runoffs and runoffs with revoting. It shows that for natural election systems, runoffs can increase manipulation complexity—even when the base system is easy to manipulate—providing the first proof that same-system runoffs can raise manipulation hardness, with examples across weighted and unweighted settings.

ABSTRACT

Do runoff elections, using the same voting rule as the initial election but just on the winning candidates, increase or decrease the complexity of manipulation? Does allowing revoting in the runoff increase or decrease the complexity relative to just having a runoff without revoting? For both weighted and unweighted voting, we show that even for election systems with simple winner problems the complexity of manipulation, manipulation with runoffs, and manipulation with revoting runoffs are independent, in the abstract. On the other hand, for some important, well-known election systems we determine what holds for each of these cases. For no such systems do we find runoffs lowering complexity, and for some we find that runoffs raise complexity. Ours is the first paper to show that for natural, unweighted election systems, runoffs can increase the manipulation complexity.

Motivation & Objective

• To analyze how the complexity of manipulating runoff elections compares to the original election system when the same voting rule is used in both rounds.
• To determine whether allowing revoting in the runoff increases or decreases manipulation complexity relative to standard runoffs.
• To investigate whether runoffs can raise manipulation complexity in natural, real-world election systems with tractable winner problems.
• To identify conditions under which runoffs or revoting runoffs make manipulation NP-complete even when the base system is in P.
• To provide a comprehensive classification of manipulation complexity across three scenarios: base system, runoff, and revoting runoff, for various well-known voting systems.

Proposed method

• Reduction from the NP-complete Partition problem to construct elections where manipulation complexity shifts due to runoffs.
• Use of McGarvey's construction to generate weighted majority graphs with desired pairwise margins, enabling control over runoff outcomes.
• Design of weighted and unweighted election systems where manipulation complexity changes between P and NP-complete across the three scenarios (base, runoff, revoting runoff).
• Analysis of specific systems like Plurality, Veto, HalfApproval, and Llull to demonstrate complexity shifts under runoffs.
• Proof of NP-completeness for Llull elections with runoffs and revoting by constructing majority graphs that enforce cycle structures dependent on partitionability.
• Contrast of cases where revoting reduces complexity (e.g., from NP-complete to P) to show independence between the three complexity classes.

Experimental results

Research questions

• RQ1 Can same-system runoffs increase the computational complexity of manipulating an election, even when the base system is easy to manipulate?
• RQ2 Does allowing revoting in the runoff always increase or decrease manipulation complexity, and how does it compare to standard runoffs?
• RQ3 Are there natural, real-world election systems where manipulation of the base system is in P, but manipulation with runoffs becomes NP-complete?
• RQ4 Is there a universal relationship between the complexity of manipulation with runoffs and with revoting runoffs, or can they be independent?
• RQ5 Can we construct election systems where manipulation complexity drops from NP-complete to P when revoting is allowed, despite runoffs increasing it?

Key findings

• For unweighted HalfApproval, manipulation with both runoffs and revoting runoffs remains NP-complete even with just one manipulator.
• In weighted Veto elections, manipulation is in P, but becomes NP-complete when runoffs or revoting runoffs are introduced.
• For four-candidate weighted Llull elections, manipulation with runoffs and with revoting runoffs are both NP-complete, even when the base system is in P.
• There exist systems where manipulation of the base system is NP-complete, but manipulation with runoffs is in P, demonstrating that runoffs can reduce complexity.
• The paper constructs a system where manipulation with runoffs is NP-complete, but with revoting runoffs it becomes P, proving that revoting can lower complexity.
• No universal relationship exists between the manipulation complexity of the base system, runoffs, and revoting runoffs—each can be independently P or NP-complete.