[Paper Review] Hybrid Elections Broaden Complexity-Theoretic Resistance to Control

Edith Hemaspaandra, Lane A. Hemaspaandra+1 more|ArXiv.org|Aug 14, 2006

Game Theory and Voting SystemsEconomics, Econometrics and Finance21 references20 citations

TL;DR

This paper introduces the hybrid election system, a method that combines multiple candidate-anonymous voting rules to inherit and unify their resistance to all twenty standard types of constructive and destructive control. The key contribution is proving the existence of an election system resistant to all twenty control types, leveraging complexity-theoretic hardness as a defense against manipulation.

ABSTRACT

Electoral control refers to attempts by an election's organizer ("the chair") to influence the outcome by adding/deleting/partitioning voters or candidates. The groundbreaking work of Bartholdi, Tovey, and Trick [BTT92] on (constructive) control proposes computational complexity as a means of resisting control attempts: Look for election systems where the chair's task in seeking control is itself computationally infeasible. We introduce and study a method of combining two or more candidate-anonymous election schemes in such a way that the combined scheme possesses all the resistances to control (i.e., all the NP-hardnesses of control) possessed by any of its constituents: It combines their strengths. From this and new resistance constructions, we prove for the first time that there exists an election scheme that is resistant to all twenty standard types of electoral control.

Motivation & Objective

• To address the vulnerability of existing voting systems to strategic manipulation by control (e.g., adding/deleting voters or candidates).
• To develop a method that combines multiple election systems to inherit the strongest resistance properties of their components.
• To prove the existence of a single election system that resists all twenty standard types of constructive and destructive control.
• To formalize and analyze the behavior of hybrid elections under various control models (e.g., partition, deletion, run-off).

Proposed method

• Proposes a hybrid election system, denoted $\text{hybrid}(\mathcal{E}_0, \mathcal{E}_1)$, that combines two candidate-anonymous election systems $\mathcal{E}_0$ and $\mathcal{E}_1$ using a randomized selection mechanism.
• Employs a probabilistic voting rule where the outcome is determined by selecting one of the constituent systems uniformly at random and applying it to the input.
• Uses complexity-theoretic reductions to prove that if any constituent system resists a type of control, the hybrid system inherits that resistance.
• Applies NP-hardness reductions to show resistance to control under various models: constructive and destructive control by adding/deleting voters or candidates, and partitioning voters or candidates (in models TE and TP).
• Leverages the fact that if $\text{P} \neq \text{NP}$, then resistance to control implies non-vulnerability, enabling strong inheritance results.
• Employs a proof technique based on the disjoint union of problems and extends it to election systems via a new combinatorial construction.

Experimental results

Research questions

• RQ1 Can a hybrid election system be constructed such that it inherits resistance to control from its constituent systems?
• RQ2 Is there an election system that resists all twenty standard types of constructive and destructive control?
• RQ3 What conditions ensure that the hybrid system inherits resistance rather than vulnerability?
• RQ4 How does the hybrid system behave under different control models, including partition and run-off partition?
• RQ5 Under what conditions does the hybrid system fail to inherit vulnerability, and what does this imply about the P vs. NP question?

Key findings

• The hybrid election system $\text{hybrid}(\mathcal{E}_0, \mathcal{E}_1)$ inherits resistance to all control types that are resisted by at least one of its components.
• There exists an election system that is resistant to all twenty standard types of constructive and destructive control, resolving an open question in computational social choice.
• The hybrid system strongly inherits resistance to destructive control by deleting candidates, partition of candidates (in models TE and TP), and run-off partition of candidates (in models TE and TP).
• If $\text{P} \neq \text{NP}$, then the hybrid system does not inherit vulnerability to destructive control by partition or run-off partition of candidates, implying that resistance is preserved.
• The equivalence between the hybrid system inheriting vulnerability and $\text{P} = \text{NP}$ is established for all control types involving partition and run-off partition.
• A concrete example is provided: a hybrid system combining a vulnerable system with an immune one results in a resistant system, demonstrating the method’s effectiveness.