Dichotomy for Voting Systems

TL;DR

This paper characterizes when scoring-based voting systems are computationally resistant to manipulation, showing that most are NP-complete to manipulate unless they are trivial or similar to plurality voting.

Contribution

It provides a complete dichotomy theorem classifying scoring protocols as either polynomial-time manipulable or NP-complete to manipulate based on their score vector structure.

Findings

Systems with two or more non-first score values are NP-complete to manipulate.

Systems with only one non-first score value are manipulable in polynomial time.

Trivial, plurality, and related systems are exceptions to the NP-completeness.

Abstract

Scoring protocols are a broad class of voting systems. Each is defined by a vector $(\alpha_1,\alpha_2,...,\alpha_m)$, $\alpha_1 \geq \alpha_2 \geq >... \geq \alpha_m$, of integers such that each voter contributes $\alpha_1$ points to his/her first choice, $\alpha_2$ points to his/her second choice, and so on, and any candidate receiving the most points is a winner. What is it about scoring-protocol election systems that makes some have the desirable property of being NP-complete to manipulate, while others can be manipulated in polynomial time? We find the complete, dichotomizing answer: Diversity of dislike. Every scoring-protocol election system having two or more point values assigned to candidates other than the favorite--i.e., having $||\{\alpha_i \condition 2 \leq i \leq m\}||\geq 2$--is NP-complete to manipulate. Every other scoring-protocol election system can be manipulated in polynomial time. In effect, we show that--other than trivial systems (where all candidates alway tie), plurality voting, and plurality voting's transparently disguised translations--\emph{every} scoring-protocol election system is NP-complete to manipulate.