The Complexity of Manipulative Attacks in Nearly Single-Peaked Electorates

TL;DR

This paper investigates how the complexity of manipulative actions in elections is affected when electorates are nearly single-peaked, revealing that even a few mavericks can make problems NP-hard, but some cases remain tractable.

Contribution

It analyzes the complexity of manipulative actions in nearly single-peaked electorates, identifying conditions under which complexity increases or remains manageable.

Findings

One maverick can make manipulative problems NP-hard.

Some nearly single-peaked cases remain polynomial-time solvable.

Tolerance to mavericks varies depending on election system and nearness measure.

Abstract

Many electoral bribery, control, and manipulation problems (which we will refer to in general as "manipulative actions" problems) are NP-hard in the general case. It has recently been noted that many of these problems fall into polynomial time if the electorate is single-peaked (i.e., is polarized along some axis/issue). However, real-world electorates are not truly single-peaked. There are usually some mavericks, and so real-world electorates tend to merely be nearly single-peaked. This paper studies the complexity of manipulative-action algorithms for elections over nearly single-peaked electorates, for various notions of nearness and various election systems. We provide instances where even one maverick jumps the manipulative-action complexity up to $\np$-hardness, but we also provide many instances where a reasonable number of mavericks can be tolerated without increasing the manipulative-action complexity.