# The Shield that Never Was: Societies with Single-Peaked Preferences are More Open to Manipulation and Control

## TL;DR

This paper demonstrates that in societies with single-peaked preferences, the computational complexity barriers that protect elections from manipulation and control are ineffective, making such elections more vulnerable than previously thought.

## Contribution

It shows that the NP-hardness protections against manipulation and control do not hold in societies with single-peaked preferences, challenging prior assumptions about election security.

## Key findings

- NP-hardness shields vanish with single-peaked preferences
- Single-peaked societies are more susceptible to manipulation
- Complexity-based protections are ineffective in these societies

## Abstract

Much work has been devoted, during the past twenty years, to using complexity to protect elections from manipulation and control. Many results have been obtained showing NP-hardness shields, and recently there has been much focus on whether such worst-case hardness protections can be bypassed by frequently correct heuristics or by approximations. This paper takes a very different approach: We argue that when electorates follow the canonical political science model of societal preferences the complexity shield never existed in the first place. In particular, we show that for electorates having single-peaked preferences, many existing NP-hardness results on manipulation and control evaporate.