New Complexity Results on Coalitional Manipulation of Borda

TL;DR

This paper investigates the computational complexity of manipulating Borda voting with two non-manipulators, revealing a sharp contrast: it is NP-hard with weighted votes greater than one, but efficiently solvable when weights are at most one.

Contribution

The paper provides new complexity results for Borda manipulation, showing a clear boundary based on the weight of non-manipulators' votes.

Findings

NP-hardness when non-manipulator weight > 1

Polynomial-time algorithm when weight ≤ 1

Sharp complexity contrast based on vote weight

Abstract

The Borda voting rule is a positional scoring rule for $z$ candidates such that in each vote, the first candidate receives $z-1$ points, the second $z-2$ points and so on. The winner in the Borda rule is the candidate with highest total score. We study the manipulation problem of the Borda rule in a setting with two non-manipulators while one of the non-manipulator's vote is weighted. We demonstrate a sharp contrast on computational complexity depending on the weight of the non-manipulator: the problem is NP-hard when the weight is larger than $1$ while there exists an efficient algorithm to find a manipulation when the weight is at most $1$.