How Hard Is Bribery in Elections?

TL;DR

This paper investigates the computational complexity of election bribery, revealing that it varies greatly depending on the election system and setting, with some cases being efficiently solvable and others NP-complete.

Contribution

The paper provides a comprehensive complexity analysis of bribery in various election systems, including scoring protocols and Dodgson voting, with new dichotomy results for weighted bribery cases.

Findings

Bribery complexity varies significantly across different election systems.

Certain bribery problems are NP-complete, while others are solvable in polynomial time.

A dichotomy classification for bribery in scoring protocols is established.

Abstract

We study the complexity of influencing elections through bribery: How computationally complex is it for an external actor to determine whether by a certain amount of bribing voters a specified candidate can be made the election's winner? We study this problem for election systems as varied as scoring protocols and Dodgson voting, and in a variety of settings regarding homogeneous-vs.-nonhomogeneous electorate bribability, bounded-size-vs.-arbitrary-sized candidate sets, weighted-vs.-unweighted voters, and succinct-vs.-nonsuccinct input specification. We obtain both polynomial-time bribery algorithms and proofs of the intractability of bribery, and indeed our results show that the complexity of bribery is extremely sensitive to the setting. For example, we find settings in which bribery is NP-complete but manipulation (by voters) is in P, and we find settings in which bribing weighted voters is NP-complete but bribing voters with individual bribe thresholds is in P. For the broad class of elections (including plurality, Borda, k-approval, and veto) known as scoring protocols, we prove a dichotomy result for bribery of weighted voters: We find a simple-to-evaluate condition that classifies every case as either NP-complete or in P.