  We study electoral campaign management scenarios in which an external party
can buy votes, i.e., pay the voters to promote its preferred candidate in their
preference rankings. The external party's goal is to make its preferred
candidate a winner while paying as little as possible. We describe a
2-approximation algorithm for this problem for a large class of electoral
systems known as scoring rules. Our result holds even for weighted voters, and
has applications for campaign management in commercial settings. We also give
approximation algorithms for our problem for two Condorcet-consistent rules,
namely, the Copeland rule and maximin.
