How Hard is Safe Bribery?

Bribery in an election is one of the well-studied control problems in computational social choice. In this paper, we propose and study the safe bribery problem. Here the goal of the briber is to ask the bribed voters to vote in such a way that the briber never prefers the original winner (of the unbribed election) more than the new winner, even if the bribed voters do not fully follow the briber's advice. Indeed, in many applications of bribery, campaigning for example, the briber often has limited control on whether the bribed voters eventually follow her recommendation and thus it is conceivable that the bribed voters can either partially or fully ignore the briber's recommendation. We provide a comprehensive complexity theoretic landscape of the safe bribery problem for many common voting rules in this paper.