Unweighted Coalitional Manipulation Under the Borda Rule Is NP-Hard

The Borda voting rule is a positional scoring rule where, for m candidates, for every vote the first candidate receives m-1 points, the second m-2 points and so on. A Borda winner is a candidate with highest total score. It has been a prominent open problem to determine the computational complexity of Unweighted Coalitional Manipulation under Borda: Can one add a certain number of additional votes (called manipulators) to an election such that a distinguished candidate becomes a winner? We settle this open problem by showing NP-hardness even for two manipulators and three input votes. Moreover, we discuss extensions and limitations of this hardness result.