Manipulating two stage voting rules

We study the computational complexity of computing a manipulation of a two stage voting rule. An example of a two stage voting rule is Black’s procedure. The first stage of Black’s procedure selects the Condorcet winner if they exist, otherwise the second stage selects the Borda winner. In general, we argue that there is no connection between the computational complexity of manipulating the two stages of such a voting rule and that of the whole. However, we also demonstrate that we can increase the complexity of even a very simple base rule by adding a stage to the front of the base rule. In particular, whilst Plurality is polynomial to manipulate, we show that the two stage rule that selects the Condorcet winner if they exist and otherwise computes the Plurality winner is NP-hard to manipulate with 3 or more candidates, weighted votes and a coalition of manipulators. In fact, with any scoring rule, computing a coalition manipulation of the two stage rule that selects the Condorcet winner if they exist and otherwise applies the scoring rule is NP-hard with 3 or more candidates and weighted votes. It follows that computing a coalition manipulation of Black’s procedure is NP-hard with weighted votes. With unweighted votes, we prove that the complexity of manipulating Black’s procedure is inherited from the Borda rule that it includes. More specifically, a single manipulator can compute a manipulation of Black’s procedure in polynomial time, but computing a manipulation is NP-hard for two manipulators.