We study the computational complexity of control problems for two-stage voting rules. An example of a two-stage voting rule is the Black's procedure. The first stage of the Black's procedure selects the Condorcet winner if one exists; otherwise, in the second stage the Borda winner is selected. The computational complexity of the manipulation problem of two-stage voting rules has recently been studied by Narodytska and Walsh [20] and Fitzsimmons et al. [14]. Extending their work, we consider the control problems for similar scenarios, focusing on constructive control by adding or deleting votes, denoted as CCAV and CCDV, respectively.

Let X be the voting rule applied in the first stage and Y the one in the second stage. As for the manipulation problem shown in [20, 14], we prove that there is basically no connection between the complexity of CCAV and CCDV for X or Y and the complexity of CCAV and CCDV for the two-stage election X THEN Y: CCAV and CCDV for X THEN Y could be NP-hard, while both problems are polynomial-time solvable for X and Y . On the other hand, combining two rules X and Y, both with NP-hard CCAV and CCDV, could lead to a two-stage election, where both CCAV and CCDV become polynomial-time solvable. Hereby, we also achieve some complexity results for the special case X THEN X. In addition, we show that, compared to the manipulation problem, the control problems for two-stage elections admit more diverse behaviors concerning their complexity. For example, there exist rules X and Y, for each of which CCAV and CCDV have the same complexity, but CCAV and CCDV behave differently for X THEN Y.