Closing the Gap of Control Complexity in Borda Elections: Solving ten open cases

We consider the problem of control in elections where an election chair seeks to either make a designated candidate win, or prevent her from winning, via actions such as adding, deleting, or partitioning either candidates or voters. These scenarios have been studied for many voting systems and the related control problems have been classified in terms of their complexity. However, for one of the most prominent natural voting systems, the Borda Count, complexity results are known for only half of these cases. We settle the complexity for ten missing cases in the unique-winner model, leaving just one case open. We also show that Borda is vulnerable to control for this one open case in the nonunique-winner model. An interesting consequence is that Borda is vulnerable to another type of control in the nonunique-winner model, yet it is resistant to it in the unique-winner model. This is one of the few known cases where the complexity of control problems differs depending on the winner model chosen.