When are elections with few candidates hard to manipulate?
Journal of the ACM (JACM)
Nonexistence of voting rules that are usually hard to manipulate
AAAI'06 Proceedings of the 21st national conference on Artificial intelligence - Volume 1
Junta distributions and the average-case complexity of manipulating elections
Journal of Artificial Intelligence Research
Universal voting protocol tweaks to make manipulation hard
IJCAI'03 Proceedings of the 18th international joint conference on Artificial intelligence
Where are the really hard manipulation problems? the phase transition in manipulating the veto rule
IJCAI'09 Proceedings of the 21st international jont conference on Artifical intelligence
An Empirical Study of the Manipulability of Single Transferable Voting
Proceedings of the 2010 conference on ECAI 2010: 19th European Conference on Artificial Intelligence
Hybrid voting protocols and hardness of manipulation
ISAAC'05 Proceedings of the 16th international conference on Algorithms and Computation
Unweighted coalitional manipulation under the Borda rule Is NP-hard
IJCAI'11 Proceedings of the Twenty-Second international joint conference on Artificial Intelligence - Volume Volume One
Convergence of iterative voting
Proceedings of the 11th International Conference on Autonomous Agents and Multiagent Systems - Volume 2
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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 it exists, 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 simple first 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 it exists and otherwise computes the Plurality winner is NP-hard to manipulate with three 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 three 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. With two stage voting rules, we can also allow agents to re-vote between rounds. We study the impact of such re-voting on manipulation.