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The Borda voting rule is a positional scoring rule for $z$ candidates such that in each vote, the first candidate receives $z-1$ points, the second $z-2$ points and so on. The winner in the Borda rule is the candidate with highest total score. We study the manipulation problem of the Borda rule in a setting with two non-manipulators while one of the non-manipulators vote is weighted. We demonstrate a sharp contrast on computational complexity depending on the weight of the non-manipulator: the problem is NP-hard when the weight is larger than $1$ while there exists an efficient algorithm to find a manipulation when the weight is at most $1$.
We prove that it is NP-hard for a coalition of two manipulators to compute how to manipulate the Borda voting rule. This resolves one of the last open problems in the computational complexity of manipulating common voting rules. Because of this NP-ha
Most work on manipulation assumes that all preferences are known to the manipulators. However, in many settings elections are open and sequential, and manipulators may know the already cast votes but may not know the future votes. We introduce a fram
Most work on manipulation assumes that all preferences are known to the manipulators. However, in many settings elections are open and sequential, and manipulators may know the already cast votes but may not know the future votes. We introduce a fram
Schulzes rule is used in the elections of a large number of organizations including Wikimedia and Debian. Part of the reason for its popularity is the large number of axiomatic properties, like monotonicity and Condorcet consistency, which it satisfi
A key question in cooperative game theory is that of coalitional stability, usually captured by the notion of the emph{core}--the set of outcomes such that no subgroup of players has an incentive to deviate. However, some coalitional games have empty