Various Condorcet consistent social choice functions based on majority rule (tournament solutions) are considered in the general case, when ties are allowed: the core, the weak and strong top cycle sets, versions of the uncovered andminimal weakly stable sets, the uncaptured set, the untrapped set, classes of k-stable alternatives and k-stable sets. The main focus of the paper is to construct a unified matrix-vector representation of a tournament solution in order to get a convenient algorithm for its calculation. New versions of some solutions are also proposed.
Keywords Solution concept • Majority relation • Tournament • Matrix-vector representation • Condorcet winner • Core • Top cycle • Uncovered set • Weakly stable set • Externally stable set • Uncaptured set • Untrapped set • k-stable alternative • k-stable set
