Choosing from a weighted tournament

A voting situation, in which voters are asked to rank all candidates pair by pair, induces a tournament and a weighted tournament, in which the strength of the majority matters. Each of these two tournaments induces in turn a two-player zero-sum game for which different solution concepts can be found in the literature. Four social choice correspondences for voting situations based exclusively on the simple majority relation, and called C1, correspond to four different solution concepts for the game induced by the corresponding tournament. They are the top cycle, the uncovered set, the minimal covering set, and the bipartisan set. Taking the same solution concepts for the game induced by the corresponding weighted tournament instead of the tournament and working backward from these solution concepts to the solutions for the corresponding weighted tournament and then to the voting situation, we obtain the C2 counterparts of these correspondences, i.e. correspondences that require the size of the majorities to operate. We also perform a set-theoretical comparison between the four C1 correspondences, their four C2 counterparts and three other C2 correspondences, namely the Kemeny, the Kramer-Simpson, and the Borda rules. Given two subsets selected by two correspondences, we say whether it always belongs to, always intersects or may not intersect the other one. (C) 2000 Elsevier Science B.V. All rights reserved.

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