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Choosing from a Weighted Tournament

  • De Donder, Philippe
  • Le Breton, Michel
  • Truchon, Michel

    ()

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 strenght 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 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 wheighted 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-theorical comparison between the four C1 correspondences, their four C2 couterparts 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. Un vote à la majorité où les candidats sont comparés deux à deux induit un tournoi basé sur la relation majoritaire et un tournoi pondéré, où la taille de la majorité compte. Chacun de ces tournois induit à son tour un jeu à somme nulle pour lesquels on dispose de différents concepts de solution. Quatre correspondances de choix social applicables à la relation majoritaire, dites de type C1, correspondent à quatre concepts de solution différents pour le jeu induit par le tournoi correspondant. Ce sont le top cycle, le uncovered set, le minimal covering set et le bipartisan set. En utilisant les mêmes concepts de solution pour les jeux induits par les tournois pondérés équivalents, plutôt que par les tournois, et en allant des solutions pour les jeux aux tournois pondérés et ensuite aux relations majoritaires (votes), nous obtenons l'équivalent de type C2 des quatre correspondances de type C1, i.e. des correspondances qui exigent la dimension de la majorité pour opérer. Nous effectuons également une comparaison entre les quatre correspondances de type C1, leurs quatre équivalents de type C2 et trois autres correspondances de type C2, soit les règles de Kemeny, de Simpson-Kramer et de Borda. De façon plus précise, étant donné les ensembles de décision produits par deux correspondances de choix social, nous répondons aux questions: Est-ce qu'un de ces ensembles est toujours inclus dans l'autre? Si non, y a-t-il toujours intersection entre les deux ou, au contraire, peut-il arriver que leur intersection soit vide?

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Paper provided by Université Laval - Département d'économique in its series Cahiers de recherche with number 9815.

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Date of creation: 1998
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Handle: RePEc:lvl:laeccr:9815
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