A Borda measure for social choice functions
The question addressed in this paper is the order of magnitude of the difference between the Borda rule and any given social choice function. A social choice function is a mapping that associates a subset of alternatives to any profile of individual preferences. The Borda rule consists in asking voters to order all alternatives, knowing that the last one in their ranking will receive a score of zero, the second lowest a score of 1, the third a score of 2 and so on. These scores are then weighted by the number of voters that support them to give the Borda score of each alternative. The rule then selects the alternatives with the highest Borda score. In this paper, a simple measure of the difference between the Borda rule and any given social choice function is proposed. It is given by the ratio of the best Borda score achieved by the social choice function under scrutiny over the Borda score of a Borda winner. More precisely, it is the minimum of this ratio over all possible profiles of preferences that is used. This "Borda measure" or at least bounds for this measure is also computed for well known social choice functions. Cet article se penche sur la distance entre la règle de Borda et n'importe quelle autre fonction de choix social. Ces dernières associent un sous-ensemble d'options possibles à tout profil ou configuration de préférences individuelles. La règle de Borda consiste à demander aux votants d'ordonner les options possibles, en leur disant que la dernière dans leur ordre recevra un score nul, l'avant-dernière un score égal à 1, celle qui vient au troisième pire rang un score égal à 2 et ainsi de suite. Ces scores sont ensuite pondérés par le nombre de votants qui les supportent pour donner le score de Borda de chaque option. La règle choisit les options qui ont reçu le score le plus élevé. Dans cet article, une mesure simple de la différence entre la règle de Borda et n'importe quelle autre fonction de choix social est proposée. Elle est donnée
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