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 par le rapport du meilleur score de Borda obtenu par les options que sélectionne la fonction de choix social considérée sur le score de Borda d'un gagnant de Borda. De façon plus précise, c'est le minimum de ces rapports, sur l'ensemble des profils de préférences, qui est utilisé. Cette mesure de Borda ou, à tout le moins, un intervalle pour cette mesure est calculé pour un certain nombre de fonctions de choix social bien connues.

The Art of Making Everybody Happy: How to Prevent a Secession

In this paper we examine compensation schemes that prevent a threat of secession by any of a country's regions. We prove that, under quite general assumptions on the distribution of citizens' preferences, there exist transfer schemes that are secession-proof. Moreover, we show that these compensation schemes entail a degree of partial equalization among regions: the gap between advantaged regions and disadvantaged regions has to be reduced but it should never be completely eliminated. We demonstrate that in the case of a uniform distribution of the nation's citizens, the secession-proof conditions generate the 50 percent compensation rule for disadvantaged regions. Copyright 2003, International Monetary Fund

Stable Partitions in a Model with Group-Dependent Feasible Sets.

In this paper we consider a model of group formation where group of individuals may have different feasible sets. We focus on two polar cases, increasing returns, when the set of feasible alternatives increases if a new member joins thegroup, and decreasing returns, when a new member has an opposite effect and reduces the number of alternatives available for the enlarged group. We consider two notions, stability and strong stability of group structures, that correspond to Nash and Strong Nash equilibrium of the associated non-cooperative game. We prove existence results for various classes of environments and also investigate the link between the dimensionality of the set of alternatives and the existence of stable structures.feasible sets; stable partitions; positive externality; increasing and decreasing returns