Abstract
The basic purpose of this paper is to link both theorems of impossibility and existence by introducing fuzzy relations of preference and an exogeneous requirement, the planner's one, and then proving the fundamental part played by the extremist agents, leximin and leximax. In other words, to bring out the link between the planner's requirement and the difficulty of the transition from individual to collective, as well as the theoric relation between this requirement and the extremist agents, we define a fuzzy behavior of preference which allows us to build up two determinant fuzzy coalitions. These coalitions will be the base of the planner's requirement and the link between pessimistic results (Arrow's impossibility) and optimistic ones (May's theorem of majority choice).
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Abdi, H., Barthelemy, J. P., and Luong, X.: 1980, ‘Information Préordinale et Analyse des Préférences’ in Data Analysis and Informatics, Amsterdam, North-Holland.
Arrow, K. J.: 1951, 1963, Choix Collectif et Préférences Individuelles, 2nd Edition. Calmann-Levy, Paris.
Barret, C. R., Pattanaik, P. K., and Salles, M.: 1985, ‘On the Structure of Fuzzy Social Welfare Functions’ Séminaire de L'Institut de Mathematiques Economiques, Université de Dijon.
Barthelemy, J. P.: 1979, Propriétés métriques des ensembles ordonnés. Comparaison et agrégation des relations binaires. Thèse de Doctorat d'Etat, Besançon.
Barthelemy, J. P. and Montjardet, B.: 1979, ‘Ajustement et Résumé de Données Relationnelles: les Relations Centrales’ in Data Analysis and Informatics, Amsterdam, North-Holland, pp. 645–653.
Billot, A. B.: 1986, ‘A Contribution to a Mathematical Theory of Fuzzy Games’ in Fuzzy Economics and Spatial Analysis, C. Ponsard and B. Fustier, eds., Dijon, Librairie de l'Université, pp. 47–56.
Billot, A. B.: 1987, Préférence et Utilité Floues, Travaux et Recherches de L'Université de Droit, D'Economie et de Sciences Sociales de Paris, Presses Universitaires de France, Paris.
Billot, A. B.: 1988, Economic Theory of Fuzzy Equilibria, Thèse de Doctorat de Sciences Economiques, Novembre 1988, Université de Dijon, to appear in Lecture Notes in Mathematical Economics, Springer-Verlag.
Black, D.: 1948, ‘The Decisions of a Committee Using a Simple Majority’ Econometrica 16, 246–261.
Blair, D. H., Bordes, G., Kelly, J. S., and Suzumura, K.: 1986, ‘Impossibility Theorems without Collective Rationality’ Journal of Economic Theory 13, 361–379.
Bordes, G.: 1979, ‘Some More Results on Consistency, Rationality and Collective Choice’ in Aggregation and Revelation of Preferences, J. J. Laffont, ed., Amsterdam, North-Holland.
Bordes, G.: 1980, ‘Procédures d'Agrégation et Fonctions de Choix’ in Analyse et Agrégation des Préférences (dans les Sciences Sociales, Economiques et de Gestion), Economica, Paris.
Chakraboty, M. K. and Das, M.: 1985, ‘Reduction of Fuzzy Strict Order Relations’ Fuzzy Sets and Systems 15, 33–44.
Chakraboty, M. K. and Sarkar, S.: 1987, ‘Fuzzy Antisymetry and Order’ Fuzzy Sets and Systems 21, 169–182.
Chichilinsky, G.: 1982, ‘Social Aggregation and Continuity’ The Quarterly Journal of Economics 97, 337–352.
Cholewa, W.: 1985, ‘Aggregation of Fuzzy Opinions - An Axiomatic Approach’ Fuzzy Sets and Systems 17, 249–258.
Clause, M.: 1982, Choix Collectifs dans un Contexte Imprecis, Thèse de 3ème Cycle en Economie Mathématique et Econométrie, Université de Dijon.
d'Aspremont, C. and Gevers, L.: 1977, ‘Equity and the Informational Basis of Collective Choice’ Review of Economic Studies 44, 199–209.
Deb, R. and Blau, J. H.: 1977, ‘Social Decision Functions and Veto’ Econometrica 45, 871–879.
Deschamps, R. and Gevers, L.: 1978, ‘Leximin and Utilitarian Rules: a Joint Characterization’ Journal of Economic Theory 17, 143–163.
Dutta, B.: 1977, ‘Existence of Stable Situations, Restricted Preferences and Strategic Manipulations under Democratic Group Decision Rules’ Journal of Economic Theory 15, 99–111.
Ekeland, I.: 1979, Economie Mathématique, Chap. 1, Hermann, Paris.
Fishburn, P. C.: 1973, The Theory of Social Choice, Princeton University Press.
Fung, L. W. and Fu, K. S.: 1975, ‘An Axiomatic Approach to Rational Decision-Making in a Fuzzy Environment’ in L. A. Zadeh, K. S. Fu, K. Tanaka, and M. Shimura, eds., Fuzzy Sets and Their Applications to Cognitive and Decision Processes, Academic Press, New York, pp. 227–256.
Grandmont, J. M.: 1978, ‘Intermediate Preferences and the Majority Rule’ Econometrica 46, 317–330.
Guilbaud, G. T.: 1952, ‘Les Théories de l'Intérêt Général et le Problème Logique de l'Agrégation’ Economie Appliquée 5, 502–584.
Hammond, P.: 1976, ‘Equity, Arrow's Conditions and Rawls' Difference Principle’ Econometrica 44, 793–804.
Harsanyi, J. C.: 1973, ‘Can the Maximin Principle Serve as a Basis for Morality: A Critique of John Rawls' Theory’ Document de Travail CP-351, Center for Research in Management Science, University of California, in American Political Science Review 69.
Heuchenne, C.: 1970, ‘Un Algorithme Général pour Trouver un Sous-Ensemble d'un Certain Type à Distance Minimum d'une Partie Donnée’ Mathématiques et Sciences Humaines 30, 23–33.
Hosomatu, Y.: 1978, ‘Zero-Sum Condition: A Necessary and Sufficient Condition for a Transitive Voting System’ Journal of Economic Theory18, 294–300.
Kalai, E. and Ritz, Z.: 1980, ‘Characterisation of the Private Alternatives Domains Admitting Arrow Social Welfare Functions’ Journal of Economic Theory 22, 12–22.
Kaufmann, A.: 1973–1980, Introduction à la Théorie des Sous-Ensembles Flous. Tomes I et IV, Masson, Paris.
Kim, J. B.: 1983, ‘Fuzzy Rational Choice Functions’ Fuzzy Sets and Systems 10, 37–43.
Kim, K. H. and Roush, F. W.: 1981, ‘Effective Nondictatorial Domains’ Journal of Economic Theory 24, 40–47.
Kolm, S. C.: 1973, Justice et Equité. Editions C.N.R.S., Paris.
Lemaire, J.: 1981, ‘Agrégation Typologique des Données de Préférences’ in Analyse et Agrégation des Préférences, Economica, Paris.
Mas-Colell, A. and Sonnenschein, H.: 1972, ‘General Possibility Theorems for Group Decisions’ Review of Economic Studies 39, 185–192.
May, K. O.: 1952, ‘A Set of Independent Necessary and Sufficient Conditions for Simple Majority Decision’ Econometrica 20, 680–684.
Montero, F. J.: 1986, ‘Measuring the Rationality of a Fuzzy Preference Relation’ Busefal 26, 75–83.
Montero, F. J. and Tejada, J.: 1986, ‘Some Problems on the Definition of Fuzzy Preference Relations’ Fuzzy Sets and Systems 20, 45–53.
Moulin, H.: 1979, ‘Dominance-Solvable Voting Schemes’ Econometrica 47, 249–269.
Moulin, H.: 1985, ‘Egalitarianism and Utilitarianism in Quasi-Linear Bargaining’ Econometrica 53, 49–67.
Orlovsky, S. A.: 1978, ‘Decision Making with a Fuzzy Preference Relation’ Fuzzy Sets and Systems 4, 155–167.
Orlovsky, S. A.: 1980, ‘On formalization of a General Fuzzy Mathematical Problem’ Fuzzy Sets and Systems 3, 311–321.
Ovchinnikov, S. V.: 1981, ‘Structure of Fuzzy Binary Relations’ Fuzzy Sets and Systems 6, 169–195.
Ovchinnikov, S. V.: 1988, ‘Social Choice and Lukasiewicz Logic’, working paper for the Symposium Aggregation of Fuzzy Preferences, Bruxelles, Janvier 1988.
Pattanaik, P. K. and Salles, M., eds.: 1983, Social Choice and Welfare, Contributions to Economic Analysis, no. 145, North-Holland.
Ponsard, C.: 1981, ‘L'Equilibre Spatial du Consommateur dans un Contexte Imprécis’ Sistemi Urbani 3, 107–133.
Ponsard, C.: 1986, ‘Foundations of Soft Decision Theory’ in Management Decision Support Systems Using Fuzzy Sets and Possibility Theory, J. Kacprzyk and R. R. Yager, eds., Verlag T.U.V. Rheinland, pp. 27–37.
Ponsard, C.: 1988, ‘Fuzzy Mathematical Models in Economics’ Fuzzy Sets and Systems 28, 273–283.
Prevot, M.: 1977, Sous-Ensembles Flous: Une Approche Théorique, Sirey, Collection de l'Institut de Mathématiques Economiques, Paris.
Roubens, M. and Vincke, P.: 1985, ‘Fuzzy Preferences in an Optimisation Perspective’ Colloque I.M.E. 1985, Université de Bourgogne.
Roubens, M. and Vincke, P.: 1986, Preference Modelling, Lecture Notes in Economics and Mathematical Systems, no. 250, Springer-Verlag, Berlin, New York.
Saari, D. G: 1986, ‘Price Dynamics, Social Choice, Voting Methods, Probability and Chaos’ Communication au Colloque FUR III, Aix en Provence.
Salles, M. and Wendel, R. E.: 1977, ‘A Further Result on the Core of Voting Games’ International Journal of Game Theory 6, 35–40.
Satterthwaite, M. A.: 1975, ‘Strategy-Proofness and Arrow's Conditions: Existence and Correspondence Theorems for Voting Procedures and Social Welfare Functions’ Journal of Economic Theory 10, 187–217.
Schmidt, C.: 1985, La Sémantique Economique en Question, Calmann-Lévy, Paris.
Schofield, N.: 1977, ‘Transitivity of Preferences on a Smooth Manifold of Alternatives’ Journal of Economic Theory 14, 149–171.
Schwartz, T.: 1972, ‘Rationality and the Myth of the Maximum’ Theory and Decision 1, 89–106.
Schwartz, T.: 1980, The Logic of Collective Choice. University of Texas at St Austin.
Sen, A. K.: 1969, ‘Quasi-Transitivity Rational Choice and Collective Decisions’ Review of Economic Studies 36, 381–393.
Sen, A. K.: 1970, Collective Choice and Social Welfare, San Francisco, Holden-Day.
Sen, A. K.: 1974, ‘Informational Basis of Alternative Welfare Approaches, Aggregation and Income Distribution’ Journal of Public Economics 3, 387–403.
Sen, A. K.: 1977, ‘Social Choice Theory: A Re-Examination’ Econometrica 45, 53–89.
Shimura, M.: 1973, ‘Fuzzy Sets Concepts in Rank-Ordering Objects’ Journal of Mathematical Analysis and Applications 43, 717–733.
Strasnick, S.: 1976, ‘The Problem of Social Choice: Arrow to Rawls’ Philosophy and Public Affairs 5, 241–273.
Switalsky, Z.: 1988, ‘Choice Functions Associated with Fuzzy Preference Relations in Decision Making’ in Kacprzyk, J. and Roubens, M., eds., Nonconventional Preference Relations and Decision Making, Springer-Verlag, Berlin.
Zadeh, L. A.: 1971, ‘Similarity Relations and Fuzzy Orderings’ Information Sciences 3, 177–200.
Zimmermann, H. J.: 1985, Fuzzy Set Theory - and Its Applications, Kluwer, Nijhoff Publishing, Boston.
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Billot, A. Aggregation of preferences: The fuzzy case. Theor Decis 30, 51–93 (1991). https://doi.org/10.1007/BF00134636
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DOI: https://doi.org/10.1007/BF00134636