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Social choice theory and the “Centre de Mathématique Sociale”: some historical notes

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Abstract

In this paper we describe some research directions in social choice and aggregation theory taken at the “Centre de Mathématique Sociale” since the fifties. We begin by presenting some institutional aspects concerning this center. Then we sketch a thematic history by considering the following questions about the “effet Condorcet” (“voting paradox”): What is it? How is it overcome? Why does it occur? These questions were adressed in Guilbaud's paper (Guilbaud GTh (1952) Les théories de l'intérêt général et le problème logique de l'agrégation, (Theories of the general interest and the logical problem of aggregation) which will mark the beginning of our inquiry. The conclusion outlines some more recent research developments linked to these questions.

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Notes

  1. Économie appliquée 5(4): 469–484 (October–December 1952).

  2. Économie appliquée 5(4): 501–584 (October–December 1952).1 (The other contributors of this issue were B. de Jouvenel, P. Streeten, I.M.D. Little, G. Nyblen, L. Buquet, G. Bernacer, P. Massé and J. Akerman). This paper was reprinted in Eléments de la théorie mathématique des jeux, Dunod Paris, 1967, and partially translated into English under the title Theories of the general interest and the logical problem of aggregation, in Lazarsfeld PF and Henry NW (eds) Readings in Mathematical Social Sciences, Science Research Associates, Inc., Chicago (1966), pp. 262–307. Since the original paper is difficult to find we will give the numbering of pages of this paper according to this translation.

  3. He was a student of the “Ecole Normale Supérieure” in the thirties.

  4. Below the “École des Hautes Études en Sciences Sociales” will be simply called the École.

  5. The EPHE was founded by Victor Duruy (then the french minister of public education) in 1868 as an institute of higher education initially charged to promote a more practical teaching of the sciences than the one given by the universities. In 1975 it contained six sections of which the most important were the sixth (created in 1947) and in 2002 it has only three sections (life and earth sciences, history and philology sciences, religious sciences).

  6. See Mathématique sociale, Entretien avec G.Th. Guilbaud, Savoir et Mémoire n°4, Éditions de l'EHESS, Paris, 1993, and Rosenstiehl P, La mathématique et l'École. In Revel J, Wachtel N (eds) Une École pour les sciences sociales. Cerf and Éditions de l'EHESS, Paris, 1996.

  7. When the “groupe” grew it became a “centre”, and after some terminological variations it is now called “Centre d'Analyse et de Mathématique Sociale (CAMS)”. Henceforth we will use only (so sometimes anachronistically) this acronym CAMS in our text.

  8. The center has presently 25 members, plus some retired former members but still working at the center.

  9. See for instance Armatte M, Feldman J, Leclerc B, Monjardet B, Schiltz MA, Selz Laurière M, (1989) Mathématiques et Sciences Humaines : des années soixante aux années quatre vingts La Vie des Sciences 6(1): 59-76, 6(2): 139-165.

  10. For the relations between the mathematicians and some social scientists of the École see the references given in footnote 6.

  11. The present organizers of this seminar are J.P. Barthélemy, O. Hudry, Marc Demange, B. Leclerc, and B. Monjardet.

  12. Presently Michel Armatte, Bernard Bru and Thierry Martin are also organizers of this seminar.

  13. TRAP is the acronym of “Table Ronde sur l'Agrégation des Préférences”.

  14. Although J. Feldman was no longer a member of CAMS at this time, one can mention also the Conference organized in 1991 at the École: “Moyenne, Milieu, Centre” (J. Feldman, G. Lagneau and B. Matalon, Paris).

  15. An aim of this jornal was to promote exchanges between people teaching mathematics and statistics for social sciences and those teaching these sciences. In fact CAMS members especially Guilbaud and Barbut played a significant role in the starting and the development mathematical training courses for students in economics, psychology and sociology. In particular Barbut was one of the initiators of the creation of a new curriculum “Applied Mathematics and Social Sciences” presently existing in more than 30 french universities.

  16. One must point out that during these years the quasi totality of these papers were published in French, a fact that didn't favor their knowledge by an international audience.

  17. Henceforth we will call Condorcet 's book simply the Essai.

  18. The famous “Annales's school” of historians of the École (and in particular Braudel) supported the notion of the “long history”.

  19. Daunou and Lhuillier worked on voting procedures at the end of 18th century and the beginning of 19th century and both quote the Essai (see McLean I (1995) The first golden age of social choice 1784–1803. In: Barnett W, Moulin H, Salles M, Schofield N (eds) Social choice, Welfare and Ethics. Cambridge University Press, pp 13–33).

  20. In his book An history of the mathematical theory of probability from the time of Pascal to that of Laplace (MacMillan, London, 1865), In fact Todhunter devotes a chapter to a detailed analysis of the Essai but he completely misses the significance of Condorcet's study on the systems of propositions and their possible contradictions (“these results however appear of too little value to detain us any longer”, page 375).

  21. Economists (with sufficient mathematical training) appreciated the paper (in the second edition of his book, Arrow describes it as a “remarkable exposition of the theory of collective choice and the general problem of aggregation”). But few mathematicians,—too often unaware or even contemptuous of social sciences—read it.

  22. It is interesting to mention that Arrow had been introduced to such mathematics by Tarski.

  23. Condorcet and most of his followers expressed the preference of a voter by such a linear order, i.e. by a transitive, antisymmetric and complete binary relation. Arrow expressed this preference by a weak order (called by him ordering and by others quasi-ordering or complete preorder or etc.), i.e. by a transitive and complete binary relation.

  24. This method had been already proposed in the thirteenth century by Ramon Lull (see McLean I, London J (1990) The Borda and Condorcet principles: three medieval applications. Social Choice and Welfare 7: 99–108).

  25. In Black's book (The Theory of Committees and Elections (1958) Cambridge University Press, Cambridge) the answer is given for three voters and three alternatives. In 1803 Daunou writes that the Condorcet effect “is by no means a rare occurrence”, but his assertion is based on wrong computations (See Mémoire sur les élections au scrutin, published in English translation In McLean I, Urken AB (eds) Classics of social choice. The University of Michigan Press, Ann Arbor, 1995, pp 237–276, page 243).

  26. See for instance a review of such works in Gehrlein WV (1983) Condorcet's paradox. Theory and Decision 15: 161–197. Note that the computation of the frequency is equivalent to computing the probability of the effet Condorcet under the so-called probabilistic model of “impartial culture” where each linear order has the same probability (1/m!) of being adopted by each voter. In this case Guilbaud's formula has been generalized (for instance in Gehrlein WV, Fishburn PC (1976) Probabilities of election outcomes for large electorates. Journal of Economic Theory 13: 14–25). More generally the probability of the effect has been studied when the preferences of the voters follow various probability models (see again Gehrlein's 1983 paper).

  27. in Von Neumann J, Morgenstern O (1944) Theory of games and economic behaviour. Princeton, University Press.

  28. The term simple game to name this mathematical structure was justified in the context of Von Neumann and Morgenstern's book. But in fact such a structure appears in many fields of mathematics where to use the term simple game would be absurd. Unfortunately such a terminological absurdity has became unavoidable in social choice theory (note that from a mathematical point of view a simple game is nothing more than an order filter in the Boolean lattice of all subsets of S).

  29. In fact this theorem is a consequence of a more general result proved by Guilbaud and concerning a logical problem already raised by Condorcet. Let a set of binary (“yes or no”) questions be logically linked in the sense that the answers to some imply the answers to others. A coherent opinion of an individual is defined as a set of answers to these questions respecting their links. What are the rules allowing one to aggregate several coherent individual opinions into a coherent collective opinion ? Guilbaud proves that if the aggregation rule must preserve all the possible logical links between these questions, then it must be dictatorial (page 306). A contrario, the use of, for instance, the majority rule can lead to an incoherent collective opinion, a fact called also “Condorcet effect” by Guilbaud. It is interesting to observe that this much more general Condorcet effect has been rediscovered in the eighties under the name of the “doctrinal paradox” and has led to results similar to Guilbaud's results (see List C, Pettit P (2002) Aggregating Sets of Judgments: An Impossibility Result. Economics and Philosophy 18: 89–110 and a bibliography at http://www.nuff.ox.ac.uk/users/list/ doctrinalparadox.htm).

  30. A filter \(\mathfrak{F}\) on a set N is an order ideal (U ∈ \(\mathfrak{F}\) and U \u2286 V imply V ∈ \(\mathfrak{F}\)), is stable by intersection (U, V ∈ \(\mathfrak{F}\) imply U∩V ∈ \( {\user1{\mathcal{F}}} \)) and does not contain the empty set. An ultrafilter is a maximal filter, i.e. a filter which is not strictly contained in another filter.

  31. A social welfare function is independent (of irrelevant alternatives) if the social preference on two alternatives depends only on the individual preferences on these alternatives and it is Paretian if x is socially (strictly) preferred to y if all the voters prefer (strictly) x to y.

  32. See Blau JH (1979) Semiorders and collective choice. Journal of Economic Theory 29: 195–206. Wilson (Wilson RB (1972) Social choice theory without the Pareto principle. Journal of Economic Theory 5: 478–486) proved also that Arrow's theorem could be obtained from the linear order version of this theorem, but his proof of this version which results from his general theory of aggregation is remarkably complicated.

  33. See Blau's paper in footnote 32.

  34. Translation of my sentence “Il est alors immédiat que dans l'algèbre de Boole des parties de N, \(\mathfrak{F}\) doit être un filtre maximal” in Monjardet (1969) Remarques sur une classe de procédures de vote et les théorèmes de possibilité In La décision, Éditions du CNRS, Paris, pp 177–184.

  35. See Monjardet (1978) Une autre preuve du théorème d'Arrow. R.A.I.R.O. 12: 291–296 and Monjardet B (1981) On the use of ultrafilters in social choice theory In Pattanaik PK, Salles M (eds) Social Choice and Welfare. Amsterdam, North-Holland, pp 73–78.

  36. See for instance Pouzet M (1998) A projection property and Arrow's impossibility theorem. Discrete Mathematics 192(1–3): 293–308.

  37. See for instance Black's book quoted in footnote 25.

  38. In the essay coherent opinion means linear order.

  39. Page 265 in the English translation of Guilbaud's paper quoted in footnote 2.

  40. The fact that the median problem is NP-difficult can be found for example in Orlin (1981) unpublished note or Hudry (1989) Recherche d'ordres médians : complexité, algorithmique et problèmes combinatoires. Thèse ENST, Paris.

  41. See Young HP (1988) Condorcet's Theory of Voting. American Political Science Review 82: 1231–1244.

  42. We denote this distance by d K since it has been implicitly used by Kendall as early as 1938 (the well-known Kendall “correlation coefficient” tau is nothing more than the normalization of this distance between −1 and +1) as well that by Kemeny (see below).

  43. Barbut (1967) Médianes, Condorcet et Kendall. Note SEMA, Paris and (1980) Mathématiques et Sciences Humaines 69: 5–13.

  44. Kemeny JG (1959) Mathematics without numbers. Daedalus 88: 577–591. Kemeny JG, Snell JC (1961) Mathematical Models in the Social Sciences. Ginand Co, New York.

  45. In fact this procedure has many other equivalent forms and so it has been very often (re)discovered, in particular by Brunk and independently by Hays as early as in 1960. See Monjardet B (1991) Sur diverses formes de la “Règle de Condorcet” d'agrégation des préférences. Mathématiques et Sciences humaines 111: 61–71 or Monjardet B (1997) Concordance between two linear orders: The Spearman and Kendall coefficients revisited. Journal of Classification 14 (2): 269–295.

  46. This distance is nothing more than the “path length” distance in the unoriented graph associated with the covering relation of the lattice (see for instance Birkhoff G (1967) Lattice theory. Amer. Math. Soc., Providence or Barbut and Monjardet (1970) Ordre et Classification, Algèbre et Combinatoire, tomes I et II. Hachette, Paris).

  47. Barbut (1961) Médianes, distributivité, éloignements. Note CAMS, EHESS, Paris and (1980) Mathématiques et Sciences Humaines 70: 5–31, Monjardet (1980) Théorie et application de la médiane dans les treillis distributifs finis. Annals of Discrete Mathematics 9: 87–91.

  48. This definition is also equivalent to saying that \({\user1{\mathcal{D}}}\) has no cyclic triples, i.e. that there do not exist a subset {x,y,z} of three alternatives and three linear orders in \({\user1{\mathcal{D}}}\) such that the restrictions of these orders to {x,y,z} is a cyclic permutation like xyz, yzx and zxy (such a set \({\user1{\mathcal{D}}}\) has been also called a consistent or an acyclic or a majority-consistent set, see the references in footnotes 53 and 54).

  49. See Black D (1948) On the rationale of group decision-making. Journal of Political Economy 56: 23–34 and Black's book quoted in footnote 25.

  50. Guilbaud GT, Rosenstiehl P (1970) Analyse algébrique d'un scrutin. Mathématiques et Sciences Humaines 4: 9–33.

  51. Chameni-Nembua C (1989) Règle majoritaire et distributivité dans le permutoèdre. Mathématiques et Sciences Humaines 108: 5–22.

  52. It was published in Chameni-Nembua's paper quoted in the above footnote.

  53. Abello JM, Johnson CR (1984) How large are transitive simple majority domains? SIAM J. Algebraic and Discrete Methods 3(4): 603–618 (the bound was 3.2(n−2)−4).

  54. Craven J (1996) Majority consistent preference orderings. Social Choice and Welfare 13: 259–267. Fishburn PC (1997) Acyclic sets of linear orders. Social Choice and Welfare 14: 113–124.

  55. Quetelet A (1835) Sur l'homme et le développement de ses facultés ou Essai de physique sociale. Paris.

  56. Free translation of the Cournot sentence quoted by Guilbaud and taken from Cournot's book Exposition de la théorie des chances, Paris, 1843.

  57. The following first such result has been followed by many others: Mirkin BG (1975) On the problem of reconciling partitions In Quantitative Sociology, International Perspectives on mathematical and Statistical Modelling. Academic Press, New-York, pp 441–449 (in the case of partitions as in the case of partial orders the independence and unanimity axioms lead to “oligarchic” consensus functions).

  58. Les Conférences du Palais de la Découverte, Paris, 1949.

  59. Recall that the notion of distance i.e. of metric space goes back to Frechet in 1904.

  60. Gini C (1914) L'uomo medio. Giornali degli economiste e revista de statistica 48: 1–24. The notion of the metric median in Euclidean spaces goes back to a problem raised by Fermat in 1629 and it has been used for location problems since Weber's 1909 book Uber den Standort der Industrien (see Monjardet B., Éléments pour une histoire de la médiane métrique, In Moyenne, Milieu, Centre. Histoires et usages (1991) Coll. Histoire des Sciences et Techniques, n°5. Éditions de l'EHESS, Paris).

  61. See references in Barthélemy et al. 1986 paper in the Annex.

  62. See for instance McMorris FR, Mulder HM, Powers RC (2000) The median function on median graphs and semilattices. Discrete Applied Mathematics 101: 221–230.

  63. See Leclerc's 2002 paper in the Annex.

  64. For instance the oligarchic results mentioned in footnote 57 are special cases of a meet-projection result on a meet semilattice (see Leclerc and Monjardet's 1995 paper in the Annex).

Acknowledgements

The author wish to thank M. Barbut, J.P. Barthélemy, B. Leclerc and M. Salles for their suggestions. He is particularly grateful to G.Th. Guilbaud and W. H.E. Day to have corrected many content and/or form errors contained in a preliminary version of this paper. All these contributions have greatly improved the paper.

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Authors and Affiliations

  1. CERMSEM, Université Paris I Panthéon Sorbonne, Maison des Sciences Économiques, 106-112 bd de l'Hopital, 75647, Paris Cédex 13, France

    Bernard Monjardet

  2. CAMS, EHESS, DUMMY, DUMMY, DUMMY, DUMMY

    Bernard Monjardet

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Correspondence to Bernard Monjardet.

ANNEX

ANNEX

1952

G.Th. Guilbaud, Les théories de l'intérêt général et le problème logique de l'agrégation, Economie appliquée, 5, 501–584. (Partial) English translation: Theories of the general interest and the logical problem of aggregation. In, P.F. Lazarsfeld and N.W. Henry (eds) Readings in Mathematical Social Sciences, Science Research Association, Inc., Chicago, 1966, pp. 262–307.

1959

M. Barbut Quelques aspects mathématiques de la décision rationnelle, Les temps modernes, 15, 725–745. Translated as: Does the majority ever rule? Portfolio and Art News Annual, 4, 1961, 79–83 and 161–168.

1961

M. Barbut, Médianes, distributivité, éloignements, Note CAMS, Paris, reprinted (1980) in Mathématiques et Sciences Humaines 70, 5–31.

1963

G.Th. Guilbaud and P. Rosenstiehl, Analyse algébrique d'un scrutin, Mathématiques et Sciences Humaines 4, 9–33.

1966

M. Barbut, Note sur les ordres totaux à distance minimum d'une relation binaire donnée, Mathématiques et Sciences Humaines 17, 47–48.

1967

M. Barbut, Médianes, Condorcet et Kendall, note SEMA, Paris, reprinted (1980) in Mathématiques et Sciences Humaines 69, 5–13.

1969

M. Barbut, Une classe de demi-treillis ordonnés pouvant servir à des agrégations de critères. In La décision, Agrégation et dynamique des ordres de préférence, Éditions du CNRS, Paris, pp 27–35.

J. Feldman, Ordres partiels et permutoèdre, Mathématiques et Sciences Humaines 28, 27–38.

B. Monjardet, Remarque sur une classe de procédures de vote et les “théorèmes de possibilité”. In La décision, Agrégation et dynamique des ordres de préférence, Éditions du CNRS, Paris, pp 177–184.

1970

M. Barbut and B. Monjardet, Ordre et Classification, Algèbre et Combinatoire, tomes I et II, Hachette, Paris.

1971

L. Frey and M. Barbut, Techniques Ordinales en Analyse des Données: Algèbre et Combinatoire, Hachette, Paris.

B. Monjardet, Correspondance de Galois et procédures de vote, Comptes-Rendus de l'Académie des Sciences Paris, Série A, 272, 1522–1525.

1973

J. Feldman, Pôles, intermédiaires et centres dans un groupe d'opinions, Mathématiques et Sciences Humaines 43, 39–54.

B. Monjardet, Tournois et ordres médians pour une opinion, Mathématiques et Sciences Humaines 43, 55–70.

1976

J.-P. Barthélemy, Sur les éloignements symétriques et le principe de Pareto, Mathématiques et Sciences Humaines 56, 97–125.

B. Monjardet, Lhuillier contre Condorcet au pays des paradoxes, Mathématiques et Sciences Humaines 54, 33–43.

1977

J.-P. Barthélemy, Sur les préférences individuelles centrales, Comptes-Rendus de l'Académie des Sciences Paris, Série A, 1493–1494.

J.-P. Barthélemy, A propos des partitions centrales sur un ensemble non nécessairement fini, Statistique et Analyse des données 3, 4–62.

J.-P. Barthélemy, Comparaison et agrégation des partitions et des préordres totaux, Comptes Rendus de l'Académie des Sciences Paris, Série A, 285, 985–987.

1978

B. Monjardet, Une autre preuve du théorème d'Arrow, R.A.I.R.O. 12, 291–296.

B. Monjardet, An axiomatic theory of tournament aggregation, Mathematics of Operation Research 3 (4), 334–351.

1979

J.-P. Barthélemy, Propriétés métriques des ensembles ordonnés. Comparaison et agrégation des relations binaires. Thèse d'Etat, Université de Franche–Comté, Besançon.

J.-P. Barthélemy, Caractérisations axiomatiques de la distance de la différence symétrique entre des relations binaires, Mathématiques et Sciences Humaines 67, 85–113.

J.P. Barthélemy and B. Monjardet, Ajustement et résumé de données relationnelles: les relations médianes. In E. Diday and al (eds) Data Analysis and Informatics North-Holland, pp 645–654.

B. Monjardet, Duality in the theory of social choice. In J.J. Laffont (ed) Aggregation and revelation of preferences, Workshop of the Econometric Society, North-Holland, Amsterdam, pp 131–145.

1980

B. Monjardet, Théorie et applications de la médiane dans les treillis distributifs finis, Annals of Discrete Mathematics 9, 87–91.

B. Monjardet, Variations sur l'effet Condorcet, ou Condorcet, Black, Arrow, Guilbaud et les autres, Cahiers du Centre d'Etudes de Recherche Opérationnelle 22, 1, 7–15.

1981

J.P.Barthélemy, Trois propriétés des médianes dans un treillis modulaire, Mathématiques et Sciences Humaines 75, 83–91.

J.-P. Barthélemy, Procédures métriques d'agrégation. In P. Batteau, E. Jacquet-Lagrèze and B. Monjardet (eds) Analyse et agrégation des préférences pour les sciences économiques, sociales et de gestion, Economica, Paris, pp 123–151.

J.-P. Barthélemy and B. Monjardet, The median procedure in cluster analysis and social choice theory, Mathematical Social Science 1, 235–267.

P. Batteau, J.M. Blin and B. Monjardet, Stability of aggregation procedures, ultrafilters and simple games, Econometrica 49, 527–534.

1982

J.-P. Barthélemy, Arrow's theorem: Unusual domains and extended codomains, Mathematical Social Science 3, 79–89.

J.-P. Barthélemy, Cl. Flament and B. Monjardet, Ordered sets and social sciences. In I. Rival (ed) Ordered sets, D. Reidel, Boston, pp 721–758.

1983

J.-P. Barthélemy, Médianes dans les graphes et localisation, Cahiers du CERO 23, 163–182.

B. Monjardet, On the use of ultrafilters in social choice theory. In P. K. Pattanaik and M. Salles (eds) Social Choice and Welfare, North-Holland, Amsterdam, pp 73–78.

1984

H.J. Bandelt and J.-P. Barthélemy, Medians in median graphs, Discrete Applied Mathematics, 8, 131–142.

J.-P. Barthélemy, B. Leclerc and B. Monjardet, Quelques aspects du consensus en classification. In E. Diday and al. (eds) Data analysis and informatics, Elsevier, Amsterdam, pp 307–315.

J.-P. Barthélemy, B. Leclerc and B. Monjardet, Ensembles ordonnés et taxonomie mathématique. In M. Pouzet and D. Richard (eds) Order, description and roles, North-Holland, Amsterdam, pp 523–548.

B. Leclerc, Efficient and binary consensus functions on transitively valued relations, Mathematical Social Sciences 8, 45–61.

1985

B. Monjardet, Concordance et consensus d'ordres totaux : les coefficients K et W, Revue de Statistique Appliquée 33 (2), 55–87.

1986

J.-P. Barthélemy, B. Leclerc and B. Monjardet, On the use of ordered sets in problems of comparison and consensus of classifications, Journal of Classification 3, 187–224.

J.-P.Barthélemy and F.R. McMorris, The median procedure for n-trees, Journal of Classification, 3, 329–334.

1988

J.-P. Barthélemy, Thresholded consensus for n-trees, Journal of Classification 5, 229–236.

J.-P. Barthélemy, Comments on “Aggregation of equivalence relations” by P.C. Fishburn and A. Rubinstein, Journal of Classification 5, 85–87.

J.-P. Barthélemy and B. Monjardet, The median procedure in data analysis: New results and open problems. In H.H. Bock (ed) Classification and related methods of data analysis, North-Holland, Amsterdam, pp 309–316.

B. Leclerc, Consensus applications in the social sciences. In H.H. Bock (ed) Classification and related methods of data analysis, North-Holland, Amsterdam, pp 333–340.

1989

J.-P. Barthélemy, Social welfare and aggregation procedures: combinatorial and algorithmic aspects. In F. Roberts (ed) Applications of combinatorics and graph theory to the biological and social sciences, Springer, Berlin, pp 39–73.

J.-P. Barthélemy and F.R. McMorris, On an independence condition for consensus n-trees, Applied Mathematics Letters 2, 75–78.

J.-P. Barthélemy, A. Guénoche and O. Hudry, Median linear orders: Heuristics and a branch and bound algorithm, European Journal of Operational Research 42, 313–325.

C. Chameni-Nembua, Régle majoritaire et distributivité dans le permutoèdre, Mathématiques et Sciences Humaines 108, 5–22.

B. Leclerc, Consensus approaches for multiple categorical or relational data. In R. Coppi, S. Bolasco (eds) Multiway Data Analysis, Elsevier Sciences, Amsterdam, pp 65–75.

Hudry, O., Recherche d'ordres médians : complexité, algorithmique, et problèmes combinatoires, thèse de doctorat de mathématiques, E.N.S.T., Paris.

1990

B. Leclerc, Medians and majorities in semimodular lattices, SIAM Journal on Discrete Mathematics 3, 266–276.

B. Monjardet, Sur diverses formes de la “Règle de Condorcet” d'agrégation des préférences, Mathématiques et Sciences Humaines 111, 61–71.

B. Monjardet, Arrowian characterizations of latticial federation consensus functions, Mathematical Social Sciences 20 (1), 51–71.

1991

J.-P. Barthélemy and M.F.Janowitz, A formal theory of consensus, Siam Journal on Discrete Mathematics 4, 305–322.

J.-P. Barthélemy, F.R. McMorris and R.C. Powers, Independence condition for consensus n-trees revisited, Applied Mathematics Letters 4 (5), 43–46

B. Leclerc, Aggregation of fuzzy preferences: a theoretic Arrow-like approach, Fuzzy Sets and Systems 43 (3), 291–309.

B. Monjardet, Éléments pour une histoire de la médiane métrique. In Moyenne, Milieu, Centre. Histoire et usages. Coll. Histoire des Sciences et Techniques, n°5, Éditions de l'EHESS, Paris.

1992

J.-P. Barthélemy, F. R. McMorris and R.C. Powers, Dictatorial consensus functions on n-trees, Mathematical Social Sciences 25, 59–64.

I. Charon, A. Germa and O. Hudry, Utilisation des scores dans des méthodes exactes déterminant les ordres médians des tournois, Mathématiques, Informatique et Sciences humaines 119, 53–74.

1993

B. Leclerc, Lattice valuations, medians and majorities, Discrete Mathematics 111, 345–356.

1994

B. Leclerc, Medians for weight metrics in the covering graphs of semilattices, Discrete Applied Mathematics 49, 281–297.

1995

J.-P. Barthélemy and B. Leclerc, The median procedure for partitions. In P. Hansen (ed) Partitioning data sets, American Mathematical Society, Providence, pp. 3–34.

J.P. Barthélemy, F.R. McMorris and R.C. Powers, Stability conditions for consensus functions defined on n-trees, Mathematical and Computer Modelling 22 (1), 79–87.

B. Leclerc and B. Monjardet, Latticial theory of consensus. In W. Barnett, H. Moulin, M. Salles and N. Schofield (eds) Social choice, Welfare and Ethics, Cambridge University Press, pp 145–160.

1997

I. Charon, A. Guenoche, O. Hudry and F. Woirgard, New results on the computation of median orders, Discrete Mathematics 165–166, 139–154.

1998

I. Charon and O. Hudry, Lamarckian genetic algorithms applied to the aggregation of preferences, Annals of Operations Research 80, 281–297.

2000

J.-P. Barthélemy and F. Brucker, Average consensus in numerical taxonomy. In W. Gaul and al. (eds) Data Analysis, Springer, Berlin, pp 95–104.

2003

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Monjardet, B. Social choice theory and the “Centre de Mathématique Sociale”: some historical notes. Soc Choice Welfare 25, 433–456 (2005). https://doi.org/10.1007/s00355-005-0012-z

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