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The median procedure in the semilattice of orders. (English) Zbl 1026.06006
Summary: Let $X$ be a finite set; we are concerned with the problem of finding a consensus order $P$ that summarizes an $m$-tuple (profile) $P^*$ of (partial) orders on $X$. A classical approach is to consider a distance function $d$ on the set $O$ of all the orders of $X$ and to search to minimize the remoteness $\sum_{1\le i\le m} d(P,P_i)$. We study some properties of this median procedure, and compare it with some other consensus approaches. Besides the classical symmetric difference metric, other distances are considered, and we particularly address the consequences for the consensus problem of the existence of a semilattice structure on the set $O$.

MSC:
06A12Semilattices
91B14Social choice
06A06Partial order
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References:
[1] Arrow, K. J.: Social choice and individual values. (1951) · Zbl 0984.91513
[2] Barbut, M.; Monjardet, B.: Ordre et classification, algèbre et combinatoire. (1970) · Zbl 0267.06001
[3] Barthélemy, J. P.: Remarques sur LES propriétés métriques des ensembles ordonnés. Math. sci. Hum. 61, 39-60 (1978)
[4] Barthélemy, J. P.: Caractérisations axiomatiques de la distance de la différence symétrique entre des relations binaires. Math. sci. Hum. 67, 85-113 (1979)
[5] Barthélemy, J. P.: Arrow’s theoremunusual domain and extended codomain. Math. social sci. 3, 79-89 (1982)
[6] Barthélemy, J. P.; Janowitz, M. F.: A formal theory of consensus. SIAM J. Discrete math. 4, 305-322 (1991) · Zbl 0734.90008
[7] Barthélemy, J. P.; Leclerc, B.: The median procedure for partitions. Partitioning data sets, 3-34 (1995) · Zbl 0814.62031
[8] Barthélemy, J. P.; Monjardet, B.: The median procedure in cluster analysis and social choice theory. Math. social sci. 1, 235-268 (1981) · Zbl 0486.62057
[9] C. Birfet, Médianes dans le demi-treillis des ordres, Mémoire de DEA MIASH, 1995.
[10] Birkhoff, G.: Lattice theory. (1967) · Zbl 0153.02501
[11] Brown, D. J.: Aggregation of preferences. Quart. J. Econom. 89, 456-469 (1975)
[12] N. Caspard, B. Monjardet, The lattice of Moore families and closure operators on a finite set: a survey, Discrete Appl. Math., this issue. · Zbl 0971.06009
[13] Charon, I.; Hudry, O.; Woirgard, F.: Ordres médians et ordres de Slater des tournois. Math. inf. Sci. hum. 133, 23-56 (1996)
[14] Doignon, J. -P; Falmagne, J. C.: Well-graded families of relations. Discrete math. 173, 35-44 (1997) · Zbl 0877.06002
[15] O. Hudry, Tournois et optimisation combinatoire, Thèse d’habilitation, Université Pierre et Marie Curie, Paris, 1998.
[16] Kemeny, J.: Mathematics without numbers. Daedalus 88, 577-591 (1959)
[17] Leclerc, B.: Efficient and binary consensus functions on transitively valued relations. Math. social sci. 8, 45-61 (1984) · Zbl 0566.90003
[18] Leclerc, B.: Lattice valuations, medians and majorities. Discrete math. 111, 345-356 (1993) · Zbl 0785.06006
[19] Leclerc, B.: Medians for weight metrics in the covering graphs of semilattices. Discrete appl. Math. 49, 281-297 (1994) · Zbl 0794.06002
[20] Leclerc, B.; Monjardet, B.: Latticial theory of consensus. Social choice, welfare, and ethics, 145-160 (1995) · Zbl 0941.91029
[21] Li Jinlu, Singular points and an upper bound of medians in upper semimodular semilattices, Department of Mathematics, Shawnee State University, preprint. · Zbl 0968.06008
[22] Mas-Collel, A.; Sonnenschein, H.: General possibility theorems for group decisions. Rev. econom. Stud. 39, 185-192 (1972) · Zbl 0239.90003
[23] F.R. McMorris, H.M. Mulder, R.C. Powers, The median function on median graphs and semilattices, 1996, submitted for publication. · Zbl 0951.05032
[24] B.G. Mirkin, On the problem of reconciling partitions, in: Quantitative Sociology, International Perspectives on Mathematical and Statistical Modelling, Academic Press, New York, 1975, 441--449.
[25] Monjardet, B.: Théorie et applications de la médiane dans LES treillis distributifs finis. Ann. discrete math. 9, 87-91 (1980) · Zbl 0451.06012
[26] Monjardet, B.: Metrics on partially ordered sets--a survey. Discrete math. 35, 173-184 (1981) · Zbl 0463.46016
[27] Monjardet, B.: Comments on R.P. Dilworth’s lattices with unique irreducible decompositions. The dilworth theorems; selected works of robert P. Dilworth, 192-201 (1990)
[28] Monjardet, B.: Arrowian characterization of latticial federation consensus functions. Math. social sci. 20, 51-71 (1990) · Zbl 0746.90002
[29] Pirlot, M.; Vincke, Ph: Semiorders. properties, representations, applications. (1997) · Zbl 0897.06002
[30] Reinelt, G.: The linear ordering problem: algorithm and applications. (1985) · Zbl 0565.68058
[31] Y. Wakabayashi, Aggregation of binary relations: algorithmic and polyhedral investigations, Doctoral Thesis, Universität Augsburg, 1986. · Zbl 0606.68036
[32] Young, H. P.: An axiomatization of borda’s rule. J. econom. Theory 9, 43-52 (1974)
[33] Young, H. P.; Levenglick, A.: A consistent extension of Condorcet’s election principle. SIAM J. Appl. math. 35, 285-300 (1978) · Zbl 0385.90010