Medians in LLD Meet Semilattices, and the Case of Orders

Bruno Leclerc

Let X be an n-set; we are concerned with the problem of �nding a consensus order

P on X that summarizes a k-tuple (pro�le) (Pi)1�i�k of orders on X. This problem is a

basic one in the domain preference aggregation, but has been mainly studied in the case

of linear orders. A classical approach is to consider a distance function d on the set O of

all the orders on X and to search to minimize the remotenessP

1�i�kd(P; Pi). The fact

that O is a semilattice allows us to apply some already recognized properties of consensus

functions in such structures (Monjardet 1990, Leclerc 1994). Firstly, we particularize these

properties to the class of lower locally distributive (LLD) semilattices, which includes O.

Such semilattices are characterized by the identity of two metrics, the most extensively used

ones in studies about medians in lattice structures. Then, we consider the speci�c case

of orders, where this lattice approach allows us to extend some results about the role of

majority pairs already obtained, by very di�erent considerations, in the case of linear orders

(see Charon et al. 1996). We obtain other properties of the median procedure for orders,

like the Pareto property of medians with the symmetric di�erence metric. We compare

the median procedure, for various metrics, with other consensus approaches: quota rules,

Arrowian axiomatics (Brown 1975, Leclerc 1984).

References

D.J. Brown (1975): Aggregation of preferences, Quaterly J. of Economics, 89, 456-469.

I. Charon, O. Hudry, F. Woirgard (1996): Ordres m�edians et ordres de Slater des tournois,

Math. Inf. Sci. hum., 133, 23-56.

B. Leclerc (1984): E�cient and binary consensus functions on transitively valued relations,Math.

Soc. Sci., 8, 45-61.

B. Leclerc (1994): Medians for weight metrics in the covering graphs of semilattices, Discrete

Applied Math., 49, 281-297.

B. Monjardet (1990): Arrowian characterization of latticial federation consensus functions,Math.

Soc. Sci., 20, 51-71.

Bruno Leclerc

CAMS-EHESS, 54 bd Raspail

F-75270 Paris cedex 06, FRANCE

phone: (33) (0)1 49 54 20 39

fax: (33) (0)1 49 54 21 09

http : ==www:ehess:fr=centres=cams=index:html

email: leclerc@ehess:fr

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