The Duality between the Path Independent Choice Operators and the Antiexchange Closure Operators
by
B. Monjardet
CERMSEM, Université Paris I Panthéon Sorbonne, Maison des Sciences Économiques

In this talk we consider operators on a finite set S, i.e. maps from 2^S into 2^S (where 2^S denotes the set of all subsets of S). A set of operators is ordered by the pointwise order between maps. A choice operator c on S is an operator on S satisfying \emptyset subset c(A) subset or equal A, for every non empty A subset or equal S, and c(\emptyset)=\emptyset. Choice operators appear in several fields of mathematics, but they have been especially studied in the theory of the revealed preference in microeconomics where, under the name of choice functions, they are models of consumer'choice behavior. In particular this theory characterizes "rational" (in some sense) mechanisms of choice by means of axioms on choice functions. Three axioms called Heritage (H), Outcast (O) and Concordance (C) are particularly significant since their combination allows to characterize such rational choice mechanisms. A choice operator c satisfies the axioms H and O if and only if it satisfies the path independence property (PI): c(A \cup B) = c(c(A) \cup c(B)), for all A, B subset or equal S. The three sets of choice operators satisfying H, C or O (ordered by the pointwise order) are lattices whereas the set of path independent choice operators is a join-semilattice. A closure operator k is an idempotent, extensive and isotone operator. A closure operator k is said to be anti-exchange if k(\emptyset) = \emptyset and for all A subset or equal S and x, y (x =/= y) in S\k(A), y in k(A+x) implies x not in k(A+y). The family of closed sets defined by an anti-exchange closure operator is a convex geometry and such a family is a lower locally distributive (called also meet-distributive) lattice. The set of all anti-exchange closure operators (ordered by the pointwise order) is a meet semilattice. Johnson and Dean and independently Koshevoy have shown that there exists a correspondence between path independent choice operators and anti-exchange closure operators. In fact this correspondence induces a duality between the join-semilattice of the path independent choice operators and the meet-semilattice of the anti-exchange closure operators. Then this duality can be used to obtain immediately, from classical or non-classical results in the theory of anti-exchange closure operators, old or new results in the theory of choice operators (and conversely). For instance one can use this duality to reobtain the Aizerman and Malishevski representation result of path independent choice operators by linear orders and to obtain the minimum number of linear orders required in such a representation.

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Date received: March 11, 2003