The congruence axiom and path independence
Let \(X\) be a universal set and let \([X]\) denote the collection of non-empty subsets of \(X\). The authors consider choice functions \(C:S \to [X]\), where \(S\) is a non-empty subcollection of \(X\). Let \(A \in S\). A path in \(A\) is a finite ordered collection of non-empty subsets of \(A\) covering \(A\). A choice function is called path independent on \(A\) if the ultimate choice in some sequential procedure does not depend upon the path. In the general setting of the authors, the domain of the choice function may not be all of \([X]\). In this case, it is possible that given a set \(A\in S\), some of the subsets of a path in \(A\), their images under the choice function, or indeed the image of \(A\), may not be in \(S\). Hence, in order to define sequential choice procedures, the authors introduce various technical admissibility criteria. Under these admissibility criteria, the main result of the paper is that a sequential choice function is path independent if and only if it satisfies Richter's congruence axiom on revealed preference, which in turn is a necessary and sufficient condition for \(C\) to have a transitive rationalization.
scientific article; zbMATH DE number 1348018
scientific article