Selection closedness and scoring correspondences

Abstract

Universal self-selectivity of a social choice function (SCF) F defined in Koray (Econometrica 68:981–995, 2000) implies that F is either dictatorial or anti-dictatorial. In an attempt to escape this impossibility, here we weaken self-selectivity of an SCF by introducing the notion of selection-closedness pertaining to families of SCFs. As in the self-selectivity setting, a society, which is to make a choice from a set A of alternatives, is also to choose the choice rule that will be employed in making that choice. Self-selectivity of an SCF F requires that F outrivals all available SCFs by selecting itself from among them if it is also used in choosing the choice rule, where the societal preferences on the available SCFs are induced from those on the set A of alternatives in a consequentialist way. Given a collection \({\mathscr {F}}\) of SCFs and a nonempty finite set \({\mathcal {A}}\) of available SCFs containing also members of \({\mathscr {F}},\) an SCF in \({\mathscr {F}} \cap {\mathcal {A}}\) is now not required any more to select itself from \({\mathcal {A}},\) but it suffices that it selects some member of \({\mathscr {F}}\) for \({\mathscr {F}}\) to be selection-closed. It is shown that a proper subset of the collection of all neutral SCFs is selection-closed if and only if all its members are either dictatorships or anti-dictatorships. We further weaken the notion of selection-closedness to an extent that not only enables us to escape the impossibility result, but also equips us with a yardstick to compare correspondences as to whether or not their singleton valued refinements form a weakly selection-closed family. A rich family of scoring correspondences with strict scoring vectors are shown to pass the test of weak selection-closedness, while the Pareto and Condorcet correspondences fail.