Some Further Remarks on Preference Proximity

Abstract

In an interesting paper Nitzan [1988] suggests an alternative formulation of preservation of preference proximity. He shows that his formulation is consistent with anonymity and respect for unanimity. This contrasts with the negative results that have been obtained with the formulations of preservation of preference proximity formulated by Baigent [1987] and Chichilnisky [1982]. The definitions and notation are taken from Baigent [1987] and Nitzan [1988]. The formal framework used by Nitzan is less general than that used in Baigent [1987] in the following ways. The set of all preferences P is restricted to the set of all linear preferences P*; the set of all profiles Pr is restricted to the set of all profiles of linear preferences Pr*; the set of all nonempty subsets of alternatives K is restricted to a single fixed nonempty subset of alternatives that will be taken to be X, the subset consisting of all alternatives. The class of social choice procedures considered by Nitzan consists of social decision functions F*:Pr* K, which assign a nonempty subset of alternatives to every profile of linear preferences. The essence of this condition may be explained as follows. If, according to some metric 6 on P*, profile p is closer to some profile for which x is the unanimous winner than profile q is to any profile for which y is the unanimous winner, then the choice resulting from p should be closer to simply choosing x than the choice resulting from q is to simply choosing y. This formulation gives a special significance to unanimity in situations in which unanimity is not present. It is not clear why choice should be restricted by considering what would be chosen unanimously if preferences in fact lack unanimity. This is not to deny that unanimous winners should be chosen when they exist. The point being made is that arguments for choosing unanimously when unanimity exists are not sufficient to establish that unanimity considerations should restrict choice when unanimity does not in fact exist. It can in fact be argued that there are other considerations which are important apart from unanimity and that these should be brought into play whenever unanimity is lacking. Such lexical arguments are very familiar. Generations of students of welfare economics have been told that only Pareto optimal alternatives