The structure of continuous-valued neutral monotonic social functions

This article consists of several results characterizing neutral monotonic social functions over alternative sets that are compact, convex subsets of Euclidean space. One major result is that a neutral monotonic social function is continuous-valued (i.e., does not make abrupt reversals) for all profiles of continuous-valued weak orderings if and only if it is simple (i.e., completely determined by a single set of decisive coalitions). A second major result is that a continuous-valued neutral monotonic social function will guarantee the existence of a socially undominated alternative for all profiles of continuousvalued, almost convex weak orderings if and only if the smallest empty intersection within the set of decisive coalitions is at least m+2 in size where m is the dimension of the alternative set.Associate Professor of Law, University of Southern California and Assistant Professor of Law and Economics, California Institute of Technology.This article is taken almost directly from a chapter of my Ph.D. dissertation (Strnad 1982). That chapter itself is substantially the same as an earlier paper, Strnad (1981). I have profited greatly from the suggestions, guidance and encouragement of Donald Brown in this work. Strnad (1981) was presented at the California Institute of Technology Theory Workshop in January 1982 and at the annual Public Choice Society Meetings in San Antonio, Texas on March 5–7, 1982. I am grateful to participants at both the Workshop and the Meetings for their comments. Comments by the referee and by Jerry Kramer, the editor, also were very helpful. Any remaining errors are solely my responsibility.Substantial financial support for this work has been provided by Summer Research Grants for 1981 and 1982 from the University of Southern California Law Center