Scaling invariance and a characterization of linear objective functions

Abstract

A decision-maker who aims to select the "best" collection of alternatives from the finite set of available ones might be severely restricted in the design of the selection method. If the representation of valuations of available alternatives is subject to invariance under linear scaling, such as the choice of the unit of measurement, a sensible way to compare choices is to compare weighted sums of individual valuations corresponding to these choices. This scaling invariance, in conjunction with additional reasonable axioms, provides a characterization of linear 0-1 programming objective functions. The problem of finding an optimal subset of available data to be aggregated, allowing for use of different aggregation methods for different subsets of data, is also addressed. If the input data in the optimal aggregation problem are measured on a ratio scale and if the aggregation must be unanimous and symmetric, the arithmetic mean is the only sensible aggregation method. © 2011 Springer-Verlag.