IN THIS PAPER we consider procedures for going from several individual preferences among several alternatives, called candidates, to something which may be called a collective preference. The individual preferences take the form of (total) orderings of the alternatives, and the collective preference is to take the form of a (total) weak ordering (i.e., ties allowed). We consider certain properties which seem desirable in such and investigate which have these properties. The of view taken here differs from that of other work in this area (e.g., [1, 2, 3, 4]) chiefly in asking that the procedure work for all possible sizes of the voting population, rather than for a fixed population, given in advance. This permits us to require, for example, that if each of two bodies of voters prefers candidate A to candidate B under a given procedure, then the combination of these bodies should prefer A to B under the same procedure. In Section 1 we give the formal definitions of an aggregation procedure and discuss certain desirable features, namely neutrality (treats candidates symmetrically), (the condition mentioned above), monotonicity, and an Archimedean property which says, roughly, that a sufficiently large body with a given distribution of preferences can impose its will on any body of fixed size. In Section 2 we introduce certain procedures: point systems and systems (roughly, allowing infinitesimal points), which are neutral and separable. They are monotonic if and only if the points are arranged in the natural order, and the are, in addition, Archimedean. In Section 3 we prove a converse, namely that any neutral and separable procedure can be realized by a generalized system and, if it is Archimedean, by a system. This part requires some familiarity with the notions of least upper bound of a set of real numbers and bases of vector spaces. In Section 4 (which is largely independent of Section 3), we consider point which use in a succession of eliminations. Such are neither separable nor monotonic but do satisfy some very weak separability and monotonicity conditions. While these probably do not characterize runoff systems, we know of no other satisfying them.

IF the variable X represents the price of an active stock, it is well known that the tendency of X to change is an increasing function of price itself, say f(X). If the tendency to relative change with respect to price were constant, graphs whose amplitude of fluctuation is uncorrelated with price could be constructed by (1) plotting log X on ordinary arithmetic scale or (2) plotting X on semilogarithmic scale. However, the tendency to relative change f(X)/X has been found to be a decreasing function of X and hence any constant-amplitude scale for plotting stock prices must be based on some function other than log X, say F(X), whose derivative is inversely proportional to the tendency to change with respect to price. In other words, the desired function is any solution of the differential equation