Arrovian Social Choice
A standard view of the British economists at the end of the nineteen century and the beginning of the twentieth century was that welfare, utility, satisfaction, and so on had a money measure (Pigou 1932).
It seems clear that in this case a utilitarian social welfare function could be used where the social welfare is the addition of the individual welfares. Since individual utility was measured in monetary terms, the problems of scales, origin, comparability, and so on were assumed to have already been solved. Maximizing social welfare amounted to finding a maximum for the utilitarian social welfare function, given that individual utility functions were fixed, the variables being the social states, whatever the term of social state covers. However, at the same time under the influence of Leon Walras and, above all, Vilfredo Pareto, economists wanted to get rid of the measurement of utility problem. The solution was to use ordinal utility or even the underlying preference relation. With ordinal utility functions, still numerical functions, the real numbers/ utilities could only be meaningfully compared according to the relation ≥. All the other mathematical properties defining the field of real numbers were rejected. A kind of corollary to the ordinalism thesis was that interpersonal comparisons have to be excluded too, even when these comparisons are limited to the relation ≥, that is, it could not be asserted that the utility of individual i in state x is, say, greater than the utility of individual j in state y. On this basis, the only possible concept relative to the social goodness of a social state was Pareto optimality: a social state x is optimal if there is no other feasible social state y such that all individual utilities are greater for y than for x (or in its strong version: all individual utilities are at least as great for y than for x and one is greater).In the 1930s, interpersonal comparisons were, however, reintroduced as (virtual) compensations by Hicks, Kaldor, Harrod and Scitovsky (see Arrow and Scitovsky 1969; Baumol and Wilson 2001). The principle of compensation is that in a change of social states, say from x to y, individuals who gain in the change could virtually compensate those who lose, making the change a Pareto improvement, that is, after compensation every individual has a greater utility due to the change. It is obvious that this procedure entails interpersonal comparisons. Furthermore, it has been shown that it was not immune to paradoxes. It is in this context that Bergson proposed the new notion of social welfare function in 1938. The form of the function has been modified by Samuelson (1947) and it is in this form proposed by Samuelson that the function is generally presented. In Samuelson’s version, the social welfare function, say f, associates a real number to a list of individual utilities u1,..., un of individuals 1,..., n for some social state belonging to a fixed set of social states. The individual utility functions are fixed. For instance, if we have a Cobb-Douglas utility function defined over the positive orthant of a ^-dimensional Euclidean space for individual i, say, ui(x) = (3/5) x11ft x21/k... xkιzk, the parameters 3/5 and 1/k are fixed, whatever the variables x1,..., xk are. To impose upon such a social welfare function a Paretian property is to assume that ∂f∕∂ui > 0 (loosely speaking, social welfare increases, or decreases, when individual i’s utility increases, or decreases, all other things being equal). The purpose of the function is then to select some Pareto-optimal social state through classical maximization, and this for public policy. However, it remains to be known how and who will construct the function. In some sense, Arrow provided a reply to this question, doubly negative, with his impossibility theorem. Arrow’s analysis marks the birth of modern social choice theory.