## Social choice and individual values.(English)Zbl 0984.91513

Cowles Commission Monograph. 12. New York, NY: Wiley, London: Chapman & Hall, xi, 99 p. (1951).
This book is concerned with the following problem. A collection of individuals and a set of social alternatives are given and it is assumed that each individual ranks the alternatives in accordance with his preference. Problem: to obtain a ”social ordering” of the alternatives as a function of the individual orderings, which will represent the preferences of the community as a whole and which will satisfy certain requirements of compatibility with the preferences of the individuals. The problem is formalized as follows. A set $$S$$ of alternatives is given. A weak ordering on $$S$$ is defined to be a relation $$R$$ which is transitive and such that any two alternatives are comparable. A set of $$n$$ weak orderings $$R_ 1,\cdots,R_ n$$ is given corresponding to $$n$$ individuals. The problem is then to find a function which attaches to each such $$n$$-tuple of orderings an ordering $$R$$. Such a function is called a ”social welfare function,” and the ordering $$R$$ is called the ”social ordering.” The author now gives a number of requirements which the social welfare function must satisfy, which we paraphrase roughly. (1) If two different sets of individual orderings $$R_ 1,\cdots,R_ n$$ and $$R_ 1{}',\cdots,R_ n{}'$$ are identical except that a particular alternative $$x$$ is raised in preference by some of the individuals in the second set of orderings $$R_ i{}'$$, then this alternative will not be lowered in the corresponding social ordering $$R'$$. (2) The relative positions of two alternatives $$x$$ and $$y$$ in the social ordering $$R$$ shall depend only on their relative positions in the individual orderings $$R_ 1,\cdots,R_ n$$ and not on the positions of alternatives distinct from $$x$$ and $$y$$. A social welfare function is called ”imposed” if for some pair of alternatives $$x$$ and $$y$$, $$xRy$$ for every social ordering $$R$$. A social welfare function is called ”dictatorial,” if there exists an integer $$i$$, $$1\leq i\leq n$$, such that for any $$R_ 1,\cdots,R_ n$$, the social ordering $$R$$ is the same as $$R_ i$$. The main result of the book can now be stated (General Possibility Theorem): If $$S$$ contains more than two alternatives then any social welfare function satisfying the first and second conditions must be either imposed or dictatorial. A similar theorem is also proved for cases where restrictions are placed on the allowable individual orderings $$R_ 1,\cdots,R_ n$$. A large portion of the book is taken up with giving economic justifications for the various axioms and conditions used in setting up the problem. However, the argument for (2) is not convincing. The following simple example may illustrate the difficulty. Two individuals are ranking 100 alternatives. Suppose $$x$$ and $$y$$ are two alternatives and suppose the first individual ranks $$x$$ first and $$y$$ last, the second ranks $$y$$ first and $$x$$ second. It then seems reasonable that the social ordering should rank $$x$$ above $$y$$. On the other hand if the first individual ranks $$x$$ first and $$y$$ second, while the second ranks $$y$$ first and $$x$$ last the same reasoning would rank $$y$$ above $$x$$ in the social ordering. However, the author’s second condition requires that $$x$$ must also be ranked above $$y$$ in this second case, which seems to contradict common sense. Thus if one accepts the author’s remark that the result of the main theorem is ”paradoxical” it would seem that paradoxes are already evident in his basic assumptions.

### MSC:

 91-02 Research exposition (monographs, survey articles) pertaining to game theory, economics, and finance 91B14 Social choice 91B15 Welfare economics