¹ These problems together constitute the fundamental social problem statically conceived. No attention is given in this note to considerations of growth.

² Principal references are: Kaldor, N., “Welfare Propositions of Economics and Interpersonal Comparisons of Utility,” Economic Journal, 09, 1939 CrossRefGoogle Scholar; de Scitovszky, T., “A Note on Welfare Propositions in Economics,” Review of Economic Studies, 11, 1941 Google Scholar; I.M.D. Little, A Critique of Welfare Economics; and Samuelson, Paul A., “Evaluation of Real National Income,” Oxford Economic Papers, New Series, 01, 1950.Google Scholar

³ Social Choice and Individual Values (New York, 1951)Google Scholar; “A Difficulty in the Concept of Social Welfare,” Journal of Political Economy, 08, 1950.Google Scholar References will be to the book.

⁴ Arrow calls such a procedure a Social Welfare Function. Following Bergson, economists commonly use this term to describe a related but distinct concept and it would have been preferable to have used a term like “Constitution” which, moreover, does, in its ordinary usage, have the required connotation. (Cf. 23–4). In this note we shall use this term.

⁵ We have endeavoured to give a non-technical account of the Arrow conditions. The reader is referred to chapter 3 for the full statement.

⁶ The first three are explicitly stated (p. 6–8), the fourth seems to be implicit in the analysis.

⁷ The Theory of Games and Economic Behavior (2nd ed.), 15–30 Google Scholar and Appendix.

⁸ The meaningfulness of these statements is derived from the fundamental assumption about interpersonal utilities.

⁹ Arrow presents first the proof of the 2-individual, 3-alternative case. The key step is the derivation of consequence 3: If x′Pxy′ and y′P2x′, then x′ly′. This is based on the following assumption: x′P₁y′ and y′P₂x′ implies either always x′ly′ or always x′Py′ or always y′Px′. This is merely the spelling out of the condition that the social ordering depends only on individual orderings. The either … or is used in the exclusive sense. On this point Arrow is not entirely consistent: compare his definition (p. 13) and his usage in the statement of Lemma 1 (e) (p. 14). It will be found impossible to derive all of statements (2) - (6) (p. 50–1) if it is allowed, in our manner, that orderings are not all-important. Constitution 1 does not commit itself in advance to deciding whether x′P₁y′ and y′P₂x′ implies x′Py′ or x′ly′ or y′Px′. A one-to-one correspondence between two sets of individual orderings is neither necessary nor sufficient for the same social ordering.

Arrow next presents the proof of the many-individual, three-alternatives case. The key step is the derivation of consequence 4: If V′ is decisive for either x against y or y against z, V′ is decisive for x against z, where x, y, z are distinct alternatives (p. 54). Again, the proof is based on generalization from a special set of assumptions about individual orderings with the aid of the condition that only individual orderings are to count, (p. 55–6). Under Constitution 1 no one set of individuals can be always “decisive”. Indeed, there is no decisive set under Constitution 1. Whether or not a particular subset of voters can be “decisive” in any particular election depends on the preferences of other voters.