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On the Problem of Social Welfare Functions*

Published online by Cambridge University Press:  07 November 2014

J. C. Weldon
J. C. Weldon*
Affiliation:
McGill University
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Extract

This paper is concerned with the “General Possibility Theorem” of Dr. K. J. Arrow. Since the theorem is certainly elegant and probably important, it is proper as well as convenient to refer to it from this point on as Arrow's theorem. The theorem deals with the problem of discovering a rule by which social preferences can be constructed from individual preferences. Such a rule, or social welfare function, gives a social ordering of alternatives of any kind for every possible arrangement of the corresponding individual orderings. Arrow's theorem declares that no social welfare function exists that satisfies those conditions that most of us would consider essential to a satisfactory rule. The outcome is something of a shock to preconceptions.

In what follows two aspects of Arrow's theorem are considered. In the early part of the paper the theorem as such is examined. A brief and informal statement of Arrow's argument is followed by a formal statement of the conditions on which the argument depends. A proof of the theorem is offered that seems to be rather more naturally constructed than the original. A set of conditions is proposed (derived from Arrow's proof) that is weaker than the original set, that seems to be at least as plausible, and that leads to the same conclusion by a short and direct route. In the later part of the paper, methods of circumventing the theorem (there seems to be no way of removing it) are explored.

Type
Research Article
Information
Canadian Journal of Economics and Political Science/Revue canadienne de economiques et science politique , Volume 18 , Issue 4 , November 1952 , pp. 452 - 463
Copyright
Copyright © Canadian Political Science Association 1952

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Footnotes

*

This paper was presented at the annual meeting of the Canadian Political Science Association in Quebec, June 4, 1952.

References

1 For Arrow's statement of the theorem, see his Social Choice and Individual Values, Cowles Commission for Research in Economics, Monograph no. 12, (New York: Wiley and Sons, 1951), 59.Google Scholar Though Arrow has discussed the theorem in other places, all references below are to the monograph. The present paper is really no more than a commentary on Arrow's work.

2 The reader should be cautioned that the illustration is by a novice for novices. The pure mathematician can probably point to a good many places where style and even substance are deficient.

3 The social welfare function and the institutional mechanism are rarely completely analogous, for the latter depends on expressed and not on actual preferences.

4 Here, and wherever else Arrow is paraphrased, it is quite possible that Arrow himself would disclaim responsibility for at least part of what is said.

5 The reader to whom the notation that follows is unfamiliar can find it (or a very similar notation) described in any text on symbolic logic. Short and clear accounts, fascinating in their own right, are available in Tarski's, A. Introduction to Logic (New York, 1941)Google Scholar and Quine's, W. V. Methods of Logic (New York, 1950).Google Scholar Although the axioms are given formally, the proof is developed in relatively informal style and can be followed with the aid of the informal statement of the axioms already presented.

6 If X̄ has one element only, the corresponding theorem, though trivial, is still true, chiefly because of the way V is framed. If X̄ has two elements, the corresponding theorem is true if ̄K has only one element, but is otherwise false. It is therefore possible to construct an f̄ satisfying I to V if there are precisely two alternatives and more than one person. Arrow, incidentally, seems to overlook the Robinson Crusoe exception to the two-alternative case. Cf. Arrow, , Social Choice, 46–8.Google Scholar

7 The argument used here is needed below in circumstances where inspection is hardly feasible. It is given in greater detail there.

8 Thus, in this block the (m + 1)th person prefers x̄ 1 to x̄ 2, to x̄ 3, etc.

9 By II, the absence of 〈ḡh̄〉 from any one of the social orderings would imply its absence from the given ordering, given that III holds. This contradicts the hypothesis.

10 This argument, with obvious modifications, is needed at one or two other points in what follows. The reader will be referred to this footnote. Incidentally, the word “argument” has to do double duty throughout. It is used in its technical sense, the “argument” of a function, and in its literary sense. The context indicates the sense intended.

11 See footnote 10.

12 It is here that the need for three alternatives arises. Cf. footnote 6. The substance of the construction that follows is its use of the transitivity property of I to extend, in step-by-step fashion, the known powers of k̄ m+1 to all circumstances. Cf. Arrow, , Social Choice, 54 ff.Google Scholar

13 If it were present in this least favourable circumstance, it would be present in all circumstances, so entailing a violation of IV. See footnote 10.

14 Again see footnote 10.

15 The sequence of steps by which the powers of k̄ m+1 are generalized can be varied in a number of ways, but as far as one can see, at least three steps are needed. Here these steps concern, in turn, the pairs 〈h̄i〉), i ≠ ḡ, h̄, the pairs 〈ij〉), i ≠ j; i, j ≠ h̄, and all other pairs. Cf. Arrow, , Social Choice, 55.Google Scholar

16 In Arrow's argument, the change amounts to omitting “Definition 10” and substituting “If and only if … “ for “If … then …” in “Consequence 2.” Ibid., 52-3.

17 There is a disadvantage in using a weaker system of axioms. If relaxing the restrictions is proposed, something that should not pass may get by the second set when it would not have got by the first.

18 See “Consequence 3” and “Consequence 5,” Arrow, , Social Choice, 54 and 56.Google Scholar

19 Though this example is not considered further here, one believes that it may merit attention. A good many cases exist in which individual orderings are probably related as dependent and independent variable: thus, the orderings of members of a family or members of a board of directors. Along the same line, it would be interesting to explore the social orderings of a society that forms itself into sub-societies, the ultimate social decisions being made by representatives of the sub-societies. The democratic process in British Columbia gives an example of unusual interest. In many cases of this kind, the distinction between actual and expressed preferences is likely to be critical. The problems become almost impossible to separate from those of the theory of games.

20 If reference is made to the table given earlier, it will be seen that violations of I can be discovered in line-by-line inspection of the left-hand column; of II and III, in two-by-two comparisons of the rows; of IV, in inspection of the left-hand column as a whole; and of V, in comparison of the left-hand column with each of the other columns (the centre column here being broken down into its components). These facts are the source of the present distinctions.

21 Arrow's commentary on his theorem occupies the last thirty pages or so of his book. The reader should consult in particular chap, vii, 74-91.

22 Ibid., 81.

23 See in particular Black, D., “On the Rationale of Group Decision-Making,” Journal of Political Economy, 02, 1948.CrossRefGoogle Scholar

24 Ibid.

25 If the meaning of the conditions is kept in mind the result is not surprising.

26 The discussion is rather loose here. Majority voting must be interpreted in a way that ensures that the number of votes is odd, perhaps by assigning two votes to one of the voters if the number of voters is even. Arrow feels that it is necessary to show that Black's result holds, even when the number of alternatives is infinite. One does not accept the need for this refinement, though the matter is not very important. See Arrow, , Social Choice, 77–80.Google Scholar

27 Ibid., 89.

28 Thus, the required property of the opportunity-line is obtained in these circumstances by removing from the list of social alternatives all items that will never be encountered in an actual social choice because of technical reasons. On the one hand those items are removed that are generally believed to be technically impossible, and on the other, those items that are generally believed to contain less of both “pure” alternatives than a technically possible alternative. The conditions given are more restrictive than is formally necessary, but seem to cover most instances of practical significance.

29 An analogy might be drawn with the composite events used by Professors von Neumann and Morgenstern in their construction of a measurable utility. See the early sections of the Theory of Games and Economic Behaviour (Princeton, N.J., 1947).Google Scholar

30 For an interesting development along these lines see Kemp, M. and Asimakopulos, T., “A Note on ‘Social Welfare Functions’ and Cardinal Utility,” Canadian Journal of Economics and Political Science, XVIII, no. 2, 05, 1952.Google Scholar It is possible to disagree with certain details in the “Note” while subscribing to the intention of its argument.

31 Even here one has to take into account such things as the fanaticism (or, if one prefers, the enthusiasm) of highly organized groups. This, of course, is a practical and not a formal point.

32 Arrow, , Social Choice, 9.Google Scholar