Abstract
The following monograph was first presented for publication in November, 1949. When, after eighteen months, it was found impossible to publish it in a journal, the authors sought to have it published as a separate monograph;2 and they have been enabled to do so by the generous financial assistance of the Carnegie Trust for Scottish Universities.
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Notes
[See introduction, p. xxii]
[The article was originally submitted to the journal Econometrica, and immediately acknowledged. However, in September 1950 the editor’s secretary wrote to Black about a change in submission procedure, with no comment on what had been happening to the paper in the meantime. Not until May 24, 1951 did William B Simpson write on behalf of the journal, accepting the paper for resubmission on condition that the authors acknowledged Arrow’s result (Arrow 1950). Black was very angry at what he saw as a denial of priority, withdrew the paper, and published it separately through a small specialist publisher, William Hodge and Company.]
Cf. Black (1948) §§ 1-5.
There is a class of cases which, throughout the text, we have chosen to disregard. As an instance, in the vote between the motions a i and a j, A might be indifferent between the two, while B preferred a i and C preferred a j. So far the result would be indeterminate; but it would become determinate if either B or C were chairman and given the right of a casting vote. It is easy enough to take into account possibilities of this kind when they crop up in particular examples (see §§ 43-4, infra), but it would embarrass the text to take them into account in the main exposition.
There may be several such points of maxima on the preference curve of any member. Cf. Fig 1
A curve has a tangent at a cusp, but such a curve cannot be a convex curve.
This excludes cases (i) and (ii) of the preceding paragraph.
The restriction to convex contours is not necessary for the present section. It would be sufficient if the contours were such that lines parallel to one or other of the co-ordinate axes intersected any particular contour in not more than two points.
Cf. Black (1948: 250).
In the case where the system starts with a =a i and the first vote is taken in relation to b, P will be reached in a single vote; or if the system starts from b = b p and the first vote is taken in relation to a, P will again be reached in a single vote. A position of unstable equilibrium might similarly be reached (and maintained) as a result of the first vote.
The relevant literature in Economics is that relating to the ‘cobweb theorem.’ See Samuelson (1937-8: 368-74). Also in Mathematics, see E. T. Whittaker and G. Robinson (1937: 81-2), on successive approximations.
The graph of F(U) is always upsloping, without any horizontal segments. Such a function is sometimes called ‘monotonie strictly increasing.’
[Sic]
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Black, D., Newing, R.A. (1998). Committee Decisions with Complementary Valuation. In: McLean, I., McMillan, A., Monroe, B.L. (eds) The Theory of Committees and Elections by Duncan Black and Committee Decisions with Complementary Valuation by Duncan Black and R.A. Newing. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-4860-3_22
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