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The Theory of Elections in Single-Member Constituencies

Published online by Cambridge University Press:  07 November 2014

Duncan Black
Duncan Black*
Affiliation:
Glasgow University
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Extract

Which candidate ought to be elected in a single-member constituency if all that we take into account is the order in which each of the electors ranks the various candidates? The most reasonable answer, I think, is that that candidate ought to be elected who, on the whole or on the average, stands highest on the electors' schedules of preferences.

At the very outset of the argument we try to move from the is to the ought and to jump the unbridgeable chasm between the universe of science and that of morals. Some discussion of this step should be given, even though it does no more than point out the assumptions under which the jump is made.

We assume that, in a constituency which is to elect a single member, we know the order in which the candidates are ranked on the schedule of preferences of each elector. If the electors had only been allowed to cast a single vote and the record of this was all the information at our disposal, inevitably, I think, we would have to conclude that the candidate most deserving of election was the one with the greatest number of first-preference votes. As against this, however, our assumption will be that each elector expresses his attitude to each of the candidates in the field, and that we have this record of the electors' attitudes.

Type
Articles
Information
Canadian Journal of Economics and Political Science/Revue canadienne de economiques et science politique , Volume 15 , Issue 2 , May 1949 , pp. 158 - 175
Copyright
Copyright © Canadian Political Science Association 1949

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References

1 In some circumstances one measure of “the” average may reasonably be held to be superior to the others. The bearing of this well-known feature on the theory that we give will be fairly obvious.

2 This criterion was proposed by Borda and Condorcet, though I had no direct acquaintance with their work until the present article was approaching the stage of completion.

3 In technical language, the argument in favour of the method of election will be stronger, the less rigorous the degree of restriction on the preference curves of the electors, by which the class of cases is defined.

4 Black, , “On the Rationale of Group Decision-Making” (Journal of Political Economy, 02, 1948)CrossRefGoogle Scholar, and “The Decisions of a Committee Using a Special Majority” (Econometrica, 07, 1948, sec. 1-8).Google Scholar

5 Cf. Black, the articles cited, where a fuller account of the theory is given.

6 A matrix is simply a table to show the number of votes cast for and against any candidate when he meets another in a vote. For example, if the candidate a 1 were to be put in a vote against a 2, the 25 voters A would support a 1, while the 20 voters B and the 15 voters C would support a 2. The figures in the cell (a 1, a 2) therefore, are (25, 35). The figures in the other cells are entered in the same way, except those on the main diagonal. In each of the cells (a 1, a 1), (a 2, a 2) … we simply enter zeros, since it would be nonsensical to put any candidate in a vote against himself.

7 I owe this section to Nanson, E. J., “Methods of Election,” in the British Government blue book Miscellaneous No. 3 (1907), Cd. 3501.Google Scholar

8 Our theory excludes the possibility that the electors make guesses as to how they might best use the machinery of voting, by casting their votes otherwise than in strict accordance with their preference schedules, so as to secure the return of their particular candidate. In any event, if all the electors were to use the machinery with equal intelligence, the same result as that to which we point would be bound to eventuate, for the number using the machinery in favour of the majority candidate is greater than the number using it in favour of any of the others.

9 A proof of this proposition could also be given in terms of the transitive properly, which holds for curves of type (b). See Black, , Journal of Political Economy, 1948, p. 30.Google Scholar

10 We will discuss it further in a subsequent paper, “Some Theoretical Schemes of Proportional Representation,” and describe it here at some length.

11 If some voters show only the order of preference in which they place certain of the candidates and not others, the quota can vary at different stages in the count.

12 He will be elected, although he still will not have obtained a quota of votes, for the number of votes that he holds will remain unchanged while the other candidates are being eliminated.

13 With the data as in Fig. 8, if two places were to be filled, the candidates elected would be a 1 and a 5.

14 Journal of Political Economy, 1948, pp. 32–3.Google Scholar

15 This criterion was accepted by Laplace.

16 This matrix had been constructed, of course, from a set of single-peaked preference curves (see Fig. 8).

17 Cf. the articles referred to.

18 For a description of the principles underlying such processes, see Connor, L. R., Statistics in Theory and Practice (2nd ed.), pp. 23–7.Google Scholar