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Blocks of Voters and the Cube ‘Law’

Published online by Cambridge University Press:  27 January 2009

Graham Upton
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Extract

The so-called cube ‘law’ has become ‘part of the political folklore of Great Britain’. Indeed it seems also to have passed into the general folklore of political science, having been applied to electoral systems having single-member constituencies contested by two major parties in the United States, New Zealand, Canada, Australia and South Africa.

Type
Notes and Comments
Information
British Journal of Political Science , Volume 15 , Issue 3 , July 1985 , pp. 388 - 398
Copyright
Copyright © Cambridge University Press 1985

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References

1 Tufte, Edward R., ‘The Relationship between Seats and Votes in Two-Party Systems’. American Political Science Review, LXVII (1973). 540–54.CrossRefGoogle Scholar

2 March, James G., ‘Party Legislative Representation as a Function of Election Results’. Public Opinion Quarterly, XXI (1957–1958), 521–42.CrossRefGoogle Scholar Several other authors have included United States elections in a comparison of the behaviour of electoral systems in various countries. See. for example. Tufte, , ‘The Relationship between Seats and Votes’Google Scholar: Linehan, W. J. and Schrodt, P. A., ‘A New Test of the Cube Law’. Political Methodology. 4 (1977). 353–67Google Scholar; Schrodt, P. A., ‘A Statistical Study of the Cube Law in Five Electoral Systems’. Political Methodology. 7 (1981). 31–53.Google Scholar

3 Brookes, Ralph H., ‘Legislative Representation and Party Vote in New Zealand’. Public Opinion Quarterly, XXIII (1959). 288–91.CrossRefGoogle Scholar See also Linehan and Schrodt: Schrodt: and Tufte. as cited in footnotes 1 and 2.

4 Quaker, Terence H., ‘Seats and Votes: An Application of the Cube Law to the Canadian Electoral System’, Canadian Journal of Political Science. 1 (1968), 336–44.Google Scholar See also Casstevens, T. W. and Morris, W. D., ‘The Cube Law and the Decomposed System’. Canadian Journal of Political Science, V (1972), 521–31CrossRefGoogle Scholar; Linehan, and Schrodt, , ‘A New Test of the Cube Law’.Google Scholar

5 Schrodt, , ‘A Statistical Study of the Cube Law’.Google Scholar

6 Theil, H., ‘The Desired Political Entropy’. American Political Science Review, LXIII (1969). 521–5CrossRefGoogle Scholar and Sankoff, D. and Mellos, K., ‘The Swing Ratio and Game Theory’. American Political Science Review, LXVI (1972), 551–4CrossRefGoogle Scholar, provide theoretical justifications for a relation of this type, although their arguments were dismissed by Tufte as being unconvincing.

7 Stanton, R. G., ‘A Result of MacMahon on Electoral Predictions’, Annals of Discrete Mathematics, VIII (1980). 163–7.CrossRefGoogle Scholar

8 Examples are to be found in Qualter, p. 339; Theil, p. 521; and Casstevens, and Morris, (p. 522)Google Scholar, cited in footnotes 4 and 5, and in Taagepera, Rein, ‘Seats and Votes: A Generalization of the Cube Law of Elections’. Social Science Research, 11 (1973), 257–75.CrossRefGoogle Scholar

9 Laakso, Thus Markku, ‘Should a Two-and-a-Half Law Replace the Cube Law in British Elections?’, British Journal of Political Science, IX (1979). 355–62CrossRefGoogle Scholar, compares the cases m = 2, 2·5 and 3 and finds 2·5 to be the preferable value for British elections.

10 Kendall, M. G. and Stuart, A., ‘The Law of Cubic Proportions in Electoral Results’, British Journal of Sociology, 1 (1950), 183–97.CrossRefGoogle Scholar

11 Using the normal approximation to the binomial, with mean np and variance np (1- p).

12 Tufte, , ‘The Relationship Between Seats and Votes’, p. 545.Google Scholar

13 That these conditions appear to no longer hold true is demonstrated in Table 2 of Curtice, John and Steed, Michael, ‘Electoral Choice and the Production of Government: The Changing Operation of the Electoral System in the United Kingdom since 1955’, British Journal of Political Science, XII (1982), 249–98.CrossRefGoogle Scholar However, the position for recent years has been clouded by the intervention of third parties and Gudgin, G. and Taylor, P. J., Seats, Votes and the Spatial Organisation of Elections (London: Pion, 1979).Google Scholar Chap. 2, have shown that the required value is a result of the arbitrary placing of constituency boundaries upon a class-based voting map. See also Gudgin, G. and Taylor, P. J., ‘The Decomposition of Electoral Bias in a Plurality Election’, British Journal of Political Science, X (1980), 515–21.CrossRefGoogle Scholar

14 The effect of social milieu on geographical variations in p has recently been taken up by geographers. See, for example, Johnston, R. J., Hay, A. M. and Rumley, D., ‘On Testing for Structural Effects in Electoral Geography, Using Entropy-maximising Methods to Estimate Voting Patterns’, Environment and Planning A, XVI (1984), 233–40.CrossRefGoogle Scholar

15 A fascinating series of examples of the results of variations in p in an artificial situation are provided by Wildgen, J. K. and Engstrom, R. L., ‘Spatial Distribution of Partisan Support and the Seats/Votes Relationship’, Legislative Studies Quarterly, V (1980), 423–35CrossRefGoogle Scholar

16 Penrose, L. S., On the Objective Study of Crowd Behaviour (London: Lewis, 1952).Google Scholar

17 Thus we could have eleven blocks of size 3,000, one block of size 4,000 and one of size 2,000 without altering any of the resulting probabilities. In this case a seven to six block majority might appear as a narrow 20,000 to 19,000 vote majority or as a thumping 24,000 to 15,000 vote majority.

18 Gudgin, and Taylor, , Seats, Votes and the Spatial Organisation of Elections, Chap. 4.Google Scholar

19 The correspondence will not be exact because real-life housing-blocks will not be equi-sized. Nevertheless, the value of n does give us an order of magnitude to work with.

20 The results were taken from The Times newspaper, which gave the aggregate numbers of votes as Conservative 550, 463 (31 per cent); Labour, 573, 207 (32 per cent); Alliance, 445, 645 (25 per cent); SNP, 228, 584 (13 per cent).

21 My attention was brought to Penrose's model by the work of Stanton, ‘A Result of McMahon on Electoral Predictions’, who points out the equivalence of the cube law and Penrose's choice of fourteen blocks.

22 Thus Table 2 refers to seven values of m, but considers up to twenty-eight values of n.

23 Thus Oualter. p. 340. reports that ‘the results were at first far from encouraging’ until he analysed each type of constituency separately. Note that Kendall and Stuart were comparatively lax. considering all seats in which one or other of the major parties was successful, without regard to the possible intervention of other parties.

24 And why stop with the United Kingdom? How about the Common Market or the world …?

25 Curtice and Steed, sections 3 and 4.

26 Curtice and Steed, section 5.

27 Note that this does not mean that there are only twelve, say, identifiable groups of people in the country! It means that within any given area people can probably be subdivided into about twelve different groups – all of which might be different from all groups in other areas.