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The Theoretical Limits of Maximum Distortion: Some Analytic Expressions for Electoral Systems

Published online by Cambridge University Press:  27 January 2009

John Loosemore  and
Victor J. Hanby
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Extract

The role of the electoral system in the political process is one which has long concerned political scientists. Traditionally, the emphasis has been on the ‘mechanical’ effects that such systems have or are assumed to have, on such features as the number and strength of parties in a given political system. More often the debate became of a polemical nature, fought on the respective merits of one or other of the major electoral systems. The culmination of all this effort was that, until recently, research in this area of the discipline consisted almost entirely of a body of material, basically unsystematic in scope and essentially descriptive by nature. More contemporary research has concentrated on discovering the precise nature of the empirical relationship between electoral systems and party systems, with some degree of success. Our concern in this paper however, is not to contribute to the knowledge of the ‘mechanical’ effects of electoral systems but, in a related fashion, to examine some possible theoretical properties that may have an equal importance for democratic theory.

Type
Articles
Information
British Journal of Political Science , Volume 1 , Issue 4 , October 1971 , pp. 467 - 477
Copyright
Copyright © Cambridge University Press 1971

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References

1 Though the classic appraisal appears in Hermens, F. A., Democracy or Anarchy (Notre Dame: Univ. of Notre Dame Press, 1938)Google Scholar, the debate has been with us since the days of Bagehot, Walter, The English Constitution (New York: Appleton, 1877)Google Scholar and Mill, John Stuart, Considerations on Representative Government (London: Longmans, 1890).Google Scholar Basically, it devolved into a plurality versus proportional representation contest fought under the guise of a ‘stability’ versus ‘instability’ argument. Later contributors did little to resolve the issue. See especially van den Burgh, G., Unity in Diversity (London: Batsford, 1956);Google ScholarLakeman, Enid and Lambert, James, Voting in Democracies (London: Faber, 1970);Google ScholarDuverger, M., Political Parties (New York: John Wiley, 1954).Google Scholar

2 Until Rae, Douglas W., The Political Consequences of Electoral Laws (New Haven: Yale Univ. Press, 1967)Google Scholar, no really systematic treatment of electoral systems existed. Probably Van Den Burgh, , Unity in Diversity, and Hogan, J., Elections and Representation (Cork: Cork Univ. Press, 1945)Google Scholar were the most comprehensive until then.

3 This is the first of two papers concerned with presenting a series of analytic propositions which are of crucial importance for evaluating and understanding some concepts basic to democratic theory.

4 March, James G., ‘Party Legislative Representation as a Function of Election Results,’ Public Opinion Quarterly, XXI (1957), 521–42.CrossRefGoogle Scholar

5 Rae, The Political Consequences. Rae's work is being followed by a more comprehensive examination, based mainly at the constituency level, which is being undertaken by one of the authors of the present paper.

6 Blondel, Jean, Introduction to Comparative Government (London: Weidenfeld and Nicholson, 1970), p. 191.Google Scholar

7 For a comprehensive description of the main ‘mechanical’ differences between these systems, see Mackenzie, W. J. M., Free Elections (London: Allen and Unwin, 1958).Google Scholar

8 Regardless of how the other parties split their vote, with one party getting all the seats with no votes there can be no worse distortion.

9 We ignore the possibility of unopposed seats of course. Our analysis is only concerned with normal cases where n 1.

10 For a discussion of this concept see Rae, D., Hanby, Victor J., Loosemore, J., ‘Thresholds of Representation and Thresholds of Exclusion: An Analytic Note on Electoral Systems’, Comparative Political Studies, III (1971), 479–89.CrossRefGoogle Scholar Basically the argument is that in an electoral district of size m, with n parties competing, the threshold of representation is the smallest value a party's vote v can take yet still permit the party to win s of m seats (where s runs from 1 to m). A series of analytic expressions are presented for these thresholds by the following formula:

For plurality formula, of course, 1. It is to these above thresholds that we refer in this paper.

11 Blondel, , Introduction to Comparative Government, p. 192.Google Scholar

12 Rae, , The Political Consequences of Electoral Laws, p. 123.Google Scholar

13 Though these assumptions are not totally realistic, it would be rather complicated and not really worth the effort to drop them.

14 See Rae, , The Political Consequences of Electoral Laws p. 84Google Scholar for a discussion of his distortion formula and its characteristics.

15 Rae, , The Political Consequences of Electoral Laws, p. 117.Google Scholar

16 Rae's formula is actually E = 12/m but as he uses percentages rather than proportions we have reduced his measure to coincide with our index.

17 All three of the assumptions are actually false, but their combined effect is minimal compared with the difficulties that would arise if we dropped them yet proceeded with this present analysis.