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Comparing Proportional Representation Electoral Systems: Quotas, Thresholds, Paradoxes and Majorities

Published online by Cambridge University Press:  27 January 2009

Michael Gallagher
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Extract

The relationship between electoral systems can be examined on a number of dimensions. Seat allocation methods are conveniently divided into two groups: those based on largest remainders and those based on highest averages. The single transferable vote has its own distinct characteristics. Focusing on certain elements – the quota, thresholds, paradoxes and the conditions under which a majority of seats can be won – enables comparisons to be drawn between seat allocation methods. Certain seat allocation methods conventionally seen as variants of proportional representation (PR) cannot be regarded as such. PR methods can be rank ordered according to whether, when complete proportionality is not attainable, they display electoral bias towards larger or smaller parties. However, a definitive ordering is elusive, since some methods that are generally more favourable to larger parties can in some circumstances set lower thresholds of representation than methods generally favourable to smaller parties.

Type
Articles
Information
British Journal of Political Science , Volume 22 , Issue 4 , October 1992 , pp. 469 - 496
Copyright
Copyright © Cambridge University Press 1992

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References

1 Amoroso, Mario, ‘Italy’, in Hand, Geoffrey, Georgel, Jacques and Sasse, Christoph, eds, European Electoral Systems Handbook (London: Butterworths, 1979), pp. 140–69, at p. 158.Google Scholar

2 The terms ‘smaller’ and ‘larger’ are entirely relative. For a more formal demonstration that larger parties benefit from a low quota, see Appendix.

3 For example, Rae, Douglas, The Political Consequences of Electoral Laws, revised edn (New Haven and London: Yale University Press, 1971), p. 34Google Scholar, says that, compared with LR-Hare, LR-Imperiali lowers the price of the first seat, thereby ‘helping weak parties’.

4 For an English translation of his article, see Lijphart, Arend and Gibberd, Robert W., ‘Thresholds and Payoffs in List Systems of Proportional Representation’, European Journal of Political Research, 5 (1977), 219–44, pp. 241–2.CrossRefGoogle Scholar

5 Carstairs, Andrew McLaren, A Short History of the Electoral Systems in Western Europe (London: George Allen and Unwin, 1980), pp. 26–9, 42.Google Scholar

6 Balinski, Michel L. and Young, H. Peyton, Fair Representation: Meeting the Ideal of One Man, One Vote (New Haven and London: Yale University Press, 1982), pp. 26–8.Google Scholar

7 For equal proportions, see Huntington, Edward V., ‘A New Method of Apportionment of Representatives’, Quarterly Publication of the American Statistical Association, 17 (1921), 859–70.CrossRefGoogle Scholar For the circumstances in which the Danish method is used, see Johansen, Lars Norby, ‘Denmark’Google Scholar, in Hand, , Georgel, and Sasse, , European Electoral Systems Handbook, pp. 29–57, p. 46.Google Scholar Pure Sainte-Laguë was used at the first four post-war Danish elections – I owe this information to Arend Lijphart. The only reference of which I am aware to the employment of Imperiali highest averages is to be found in Van Den Bergh, G., Unity in Diversity: A Systematic Critical Analysis of all Electoral Systems (London: B. T. Batsford, 1956), pp. 25, 96Google Scholar, where it is stated that the method is used for municipal elections in Belgium.

8 Carstairs, , Short History of Electoral Systems, p. 29.Google Scholar

9 Balinski, and Young, , Fair Representation, p. 104.Google Scholar

10 So do LR-Hare and LR-Droop, but LR-Imperiali does not function effectively as its low quota of 3,704 votes would ‘allocate’ twenty-six seats initially. It too, therefore, cannot be classified as a PR method. We are excluding STV from this part of the discussion as it embodies a more sophisticated conception of ‘perfect proportionality’ by considering voters' lower preferences.

11 Gallagher, Michael, ‘Proportionality, Disproportionality and Electoral Systems’, Electoral Studies, 10 (1991), 33–51.CrossRefGoogle Scholar

12 Van Den Bergh, , Unity in Diversity, p. 96.Google Scholar

13 Lijphart, Arend, ‘Degrees of Proportionality of Proportional Representation Formulas’, in Grofman, Bernard and Lijphart, Arend, eds, Electoral Laws and Their Political Consequences (New York: Agathon, 1986), pp. 170–9, at pp. 172–5Google Scholar; Balinski, and Young, , Fair Representation.Google Scholar

14 Carstairs, , Short History of Electoral Systems, p. 19.Google Scholar

15 The method is illustrated by Van Den Bergh, , Unity in Diversity, pp. 68–72.Google Scholar

16 For a full account of how STV operates in Ireland, see Hand, Geoffrey, ‘Ireland’Google Scholar, in Hand, , Georgel, and Sasse, , European Electoral Systems Handbook, pp. 121–39.Google Scholar

17 Lakeman, Enid, How Democracies Vote, 4th edn (London: Faber and Faber, 1974), p. 147.Google Scholar

18 Newland, Robert A., Comparative Electoral Systems (London: The Arthur McDougall Fund, 1982), p. 64.Google Scholar A similar point is made by Hill, I. D., ‘Some Aspects of Elections: To Fill One Seat or Many’, Journal of the Royal Statistical Society A, 151 (1988), 243–75, at pp. 253–4, 273CrossRefGoogle Scholar; and by Rokkan, Stein, ‘Electoral Systems’, in Sills, David L., ed., International Encyclopedia of the Social Sciences, Vol. 5 (New York: Crowell Collier and Macmillan, 1968), pp. 6–21, at p. 15.Google Scholar

19 Quoted in Humphreys, John H., Proportional Representation: A Study in Methods of Election (London: Methuen, 1911), p. 138.Google Scholar See also Lakeman, , How Democracies Vote, p. 148Google Scholar; Newland, , Comparative Electoral Systems, p. 64.Google Scholar

20 Vote management at this election may have affected the outcome in eleven of the forty-one constituencies. It is discussed in Gallagher, Michael, ‘The Election Results and the New Dáil’, in Gallagher, Michael and Sinnott, Richard, eds, How Ireland Voted 1989 (Galway: Centre for the Study of Irish Elections, 1990), pp. 68–93, at pp. 81–3.Google Scholar

21 This philosophy is outlined in Huntington, ‘A New Method of Apportionment’.

22 Lijphart, , ‘Degrees of Proportionality’, p. 176.Google Scholar

23 Lijphart, and Gibberd, , ‘Thresholds and Payoffs’.Google Scholar See also Rokkan, , ‘Electoral Systems’, pp. 13–15Google Scholar; Rae, , Political Consequences of Electoral Laws, pp. 149–70Google Scholar; Rae, Douglas W., Hanby, Victor and Loosemore, John, ‘Thresholds of Representation and Thresholds of Exclusion’, Comparative Political Studies, 3 (1971), 479–88CrossRefGoogle Scholar; Laakso, Markku, ‘The Maximum Distortion and the Problem of the First Divisor of Different PR Systems’, Scandinavian Political Studies, 2 (1979), 161–9CrossRefGoogle Scholar; Taagepera, Rein and Shugart, Matthew S., Seats and Votes: The Effects and Determinants of Electoral Systems (New Haven and London: Yale University Press, 1989), especially pp. 274–7.Google Scholar

24 Mackie, and Rose, , International Almanac, p. 510.Google Scholar

25 Lijphart, and Gibberd, , ‘Thresholds and Payoffs’, p. 229.Google Scholar

26 Both are demonstrated by Lijphart, and Gibberd, , ‘Thresholds and Payoffs’, pp. 228–9.Google Scholar

27 Balinski, and Young, , Fair Representation, p. 39.Google Scholar Another example of the Alabama paradox is given by Carstairs, , Short History of Electoral Systems, p. 25.Google Scholar In the case he illustrates, a hundred seats would be awarded on a 45–44–11 basis under both LR-Hare and Sainte-Laguë, while if the number of seats is increased to 101, Sainte-Laguë produces a 45–45–11 outcome whereas under LR-Hare the smallest party loses a seat and the outcome is 46–45–10.

28 Newland, , Comparative Electoral Systems, pp. 40–2.Google Scholar

29 Balinski, and Young, , Fair Representation, p. 69.Google Scholar

30 Brams, Steven J., Paradoxes in Politics (New York: Free Press, 1976), p. 157.Google Scholar

31 Balinski, Michel L. and Young, H. Peyton, ‘The Quota Method of Apportionment’, American Mathematical Monthly, 84 (1975), 450–5CrossRefGoogle Scholar; Balinski, and Young, , Fair Representation, pp. 79–81 and 129–34.Google Scholar

32 This is demonstrated by Balinski, and Young, , Fair Representation, p. 90.Google Scholar

33 For example, Van Den Bergh, , Unity in Diversity, p. 25.Google Scholar The other side of the argument is that a large party can often win a majority of seats with a minority of votes under d'Hondt, LR-Droop and STV if the other votes are scattered among a number of opponents. This may happen under Sainte-Laguë and LR-Hare, though it would require a greater degree of fragmentation of the other votes, and cannot happen under Adams.

34 Gallagher, , ‘Proportionality, Disproportionality and Electoral Systems’.Google Scholar