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