¹ Kenneth May, ‘A Set of Independent Necessary and Sufficient Conditions for Simple Majority Decision’, Econometrica, 20 (1952), 680–4.

² Iain McLean, ‘Forms of Representation and Voting Systems’, in David Held, ed., Political Theory Today (Palo Alto, Calif.: Stanford University Press, 1991), pp. 172–96, at p. 187.

³ A recent and influential example of this is Arend Lijphart’s Patterns Of Democracy: Government Forms and Performance in Thirty-Six Countries (New Haven, Conn.: Yale University Press, 1999), which contrasts ‘majoritarian’ democracy based on single member district elections, with ‘consensual’ democracy based on proportionality. See also G. Bingham Powell, Elections as Instruments of Democracy: Majoritarian and Proportional Visions (New Haven, Conn.: Yale University Press, 2000).

⁴ Joseph Schumpeter, Capitalism, Socialism, and Democracy (New York: Harper, 1942).

⁵ Anthony McGann, The Logic of Democracy: Reconciling Political Equality, Minority Protection and Deliberation (Ann Arbor: University of Michigan Press, 2006); Thomas Christiano, The Rule of the Many: Fundamental Issues in Democratic Theory (Boulder, Colo.: Westview Press, 1996).

⁶ Arend Lijphart, Electoral Systems and Party Systems: A Study of Twenty-Seven Democracies, 1945–1990 (Oxford: Oxford University Press, 1994), p. 140.

⁷ McLean, ‘Forms of Representation and Voting Systems’, p. 175; Duncan Black, The Theory of Committees and Elections (Cambridge: Cambridge University Press, 1971 [1958]), p. 92.

⁸ Hannah Pitkin, The Concept of Representation (Berkeley, Calif.: University of California Press, 1967), chap. 4; Jonathon Still, ‘Political Equality and Election Systems’, Ethics, 91 (1981), 375–94.

⁹ United States Supreme Court, Vieth et al. v. Jubilerer, President of the Pennsylvania Senate, et al., Docket No. 02-1580 (2004).

¹⁰ Justice Kennedy joined with Justices Scalia, Rehnquist, O’Connor and Thomas in affirming the decision of the District Court, but wrote a concurring opinion arguing that although no standard for determining political gerrymandering was currently available, one may be found, and therefore judicial relief should not be foreclosed.

¹¹ Adrian Blau, ‘Fairness and Electoral Reform’, British Journal of Politics and International Relations, 6 (2005), 165–81, is a recent example of such work.

¹² Nevil Johnson, ‘Electoral Reform: The Risk of Unintended Consequences’, in Report of the Independent Commission on the Voting System Volume II (London: The Stationery Office, Cm 4090-II, 1988); Conservative Party, ‘Conservative Party Submission to the Independent Commission on Voting Systems’, in Report of the Independent Commission on the Voting System Volume II.

¹³ Blau, ‘Fairness and Electoral Reform’.

¹⁴ Charles Dodgson, ‘The Principles of Parliamentary Representation’, in Iain McLean and Arnold Urken, eds, Classics of Social Choice (Ann Arbor: University of Michigan Press, 1997), pp. 300–1.

¹⁵ Hugh Ward, ‘A Contractarian Defence of Ideal Proportional Representation’, Journal of Political Philosophy, 3 (1995), 86–109.

¹⁶ Eliora van der Hout, Harrie de Swart and Annemarie ter Veer, ‘Axioms Characterizing the Plurality Ranking Rule’, Social Choice & Welfare, 27 (2006), 459–75; Eliora van der Hout, Electing Representatives: A Comparison of List PR and FPTP Systems (Enschede: PrintPartners Ipskamp, 2005).

¹⁷ Robert Dahl, Democracy and its Critics (New Haven, Conn.: Yale University Press, 1988); John Stuart Mill, Considerations on Representative Government, in John Stuart Mill: Utilitarianism, On Liberty, Considerations on Representative Government, Everyman edn (London: J.M. Dent, 1993 [1861]).

¹⁸ John Rawls, Political Liberalism (New York: Columbia University Press, 1996 [1993]).

¹⁹ See Brian Barry, Theories of Justice (Berkeley: University of California Press, 1989); Brian Barry, Justice as Impartiality (Oxford: Clarendon Press, 1995).

²⁰ Bernard Williams, ‘The Idea of Equality’, in Hugo Bedau, ed., Justice and Equality (Englewood Cliffs, N.J.: Prentice-Hall, 1971); first published in Peter Laslett and W. G. Runciman, eds, Philosophy, Politics and Society, Series II (Oxford: Basil Blackwell, 1962).

²¹ Bernard Grofman, ‘Fair and Equal Representation’, Ethics, 91 (1981), 477–85, p. 479.

²² Ronald Rogowski, ‘Representation in Political Theory and in Law’, Ethics, 91 (1981), 408–11; Christiano, The Rule of the Many, p. 234.

²³ Unless the electoral districts are of unequal size. However, it is important to note that anonymity is violated even if district size is equal.

²⁴ Michael Latner and Anthony McGann, ‘Geographical Representation under Proportional Representation: The Cases of Israel and the Netherlands’, Electoral Studies, 24 (2005), 709–34.

²⁵ It would be theoretically possible to allocate legislative weight to alternatives in direct proportion to vote share in the manner suggested by Chamberlin and Courant (John Chamberlin and Paul Courant, ‘Representative Deliberations and Representative Decisions: Proportional Representation and the Borda Rule’, American Political Science Review, 77 (1983), 718–33).

²⁶ Strictly speaking, the electoral system in the Netherlands is not simple national list PR, as the country is divided into nineteen electoral districts. However, the system functions as if it were national list, as seats are allocated to districts on a national level. See Elections in the Netherlands (Ministry of Interior and Kingdom Affairs, 2002, http://www.minbzk.nl/bzk2006uk/subjects/constitution_and/publications?ActItmIdt=4799).

²⁷ Michael Balinski and H. Peyton Young, Fair Representation: Meeting the Ideal of One Man, One Vote (Washington, D.C.: The Brookings Institution, 1982/2001), propositions 8.4 and 9.4. Thanks to Professor Iain McLean for pointing out this line of enquiry, as well as its solution.

²⁸ See Rein Taagepera and Matthew Soberg Shugart, Seats and Votes: The Effects and Determinants of Electoral Systems (New Haven, Conn.: Yale University Press, 1989), pp. 112–25.

²⁹ Iain McLean (personal communication) gives the following example. Suppose parties A, B and C receive respectively 48.03 per cent, 1.49 per cent and 0.49 per cent of the vote, while parties D, E, F and G receive 9.51 per cent, 10.51 per cent, 11.51 per cent and 18.46 per cent. Then by the Webster/Sainte-Laguë method, A, B and C receive 48, 1 and 0 seats, while D, E, F and G receive 10, 11, 12 and 18 seats. Thus the coalition of A, B and C gets 50.01 per cent of the vote, but only 49 seats, while the coalition of D, E, F and G gets 49.99 per cent of the vote and 51 seats.

³⁰ Dodgson, ‘The Principles of Parliamentary Representation’, p. 315.

³¹ A quota is defined as the total number of votes cast divided by the number of seats plus one.

³² R. Banker, ‘Equity Considerations in Traditional Full Cost Allocation Practices: An Axiomatic Approach’, in Shane Moriarty, ed., Joint Cost Allocations (Oklahoma City: University of Oklahoma Press, 1981); Barry O’Neill, ‘A Problem of Rights Allocation from the Talmud’, Mathematical Social Sciences, 2 (1982), 345–71; Robert Aumann and Michael Maschler, ‘Game Theoretic Analysis of a Distribution Problem from the Talmud’, Journal of Economic Theory, 36 (1985), 195–213; Hervé Moulin, ‘The Proportional Random Allocation of Indivisible Units’, Social Choice and Welfare, 19 (2002), 381–413.

³³ We can see that Alternative 1 has a very strong bargaining position, as the only way it can be excluded from government is for all the other parties to combine against it.

³⁴ See Matthew Soberg Shugart and Martin Wattenberg, ‘Mixed-Member Electoral Systems: A Definition and Typology’, in Matthew Soberg Shugart and Martin Wattenberg, eds, Mixed-Member Electoral Systems: The Best of Both Worlds? (Oxford: Oxford University Press, 2001), pp. 9–24.

³⁵ In Germany, if a party wins more than its proportional share in the district part of the election, it is allocated fewer seats in the proportional part. Due to the 5 per cent threshold and other institutional features, seat allocations in Germany may vary somewhat from proportionality.

³⁶ Ken Gladdish, Governing from the Center: Politics and Policy-Making in the Netherlands (Dekalb: Northern Illinois University Press, 1991).

³⁷ If there are n candidates, the Borda count award a score of n−1 for each first time an alternative is ranked first by a voter, n−2 for each time it is ranked second, etc. Scores are then summed across all voters. A scoring system gives a score of a for a first place finish, b for a second, c for a third, etc., such that a ≥ b ≥ c, etc.

³⁸ Van der Hout, de Swart and ter Veer, ‘Axioms Characterizing the Plurality Ranking Rule’.

³⁹ Dummett recognizes that fragmentation causes a problem with scoring rules, but treats it as a practical problem. We, however, see it as a fundamental violation of the principle of liberal equality (Michael Dummett, Principles of Electoral Reform (Oxford: Oxford University Press, 1997), pp. 169–73).

⁴⁰ For an introduction to STV, see David Farrell, Electoral Systems: A Comparative Introduction (Basingstoke, Hants.: Palgrave, 2001), chap. 6.

⁴¹ Dodgson, ‘The Principles of Parliamentary Representation’, pp. 313–14; Gideon Doron and Richard Cronick, ‘Single Transferable Vote: An Example of a Perverse Social Choice Function’, American Journal of Political Science, 21 (1977), 303–11.

⁴² Dummett, Principles of Electoral Reform, pp. 98–108.

⁴³ Allard argues based on random preferences that non-monotonicity will occur in only 0.028 per cent of cases. Dummett argues that preferences are not randomly distributed and provides a rough estimate of 2 per cent of cases (see Crispin Allard, ‘Lack of Monotonicity – Revisited’, Representation, 33 (1995), 48–50; Dummett, Principles of Electoral Reform, p. 103).

⁴⁴ It is notable than the ballot for the Australian Senate (where STV is implemented statewide and there can be over 250 candidates on the ballot) allows voters to vote for a party list as an alternative to ranking all candidates (see Farrell, Electoral Systems, pp. 142–3).

⁴⁵ Burt Monroe, ‘Fully Proportional Representation’, American Political Science Review, 89 (1995), 933–4.

⁴⁶ Black, The Theory of Committees and Elections, p. 95; Dummett, Principles of Electoral Reform, p. 169.

⁴⁷ Dodgson, ‘The Principles of Parliamentary Representation’, p. 315.

⁴⁸ Van der Hout, de Swart and ter Veer, ‘Axioms Characterizing the Plurality Ranking Rule’.

⁴⁹ Van der Hout, de Swart and ter Veer, ‘Axioms Characterizing the Plurality Ranking Rule’.