¹ Anthony Downs, An Economic Theory of Democracy (New York: Harper & Row, 1957).

² For a discussion of the assumptions of the basic Downsian model and the vast literature that concerns itself with the two-party case, see the excellent review by Bernard Grofman, ‘Downs and Two-Party Convergence’, Annual Review of Political Science, 7 (2004), 25–46.

³ Downs, An Economic Theory of Democracy, p. 115.

⁴ We use data published in Ian Budge, Hans-Dieter Klingemann, Andrea Volkens, Judith Bara and Eric Tanenbaum, Mapping Policy Preferences: Estimates for Parties, Electors, and Governments 1945–1998 (Oxford: Oxford University Press, 2001).

⁵ For an excellent discussion of the effects of party entry on party positioning, see Thomas R. Palfrey, ‘Spatial Equilibrium with Entry’, Review of Economic Studies, 51 (1984), 139–56. For a discussion of the effects of third parties, see James Adams and Samuel Merrill III, ‘Why Small, Centrist Third Parties Motivate Policy Divergence by Major Parties’, American Political Science Review, 100 (2006), 403–17. John Aldrich, ‘A Downsian Spatial Model with Party Activists’, American Political Science Review, 77 (1983), 974–90, was the first to consider the effect of activists on the basic Downsian model. Tim Groseclose, ‘A Model of Candidate Location When One Candidate Has a Valence Advantage’, American Journal of Political Science, 45 (2001), 862–86, and Enriqueta Aragones and Thomas R. Palfrey, ‘Mixed Equilibrium in a Downsian Model with a Favored Candidate’, Journal of Economic Theory, 103 (2002), 131–61, have analysed the effect of valence on party positioning in the basic, two-party spatial model.

⁶ For considerations of the effect of uncertainty on party positioning in multi-party competition, see Tse-Min Lin, James M. Enelow and Han Dorussen, ‘Equilibrium in Multicandidate Probabilistic Spatial Voting’, Public Choice, 98 (1999), 59–83. For the most recent and comprehensive analysis of the effect of valence in multi-party competition, see Norman Schofield and Itai Sened, Multiparty Democracy: Elections and Legislative Politics (Cambridge: Cambridge University Press, 2006). See also Norman Schofield and Itai Sened, ‘Multiparty Competition in Israel, 1988–96’, British Journal of Political Science, 35 (2005), 635–63; Norman Schofield, ‘The Mean Voter Theorem: Necessary and Sufficient Conditions for Convergent Equilibrium’, Review of Economic Studies 74 (2007), 965–80; and James Adams and Samuel Merrill III, ‘Policy-Seeking Parties in a Parliamentary Democracy with Proportional Representation: A Valence-Uncertainty Model’, British Journal of Political Science, 39 (2009), 539–58. To understand how voters’ behavioural characteristics affect party positioning, see James Adams, Samuel Merrill III and Bernard Grofman, A Unified Theory of Party Competition: A Cross-National Analysis Integrating Spatial and Behavioral Factors (Cambridge: Cambridge University Press, 2005).

⁷ For a summary of the historical development of spatial theory, see Peter Ordeshook, ‘The Spatial Analysis of Elections and Committees: Four Decades of Research’, in Dennis C. Mueller, ed., Perspectives on Public Choice: A Handbook (Cambridge: Cambridge University Press, 1990); and David Austin-Smith and Jeffrey Banks, Positive Political Theory II: Strategy and Structure (Ann Arbor: University of Michigan Press, 2005). We also recommend the first chapter of Adams, Merrill and Grofman, A Unified Theory of Party Competition, which provides a discussion of some of the most important contributions to the spatial model of party competition.

⁸ Gary W. Cox, ‘Centripetal and Centrifugal Incentives in Electoral Systems’, American Journal of Political Science, 34 (1990), 903–35.

⁹ Downs suggests that the presence of two ideologically indistinguishable parties located at or near the centre of the policy space is more likely to characterize the party system under single member district plurality electoral rules, while the presence of several parties with distinct and divergent policy positions is more likely under proportional electoral rules.

¹⁰ Cox, ‘Centripetal and Centrifugal Incentives in Electoral Systems’, p. 912–13.

¹¹ Cox, ‘Centripetal and Centrifugal Incentives in Electoral Systems’, p. 921.

¹² Cox, ‘Centripetal and Centrifugal Incentives in Electoral Systems’, p. 922.

¹³ Anthony J. McGann, ‘The Advantages of Ideological Cohesion: A Model of Constituency Representation and Electoral Competition in Multi-party Democracies’, Journal of Theoretical Politics, 14 (2002), 37–70; and Samuel Merrill III and James Adams, ‘Centrifugal Incentives in Multi-Candidate Elections’, Journal of Theoretical Politics, 14 (2002), 275–300.

¹⁴ Jay K. Dow, ‘A Comparative Spatial Analysis of Majoritarian and Proportional Elections’, Electoral Studies, 20 (2001), 109–25.

¹⁵ A number of studies have found empirical evidence for the relationship between number of political parties and proportional electoral systems, including Arend Lijphart, Electoral Systems and Party Systems: A Study of Twenty-Seven Democracies, 1945–1990 (Oxford: Oxford University Press, 1994); Peter C. Ordeshook and Olga V. Shvetsova, ‘Ethnic Heterogeneity, District Magnitude, and the Number of Parties’, American Journal of Political Science, 38 (1994), 100–23; Octavio Amorim Neto and Gary W. Cox, ‘Electoral Institutions, Cleavage Structures, and the Number of Parties’, American Journal of Political Science, 41 (1997), 149–74; and Gary W. Cox, Making Votes Count (Cambridge: Cambridge University Press, 1997).

¹⁶ Lawrence Ezrow, ‘Are Moderate Parties Rewarded in Multiparty Systems? A Pooled Analysis of Western European Elections, 1984–98’, European Journal of Political Research, 44 (2005), 881–98.

¹⁷ Giovanni Sartori, Parties and Party Systems: A Framework for Analysis (Cambridge: Cambridge University Press, 1976); Kaare Strøm, Minority Government and Majority Rule (Cambridge: Cambridge University Press, 1990).

¹⁸ Markku Laakso and Rein Taagepera, ‘ “Effective” Number of Parties: A Measure with Application to West Europe’, Comparative Political Studies, 12 (1979), 3–27.

¹⁹ Sartori, Parties and Party Systems.

²⁰ Strøm, Minority Government and Majority Rule.

²¹ Outstanding empirical studies of cabinet formation include Kaare Strøm, Ian Budge and Michael J. Laver, ‘Constraints on Cabinet Formation in Parliamentary Democracies,’ American Journal of Political Science, 38 (1994), 303–5, and Lanny Martin and Randolph Stevenson, ‘Government Formation in Parliamentary Democracies’, American Journal of Political Science, 45 (2001), 33–50. Major contributions to formal work on cabinet formation include Norman Schofield, ‘Political Competition and Multiparty Coalition Governments’, European Journal of Political Research, 23 (1993), 1–33; Michael J. Laver and Kenneth Shepsle, Making and Breaking Governments: Cabinets and Legislatures in Parliamentary Democracies (Cambridge: Cambridge University Press, 1996); and Schofield and Sened, Multiparty Democracy.

²² Martin and Stevenson, ‘Government Formation in Parliamentary Democracies’, p. 39.

²³ Schofield and Sened, Multiparty Democracy.

²⁴ Laakso and Taagepera, ‘Effective Number of Parties’, pp. 1–9.

²⁵ In Table 1 and in our comparison of means, we use the incremental version of the effective number of parties; however, in the regression analyses reported towards the end of the article, we use the effective number of parties in its original form.

²⁶ Schofield, ‘Political Competition and Multiparty Coalition Governments’. See also the paper by Lin, Enelow and Dorussen, ‘Equilibrium in Multicandidate Probabilistic Spatial Voting’, in which they derive equilibrium predictions given a multidimensional issue space and probabilistic voting. And, in their recent book, Schofield and Sened, Multiparty Democracy, provide formal and empirical solutions to multidimensional bargaining among many political parties.

²⁷ Kenneth Benoit and Michael J. Laver, Party Policy in Modern Democracies (London: Routledge, 2006).

²⁸ Michael J. Laver and W. Ben Hunt, Policy and Party Competition (New York: Routledge, 1992), p. 49.

²⁹ Alan Ware, Political Parties and Party Systems (Oxford: Oxford University Press, 1996). In their discussion of policy dimensions, Budge et al., Mapping Policy Preferences, consider only the economic and social policy dimensions.

³⁰ Benoit and Laver, Party Policy in Modern Democracies.

³¹ These numbers are based on our count measure of parties in the party system. According to the effective number of parties in parliament measure, New Zealand’s party system ranges from 1.7 to 3.8, Germany’s from 2.2 to 4.0, and Denmark’s from 3.5 to 6.9.

³² We rely on the definitions of party families offered by KlausVon Beyme, Political Parties in Western Democracies (Aldershot, Surrey: Gower, 1985).

³³ Ian Budge et al., Mapping Policy Preferences.

³⁴ For examples of productive research that use the Comparative Manifesto Project data to estimate relative policy positions of political parties, see work by Ian Budge, David Robertson and Derek Hearl, Ideology, Strategy, and Party Change: Spatial Analyses of Post-War Electoral Programmes in 19 Democracies (Cambridge: Cambridge University Press, 1987); Schofield, ‘Political Competition and Multiparty Coalition Governments’; and James Adams, Michael Clark, Lawrence Ezrow and Garrett Glasgow, ‘Understanding Change and Stability in Party Ideologies: Do Parties Respond to Public Opinion or to Past Election Results?’ British Journal of Political Science, 34 (2004), 589–610.

³⁵ Ian Budge et al., Mapping Policy Preferences, chap. 6.

³⁶ Michael J. Laver and John Garry, ‘Estimating Policy Positions from Political Texts’, American Journal of Political Science, 44 (2000), 619–34, p. 620.

³⁷ The original Comparative Manifesto Project coding categories that are included in our economic policy dimension are: 303 (government and administrative efficiency), 401 (free enterprise), 402 (incentives), 403 (market regulation), 404 (economic planning), 407 (protectionism, negative), 412 (controlled economy), 413 (nationalization), 414 (economic orthodoxy), 504 (welfare state expansion), 505 (welfare state limitation), 701 (labor groups, positive). The coding categories included in our social policy dimension are: 304 (political corruption), 305 (political authority), 501 (environmental protection), 503 (social justice), 601 (national way of life, positive), 602 (national way of life, negative), 603 (traditional morality, positive), 604 (traditional morality, negative), 605 (law and order), and 606 (social harmony).

³⁸ See Laver and Garry, ‘Estimating Policy Positions from Political Texts’, and S. Bartolini and Peter Mair, Identity, Competition and Electoral Availability: The Stabilization of European Electorates 1885–1985 (Cambridge: Cambridge University Press, 1990). The validity of our measure should be similar to the Bartolini and Mair estimates presented in Table 6.2 of Budge et al., Mapping Policy Preferences, chap. 6.

³⁹ Laver and Garry, ‘Estimating Policy Positions from Political Texts’. Our social policy dimension is correlated at the level of 0.66 with that of Laver and Garry.

⁴⁰ Matthew J. Gabel and John D. Huber, ‘Putting Parties in Their Place: Inferring Party Left–Right Ideological Positions from Party Manifestos Data’, American Journal of Political Science, 44 (2000), 94–103.

⁴¹ Anthony Downs, An Economic Theory of Democracy, p. 128.

⁴² Anthony Downs, An Economic Theory of Democracy, p. 126–7.

⁴³ Cox, ‘Centripetal and Centrifugal Incentives in Electoral Systems’, p. 904.

⁴⁴ We have adjusted the x-axis scale for Iceland’s five-party system to ensure that the policy space depicted in each graph is visually comparable. In this case, the centre of the party system is shifted rightward compared to the other systems.

⁴⁵ Nathaniel Beck and Jonathan N. Katz, ‘What to Do (and Not to Do) with Times-Series Cross-Section Data in Comparative Politics’, American Political Science Review, 89 (1995), 634–47.

⁴⁶ We based our measure of median district magnitude on number presented in Cox, Making Votes Count, Table 3.3.