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² This does not mean ‘anything goes’ theoretically. Theoretical foundations must be defensible for intended purposes, which generally is to elucidate particular problems or questions rather than to explain the observed party behaviour in particular polities or elections.

³ Schofield, Norman, Spatial Models of Politics (New York: Routledge, 2008), p. xiiiGoogle Scholar; Ezrow, Lawrence, ‘Parties’ Policy Programmes and the Dog That Didn’t Bark: No Evidence That Proportional Systems Promote Extreme Party Positioning’, British Journal of Political Science, 38 (2008), 479–497CrossRefGoogle Scholar.

⁴ The Comparative Study of Electoral Systems (www.cses.org), CSES Module 1, Full Release [dataset] (Ann Arbor: University of Michigan, Center for Political Studies, 4 August 2003); The Comparative Study of Electoral Systems. CSES Module 2, Full Release [dataset] (Ann Arbor: University of Michigan, Center for Political Studies, 27 June 2007).

⁵ For a detailed discussion of the CSES data, as well as the advantages and shortcomings of these data for particular research programmes, see Norris, Pippa, Electoral Engineering: Voting Rules and Political Behavior (Cambridge: Cambridge University Press, 2004)CrossRefGoogle Scholar.

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⁷ By definition, a proportional electoral formula must operate on multi-member districts up to and including a single nationwide district. Majoritarian formulas, by contrast, may operate on multi-member districts; however, the ‘bloc vote’ in which voters receive and must cast ballots equal to the number of seats elected in a given district is, to the best of my knowledge, the only true majoritarian electoral system that uses multi-member districts. The archetypal majoritarian system is the first-past-the-post, single-member district system.

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¹² Schofield and Sened, Multiparty Democracy.

¹³ Dow, Jay K., ‘A Comparative Spatial Analysis of Majoritarian and Proportional Elections’, Electoral Studies, 20 (2001), 109–125CrossRefGoogle Scholar; Ezrow, ‘Parties’ Policy Programmes and the Dog That Didn’t Bark’.

¹⁴ Dow, ‘A Comparative Spatial Analysis of Majoritarian and Proportional Elections’.

¹⁵ Schofield and Sened, Multiparty Democracy.

¹⁶ Schofield and Sened, Multiparty Democracy, pp. 76–7, pp. 133–6Google Scholar.

¹⁷ Ezrow, ‘Parties’ Policy Programmes and the Dog That Didn’t Bark’.

¹⁸ Karlheinz Reif and Anna Melich, Euro-Barometer 31A: European Elections, 1989: Post-Election Survey, June–July 1989 [computer file], conducted by Faits et Opinions, Paris, ICPSR edn (Ann Arbor, Mich.: Inter-university Consortium for Political and Social Research [producer and distributor], 1993), doi:10.3886/ICPSR09360.

¹⁹ Such comparability problems are intrinsic to analyses that obtain party placements from manifesto analysis or expert placements when these are superimposed on voter distributions obtained from survey responses, such as those obtained from the Eurobarometer or the American National Election Studies project. Analysis not reported here studies the relationship between electoral proportionality and party system compactness when party locations are determined by the expert placements recorded in the CSES data. Contrary to the results reported later in this article, these reveal no relationship between party-system compactness and any electoral-system characteristic.

²⁰ Budge, Ian and McDonald, Michael D., ‘Choices Parties Define: Policy Alternatives in Representative Elections, 17 Countries 1945–1998’, Party Politics, 12 (2006), 451–466CrossRefGoogle Scholar; Pierce, Roy, ‘Mass–Elite Issue Linkages and the Responsible Party Model of Representation’, in Warren Miller et al., Policy Representation in Western Democracies (New York: Oxford University Press, 1999)Google Scholar; and MacDonald, S., Listhaug, O. and Rabinowitz, G., ‘Issues and Party Support in Multiparty Systems’, American Political Science Review, 85 (1991), 1107–1132CrossRefGoogle Scholar.

²¹ Hellwig, Timothy, ‘Explaining the Salience of Left–Right Ideology in Postindustrial Democracies: The Role of Structural Economic Change’, European Journal of Political Research, 47 (2008), 687–709CrossRefGoogle Scholar; Kollman, Ken, Miller, John H. and Page, Scott E., ‘Adaptive Parties in Spatial Elections’, American Political Science Review, 86 (1992), 929–937CrossRefGoogle Scholar; and Kollman, Ken, Miller, John H. and Page, Scott E., ‘Political Parties and Electoral Landscapes’, British Journal of Political Science, 28 (1998), 139–158CrossRefGoogle Scholar.

²² Alvarez, R. M. and Nagler, J., ‘Party System Compactness: Consequences and Measures’, Political Analysis, 12 (2004), 46–62CrossRefGoogle Scholar; and Ezrow, ‘Parties’ Policy Programmes and the Dog That Didn’t Bark’.

²³ This is true so long as the distribution of voters is uni-modal and approximately symmetric around some central value.

²⁴ The United States and Iceland provide examples that illustrate this point. In the 2004 US election, the mean perceived Democratic and Republican positions on the CSES right-to-left scale are 4.09 and 6.69, respectively. The mean voter ideal point and standard deviation are 5.83 and 2.33, producing UPE_{k 05} = 0.558. In the 2003 Icelandic election, five parties received vote shares of at least 5 per cent, including the Liberal party, which had a mean perceived location of 5.49, nearly adjacent to the mean citizen left–right placement of 5.41. Combined with the remaining parties’ perceived locations of 8.31 (Independence party), 6.03 (Progressive party), 4.08 (Social Alliance) and 2.25 (Left–Green Movement), and a standard deviation of citizen ideal points of 2.22, UPE_{k 05} for Iceland equals 0.728.

²⁵ I define and calculate all independent variables including disproportionality, threshold, district magnitude and the effective number of political parties in terms of the lower chamber in the case of bicameral legislatures. I do so because in most electoral democracies the lower chamber is both the proximate and larger legislative body, it is the principal legislative body in terms of defined powers, and the party system is generally designed to compete for seats in this chamber.

²⁶ Gallagher, Michael, ‘Proportionality, Disproportionality and Electoral Systems’, Electoral Studies, 10 (1991), 33–51CrossRefGoogle Scholar; and Lijphart, Arend, Electoral Systems and Party Systems (New York: Oxford University Press, 1994)CrossRefGoogle Scholar.

²⁷ This transformation simply reverses the scale without changing the relative discrepancies in vote–seat shares across nations. This makes interpreting subsequent graphs and regressions more intuitive because more proportional electoral systems return higher values of proportionality than less proportional systems.

²⁸ For example, nationally, Democratic House candidates in 2006 received 51.97 per cent of the vote, while Republican House candidates received 44.06 per cent of the vote. These vote shares returned 53.6 per cent of the House seats to the Democratic party and 46.4 per cent of the House seats to the Republican party in the first session of the 110th Congress. This reflects proportionality being generally discussed as a national-level characteristic, and by this definition, the United States is quite proportional. The proportionality of the US case is also observed by Norris, , Electoral Engineering, p. 90Google Scholar. However, as Powell and Vanberg point out, the aggregation of vote and seat shares over, in this case, 435 districts masks that for representation purposes the aggregation averages over a very large number of highly disproportional outcomes at the district level. See Bingham Powell, G. Jr and Vanberg, Georg S., ‘Election Laws, Disproportionality and Median Correspondence: Implications for Two Versions of Democracy’, British Journal of Political Science, 30 (2000), 383–411CrossRefGoogle Scholar.

²⁹ See, for example, Norris, , Electoral Engineering, pp. 88–93Google Scholar. She reports vote–seat shares as a function of electoral system type.

³⁰ Exclusive of the sixteen ‘overhang’ Bundestag seats.

³¹ Taagepera, Rein and Soberg Shugart, Matthew, Seats and Votes (New Haven, Conn.: Yale University Press, 1989), pp. 126–133Google Scholar; Lijphart, , Electoral Systems and Party Systems, pp. 30–46Google Scholar; Gallagher, Michael and Mitchell, Paul, ‘Introduction to Electoral Systems’, in Michael Gallagher and Paul Mitchell, eds, The Politics of Electoral Systems (New York: Oxford University Press, 2005), pp. 15–17CrossRefGoogle Scholar; Soberg Shugart, Matthew and Wattenberg, Martin P., ‘Mixed Member Electoral Systems: A Definition and Typology’, in Matthew Soberg Shugart and Martin P. Wattenberg, eds, Mixed Member Electoral Systems (New York: Oxford University Press, 2001)Google Scholar.

³² For a discussion of using average district magnitude to calculate the effective threshold, see Lijphart, , Electoral Systems and Party Systems, pp. 28–9Google Scholar.

³³ Lijphart, , Electoral Systems and Party Systems, p. 27Google Scholar.

³⁴ Cox, ‘Centripetal and Centrifugal Incentives in Electoral Systems’.

³⁵ Merrill and Adams, ‘Centrifugal Incentives in Multi-Candidate Elections’.

³⁶ Adams and Merrill, ‘Why Small, Centrist Third Parties Motivate Policy Divergence by Major Parties’.

³⁷ Laakso, Markku and Taagepera, Rein, ‘Effective Number of Parties: A Measure with Application to West Europe’, Comparative Political Studies, 12 (1979), 3–27CrossRefGoogle Scholar.

³⁸ Freedom House, Freedom in the World, Country Ratings: 1972–2006.

³⁹ As a general statement, the positive relationship between party-system compactness and proportionality is robust for most subsets of the data. For example, if one restricts the sample to the most proportional national elections, those that return measured values of proportionality greater than 14 (n = 27), the fitted line capturing the relationship between party-system compactness and proportionality is positive and strongly so. It remains so if one restricts this sample to the most recent election in the established Western democracies (n = 14).

⁴⁰ For established Western democracies,

The threshold coefficient is also statistically significant in the full sample of CSES observations. It does not obtain statistical significance in the sample consisting only of the most recent election in the established Western democracies. However, this sample consists of only nineteen observations, and the sign is still in the correct direction with a standard error that is nearly half of the coefficient value.

⁴¹ For example, in the full sample of CSES cases excluding those with no sub-national legislative districts,

For regressions estimated using the sample of Western democracies, and the most recent election in these democracies, the estimated coefficient for district magnitude is still positive but fails to obtain statistical significance at even the 0.10 level.

⁴² Lijphart, , Electoral Systems and Party Systems, pp. 147–9Google Scholar; Taagepera and Shugart, Seats and Votes, pp. 93–4Google Scholar.

⁴³ One might wonder if this finding results from multicollinearity between electoral system proportionality and ENP. These variables are correlated in the expected direction, but the absolute value of this correlation ranges from 0.55 to 0.65, depending on the sample. This is well below the level that would normally raise statistical concerns. In addition, there is no reason to believe that multicollinearity would affect the signs of the estimated coefficients.

⁴⁴ For example, for the full sample of cases corresponding to the first column of Table 1,

⁴⁵ These 27 national elections are those from the CSES data that produced majority coalition governments. I did not include majoritarian systems or minority governments in this analysis, because the former are not intended to produce governing coalitions and the nature of minority government is distinct from majority coalition government.

⁴⁶ On the 11-point CSES scale, third-ranked parties in government are approximately 1.1 units from the mean voter ideal point, while third-ranked parties not in government are approximately 2.0 units from the mean voter ideal point. Fourth-ranked parties in government are approximately 1.02 units from the mean voter, while those not in government are approximately 2.3 units from the mean voter ideal point. These differences are statistically significant. Third-ranked parties in government, on average, have slightly higher pluralities than those not in governments, but there are no substantive differences in the vote shares of fourth-ranked parties in government relative to those not in government.

⁴⁷ Another way to think about this pattern is to observe that a full 70 per cent of third-ranked parties that are within two standard deviations of the mean voter ideal point are included in governing coalitions, while only 30 per cent of those parties located more than two standard deviations from the mean voter ideal point are in post-election governing coalitions. Roughly the same percentages apply to parties ranked fourth in aggregate vote share. For larger parties, spatial proximity to the mean voter presents little advantage membership in governing coalitions beyond that conferred by vote share, but for smaller parties not easily distinguished by vote or seat shares, ideological moderation presents a distinct advantage for membership in the government.

⁴⁸ Laver, Michael J., ‘Models of Government Formation’, Annual Review of Political Science, 1 (1988), 1–15CrossRefGoogle Scholar; Laver, Michael J. and Schofield, Norman, Multiparty Government: The Politics of Coalitions in Europe (Oxford: Oxford University Press, 1990)Google Scholar; Schofield and Sened, Multiparty Democracy; Martin, Lanny W. and Stevenson, Randolf T., ‘Government Formation in Parliamentary Democracies’, American Journal of Political Science, 45 (2001), 33–50CrossRefGoogle Scholar; and Volden, Craig and Carrubba, Clifford J., ‘The Formation of Oversized Coalitions in Parliamentary Democracies’, American Journal of Political Science, 48 (2004), 521–537CrossRefGoogle Scholar.

⁴⁹ The sole exception in Ezrow’s study is the 1989 Greek election. Otherwise, all observations are of West European nations.

⁵⁰ Reif and Melich, Euro-Barometer 31A: European Elections, 1989: Post-Election Survey, June–July 1989 [computer file].

⁵¹ One important difference between the Eurobarometer and the CSES left–right scales is that the former is based on 1–10 points, while the latter is based on 0–10. This means that unlike the CSES, the Eurobarometer has no middle value. It is unclear whether this contributes to the discrepancies in our findings; I suspect it does so only modestly, but the differences in these metrics should be kept in mind when comparing our relative party and voter placements.

⁵² One can also compare the party placements relative to the percentiles of the voter distribution on the left–right axis. In 1989, the Labour party and Conservatives were polarized, with mean placements at the 12th and 90th percentiles of the voter distribution. In 2005, they were located at the 25th and 75th percentiles. To conserve space, these graphs are not presented.

⁵³ Ezrow, ‘Parties’ Policy Programmes and the Dog That Didn’t Bark’.

⁵⁴ Geddes, B., ‘How the Cases You Choose Affect the Answers You Get: Selection Bias in Comparative Politics’, in J. A. Stimson, ed., Political Analysis, Vol. 2 (Ann Arbor: The University of Michigan Press, 1990)Google Scholar.