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Patrick Dunleavy and Rekha Diwakar, "Analysing multiparty competition in plurality rule elections," Party Politics, 19 (November 2013), 855-886. [Available at ]

First paragraph:
In the comparative analysis of elections and party systems we have yet to develop logically acceptable ways to chart the district-level outcomes of multiparty elections, and to assess the clustering or patterning of outcomes in systematic ways. In this article, we show that the Nagayama or 'all possibilities' triangle has major defects, but can be reformulated and re-applied in two new ways. The first captures the 'layer cake' character of general election outcomes in a reductionist fashion, showing how the number of observable parties competing for votes at district level influences outcomes. An alternative variant (the crown diagram) gives a more holistic picture of outcomes, shifting attention to the performance of the top two parties or blocs linked to the predominant political--ideological dimension in a political system. We link this second innovation to a tentative logic of evolutionary development applicable to plurality rule election systems.

Figures and Tables:
Figure 1. The simplex representation of a three-party election.
Figure 2. The 'all possibilities' triangle display (after Grofman et al.).
Figure 3. Flipping and rotating the APT.
Figure 4. The size and shape of the ECS area at 3, 10 and 50 observable parties.
Table 1. The mechanical and psychological impacts of plurality rule elections on party competition, according to Duverger
Table 2. The percentage of all election districts with different numbers of 'observable' parties in the UK, India and US elections in the mid-2000s.
Figure 5. The 2006 U.S. House of Representatives election outcomes at district level.
Figure 6. The 2004 Indian general election outcomes for districts with different numbers of observable parties:
Figure 7. The 2005 general election outcomes in Great Britain for districts with different numbers of observable parties:
Figure 8. The 'crown' within the double APT diagram.
Figure 9. The seats outcomes in the U.S. House of Representatives election, 2006.
Figure 10. The seats outcomes in the Indian general election of 2004.
Figure 11. The seats outcomes in the 2005 general election in Great Britain.
Figure 12. The seats outcomes in the 1955 general election in Great Britain.
Figure 13. Hypothesized ideal-type stages in the evolution of the UK's plurality rule system over time.
Figure 14. Regional patterns of party competition in four British regions at the 2010 general election.
Figure 15. The outcomes of the European Parliament elections in Great Britain in 1999, 2004 and 2009 using regional list PR systems.

Last Paragraph:
(First paragraph of Conclusions) Within plurality rule election systems, the district-level comparison of general election outcomes has been increasingly central in the debate about competing Duvergerian expectations and opposing mechanisms favouring party fragmentation (such as the 'invulnerable majority' effect). A few earlier analysts have accumulated election data covering many separate district contests (some 58,000 results in the case of Chhibber and Kollman, 2004: ch. 2), yet confined their analysis to means and standard deviations, presenting little of the richness of outcomes patterning that can be accessed by better methods of charting. We have demonstrated that it is helpful to visually chart both a reductionist picture of V1V2 competition in races with different numbers of parties in competition, shown in the ECS view, and giving insights into the aggregated, layer-cake nature of general election outcomes; and information about how district outcomes relate to the most important basis for political alignments and how the holistic patterns vary across election outcome, shown in the crown diagram. A careful analyst will always consider both displays.

Last updated November 2013