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Rein Taagepera, "Extension of the Nagayama Triangle for Visualization of Party Strengths," Party Politics, 10 (May 2004), 301-306.

First Paragraph:
In his study of the workings of Duverger's law in Italy, Reed (2001) uses to good effect a graphic approach devised by Nagayama (1997) for the study of candidate strengths in single-member districts (SMDs). The vote shares of the second-running contestant (s2) are plotted against the vote shares of the top contestant (s1). The total of the two shares cannot exceed 100 percent, nor can the second-largest share exceed the largest--as shown by thick lines in Figure 1. These two constraints force the data points to lie within a triangle that Reed (2001) calls the Nagayama triangle. Its left side denotes perfect parity of the two top contestants, while its right side denotes the dominance of the strongest contestant over a single opponent. At the peak, the two contestants have equal strength, and there are no others. The left corner area of the triangle corresponds to the presence of multiple contestants.

Figures and Tables:
Figure 1. Votes for second-largest parties in 17 single-member district elections: closeness to two party balance
Figure 2. Seats for second-largest parties in 17 single-member district elections: closeness to one party dominance
Figure 3. Shift from second-largest party's votes to seats in three single-member district countries: moving right, close to the logical maximum
Figure 4. Vote shares of third-largest parties in single-member district systems (white circles) and PR systems (black diamonds). Superimposed diamonds in the centre of the cloud have been omitted

Last Paragraph:
The objective of this note was to point out various ways in which the Nagayama triangle can be a useful tool for analysing party systems. I have graphed a dataset of limited extent, yet random, since the selection was done not by me but by what was included in Mackie and Rose (1997). The tentative patterns found raise a number of questions. Do nationwide second party vote shares in SMD elections always tilt to the left side of the possible area of occurrence, while their seat shares tilt to the right side? Does the conversion of vote shares into seat shares always approach the pattern outlined by France, the UK and the US in Figure 3, hugging the upper part of the Nagayama triangle? Are the third-party vote shares under SMD and PR rules always similar, at a given share for the largest party? Many further questions could be asked about the PR and SMD patterns, including fourth-ranking parties and beyond. To answer them, much more extensive data need to be processed. Here I have simply pointed out that a potentially fertile new avenue for studying Duvergerian processes in a broad sense.

updated November 2013