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Bernard Grofman, Alessandro Chiaramonte, Roberto D'Alimonte and Scott L. Feld, "Comparing and Contrasting the Uses of Two Graphical Tools for Displaying Patterns of Multiparty Competition: Nagayama Diagrams and Simplex Representations," Party Politics, 10 (May 2004), 273-299.

First Paragraph:
A picture, it is said, is worth a thousand words. Similarly, graphical representation of data can often be more visually informative than presentation of the same data in tabular form. In this article we present two useful tools for representing a set of election returns for multiparty competition in individual constituencies. One is very new, the other has a venerable history and yet remains remarkably underutilized in the party and electoral systems literature.

Figures and Tables:
Figure 1. Segmented Nagayama diagram (z = 0.20)
Figure 2. Simplex representation (partial grid for triangular coordinates)
Figure 3. Simplex representation showing midpoint bisectors
Figure 4. Simplex representation showing areas of subtriangles defined by vertices and q
Figure 5. Equilateral simplex representation showing region where B + C z
Figure 6. Segmented Nagayama triangle showing district level results of 1994, 1996 and 2001 Italian elections to the Chamber of Deputies in the single-member component of the mixed system
Figure 7. Equilateral simplex representations showing district level results of 1994, 1996 and 2001 Italian elections to the Chamber of Deputies in the singlemember component of the Italian mixed system.
Table 1. Segmented Nagayama diagram segment proportions in Italian elections 1994, 1996 and 2001 (single-member district component, n = 475)
Figure 8. Segmented Nagayama triangle showing district level results of 1994, 1996 and 2001 Italian elections to the Chamber of Deputies in the single-member component of the mixed system in numerical form
Table 2. Tabular display of information about party competition in the single-member district component of Italian elections, 1994, 1996, 2001
Figure 9. Histograms showing distribution of district vote to the top two vote-getting parties/party blocs in the district: 1994, 1996 and 2001 Italian elections to the Chamber of Deputies in the single-member component of the Italian mixed system

Last Paragraph:
(First Paragraph of Discussion) Because this article is viewed as primarily a methodological one, we leave to other work a fuller discussion of the substantive implications of the 2001 Italian election results.21 We hope to have illustrated; (1) how segmented Nagayama diagrams and simplex representations may be created and visually interpreted, (2) how they each reflect different aspects of multiparty competition, and (3) how useful each type of diagram can be in compactly presenting information in a visual field that can allow quick interpretation of key facts--aided, or so we would like to believe, by an appropriate segmentation of the triangles so that rough and ready comparisons of the density in each can readily be made.

updated November 2013