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Richard G. Niemi and John Fuh-sheng Hsieh, "Counting Candidates: An Alternative to the Effective N (With An Application to the M + 1 Rule in Japan)," Party Politics, January 2002, vol. 8, no. 1, pp. 75-99.

First Paragraph:
The number of candidates or parties in an election contest has long been of theoretical interest (Duverger, 1964; Rae, 1971). The measure has taken on added importance in recent years as researchers have studied the effects of thresholds, electoral formulas, district magnitude, presidentialism, and the like, on candidate and party numbers and on the strategic coordination that lies behind these effects (Cox, 1997; Jones, 1999; Shugart and Carey, 1992; Taagepera and Shugart, 1989). As every researcher soon learns, however, counting candidates or parties is not a trivial exercise.

Figures and Tables:
Table 1. Hypothetical vote distributions and the actual and effective numbers of candidates/parties
Table 2. Hypothetical vote distributions and alternative measures of the number of candidates
Table 3. Additional hypothetical vote distributions and alternative measures of the number of candidates
Table 4. Percentages of districts with given numbers of candidates relative to the district magnitude (M)
Figure 1. Average number of viable candidates in three-member districts Note: Elections, not years, are evenly spaced, as it is likely that learning is a function of experience with elections, not merely of the passage of time.
Figure 2. Average number of viable candidates in four-member districts Note: Elections, not years, are evenly spaced, as it is likely that learning is a function of experience with elections, not merely of the passage of time.
Figure 3. Average number of viable candidates in five-member districts Note: Elections, not years, are evenly spaced, as it is likely that learning is a function of experience with elections, not merely of the passage of time.
Table A.1. The number of candidates in elections to the Japanese diet by district size, year and alternative measures

Last Paragraph:
From a substantive point of view, our re-analysis of the Japanese results lends still further empirical support to the M + 1 rule. Further, the fact that the results hold using our two new measures, particularly the 1/2 quota rule, suggests that the Duvergerian equilibrium, under which only the top M + 1 candidates are viable and each of them obtains an equal number of votes, has considerable applicability in the real world.