Minimal prime ideal

The effective number of parties is a concept which provides for an adjusted number of political parties in a country's party system. The idea behind this measure is to count parties and, at the same time, to weight the count by their relative strength. The relative strength refers to their vote share ("effective number of electoral parties") or seat share in the parliament ("effective number of parliamentary parties"). This measure is especially useful when comparing electoral systems across countries, as is done in the field of political science. The number of parties equals the effective number of parties only when all parties have equal strength. In any other case, the effective number of parties is lower than the actual number of parties. The effective number of parties is a frequent operationalization for the fragmentation of a party system.

There are two major alternatives to the effective number of parties-measure. John K. Wildgen's index of "hyperfractionalization" accords special weight to small parties. Juan Molinar's index gives special weight to the largest party. Dunleavy and Boucek provide a useful critique of the Molinar index.

Formulas

According to Laakso and Taagepera (1979) the effective number of parties is computed by following formula:

$N=\frac{1}{\sum _{i=1}^n{p}_i^2}$

Where n is the number of parties with at least one vote/seat and $p_i^2$ the square of each party’s proportion of all votes or seats. This is also the formula for the inverse Simpson index, or the true diversity of order 2.

An alternative formula proposed by Golosov (2010) is

$N=\sum _{i=1}^n\frac{p_i}{p_i+p_1^2-p_i^2}$

which is equivalent - if we only consider parties with at least one vote/seat - to

$N=\sum _{i=1}^n\frac{1}{1+(p_1^2/p_i)-p_i}$

Here, n is the number of parties, $p_i^2$ the square of each party’s proportion of all votes or seats, and $p_1^2$ is the square of the largest party’s proportion of all votes or seats.

Values

The following table illustrates the difference between the values produced by the two formulas for eight hypothetical vote or seat constellations:

Constellation	Largest component, fractional share	Other components, fractional shares	N, Laakso-Taagepera	N, Golosov
A	0.75	0.25	1.60	1.33
B	0.75	0.1, 15 at 0.01	1.74	1.42
C	0.55	0.45	1.98	1.82
D	0.55	3 at 0.1, 15 at 0.01	2.99	2.24
E	0.35	0.35, 0.3	2.99	2.90
F	0.35	5 at 0.1, 15 at 0.01	5.75	4.49
G	0.15	5 at 0.15, 0.1	6.90	6.89
H	0.15	7 at 0.1, 15 at 0.01	10.64	11.85

External links

• Michael Gallagher providing data on the Laakso-Taagepera effective number of parties for over 900 elections in over 100 countries

• Average effective number of parties (Golosov) for 183 democratic party systems and non-systems, 1792-2009, reported in Golosov, Grigorii V., "Towards a Classification of the World's Democratic Party Systems, Step 1: Identifying the Units", Party Politics, Vol. 19, No. 1, January 2013, pp. 134-138.