Effective number of parties

Some political scientists use the effective number of parties as a diversity index. The measure as introduced by Laakso and Rein Taagepera (1979),^[1] produces an adjusted number of political parties in a country's party system, weighted by their relative size. The measure is useful when comparing party systems across countries.^[2]

A jurisdiction's party system can be measured by either:

The effective number of electoral parties (ENEP) weights parties by their share of the vote.
The effective number of parliamentary parties (ENPP) weights parties by their share of seats in the legislature.

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 smaller than the number of actual parties. The effective number of parties is a frequent operationalization for political fragmentation. Conversely, political concentration can be seen by the share of power held large political parties.^[3]

Several common alternative methods are used to define the effective number of parties.^[4] John K. Wildgen's index of "hyperfractionalization" accords special weight to small parties.^[5] Juan Molinar's index gives special weight to the largest party.^[6] Dunleavy and Boucek provide a useful critique of the Molinar index.^[7]

Measures

Quadratic

Laakso and Taagepera (1979) were the first to define the effective number of parties using the 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. This definition is still the most commonly used in political science.

This measure is equivalent to the Herfindahl–Hirschman index, used in economics; the Simpson diversity index in ecology; the inverse participation ratio (IPR) in physics; and the Rényi entropy of order $\alpha =2$ in information theory.^[8]

Alternatives

An alternative formula was proposed by Grigorii Golosov in 2010.^[9]

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

Seat product model

The effective number of parties can be predicted with the seat product model^[10]^[11] as $N=(MS)^{1/6}$ , where M is the district magnitude and S is the assembly size.