Effective number of parties

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.

Alternatives

An alternative formula was proposed by Grigorii Golosov in 2010.

: 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:

Seat product model

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

Effective number of parties by country

For individual countries the values of effective number of number of parliamentary parties (ENPP) for the last available election is shown. Some of the highest effective number of parties are in Brazil, Belgium, and Bosnia and Herzegovina. European Parliament has an even higher effective number of parties if national parties are considered, yet a much lower effective number of parties if political groups of the European Parliament are considered.

References

References

1. (1979). ""Effective" Number of Parties: A Measure with Application to West Europe". Comparative Political Studies.
2. Lijphart, Arend (1999): Patterns of Democracy. New Haven/London: Yale UP
3. (2024). "Concentration of political power: Can we improve its measurement?". Comparative European Politics.
4. Arend Lijphart. (1 January 1994). "Electoral Systems and Party Systems: A Study of Twenty-seven Democracies, 1945–1990". Oxford University Press.
5. (1971-07-01). "The Measurement of Hyperfractionalization". Cps.sagepub.com.
6. Molinar, Juan. (1 January 1991). "Counting the Number of Parties: An Alternative Index". The American Political Science Review.
7. (2003). "Constructing the Number of Parties". Party Politics.
8. (2025). ""Politics, Bit by Bit: A Formal Link Between Entropy and the Effective Number of Parties".
9. Golosov, Grigorii V.. (2010). "The Effective Number of Parties: A New Approach". Party Politics.
10. Taagepera, Rein (2007). "Predicting Party Sizes". Oxford University Press
11. (2016). "The Seat Product Model of the effective number of parties: A case for applied political science". Electoral Studies.
12. "Election Indices".