Effective number of parties#Seat product model

Laakso and Taagepera (1979) were the first to define the effective number of parties using the following formula:

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

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 $\backslash alpha\; =\; 2$ in information theory.{{Cite work|last1=Bailey |first1=Jack|date=2025|title="Politics, Bit by Bit: A Formal Link Between Entropy and the Effective Number of Parties |language=en |doi=10.33774/apsa-2025-73pkd-v3|url=https://preprints.apsanet.org/engage/apsa/article-details/6787e2a481d2151a024a5e8f }}

An alternative formula was proposed by Grigorii Golosov in 2010.{{Cite journal|last=Golosov|first=Grigorii V.|date=2010|title=The Effective Number of Parties: A New Approach|journal=Party Politics|language=en|volume=16|issue=2|pages=171–192|doi=10.1177/1354068809339538|s2cid=144503915|issn=1354-0688}}

: $N\; =\; \backslash sum\_\{i=1\}^n\; \backslash frac\{p\_i\}\{p\_i+p\_1^2-p\_i^2\}$

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

: $N\; =\; \backslash sum\_\{i=1\}^n\; \backslash frac\{1\}\{1+(p\_1^2/p\_i)-p\_i\}$

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.

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

The effective number of parties can be predicted with the seat product modelTaagepera, Rein (2007). "Predicting Party Sizes". Oxford University Press{{cite journal | url=https://doi.org/10.1016/j.electstud.2015.10.011 | doi=10.1016/j.electstud.2015.10.011 | title=The Seat Product Model of the effective number of parties: A case for applied political science | date=2016 | last1=Li | first1=Yuhui | last2=Shugart | first2=Matthew S. | journal=Electoral Studies | volume=41 | pages=23–34 | url-access=subscription }} as $N\; =\; (MS)^\{1/6\}$, where M is the district magnitude and S is the assembly size.

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For individual countries the values of effective number of number of parliamentary parties (ENPP) for the last available election is shown.{{cite web|title=Election Indices |url=https://www.tcd.ie/Political_Science/about/people/michael_gallagher/ElSystems/Docts/ElectionIndices.pdf}} 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.

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• {{annotated link|Duverger's law}}
• {{annotated link|First-past-the-post voting}}
• {{annotated link|Majoritarian representation}}
• {{annotated link|Multi-party system}}
• {{annotated link|One-party state}}
• {{annotated link|Proportional representation}}
• {{annotated link|Two-party system}}
• {{annotated link|Vote splitting}}
• {{annotated link|Electoral competition}}
• {{annotated link|Micromega rule}}

• [http://www.tcd.ie/Political_Science/people/michael_gallagher/ElSystems/Docts/ElectionIndices.pdf 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., [http://ppq.sagepub.com/content/19/1/122.abstract "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.
• [http://golosovpubs.ucoz.com/index/indices_of_the_effective_number_of_parties_and_party_system_nationalization_in_excel/0-13 How to compute Golosov’s effective number of parties in Excel]

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Category:Voting theory