# Political Diversity Indices

### Description

Computes political diversity indices or fragmentation/concetration measures like the effective number of parties for an electoral unity or across unities. The very intuition of these coefficients is to counting parties while weighting them by their relative political–or electoral strength.

### Usage

1 2 3 4 5 6 | ```
PoliticalDiversity(x, index = "laakso/taagepera", margin = 1,
base = exp(1))
## Default S3 method:
PoliticalDiversity(x, index = "laakso/taagepera",
margin = 1, base = exp(1))
``` |

### Arguments

`x` |
A data.frame, a matrix, or a vector containing values for the number of votes or seats each party received. |

`index` |
The type of index desired, one of "laakso/taagepera", "golosov", "herfindahl", "gini", "shannon", "simpson", "invsimpson". |

`margin` |
The margin for which the index is computed. |

`base` |
The logarithm base used in some indices, such as the "shannon" index. |

### Details

Very often, political analysts say things like ‘two-party system’ and
‘multi-party system’ to refer to a particular kind of political party system.
However, these terms alone does not tell exactly how fragmented–or concentrated a
party system actually is. For instance, after the 2010 general election, 22 parties
obtained representation in the Lower Chamber in Brazil. Nonetheless, among these 22
parties, nine parties together returned only 28 MPs. Thus, an index to assess the
weigh or the **Effective Number of Parties** is important and helps to go
beyond the simple count of parties in a legislative branch.
A widely accepted algorithm was proposed by M. Laakso and R. Taagepera:

*N = 1/ ∑ p_i^2*

, where **N** denotes the
effective number of parties and **p_i** denotes the *it^h* party's fraction
of the seats.

In fact, this formula may be used to compute the vote share for each party. This
formula is the reciprocal of a well-known concentration index
(**the Herfindahl-Hirschman index**) used in economics to study the degree
to which ownership of firms in an industry is concentrated. Laakso and Taagepera
correctly saw that the effective number of parties is simply an instance of the
inverse measurement problem to that one. This index makes rough but fairly
reliable international comparisons of party systems possible.

**The Inverse Simpson index**,

* 1/ λ = {1 \over∑_{i=1}^R p_i^2} = {}^2D*

Where *λ* equals the probability that two types taken at random from
the dataset (with replacement) represent the same type. This simply equals true
fragmentation of order 2, i.e. the effective number of parties that is obtained
when the weighted arithmetic mean is used to quantify average proportional
diversity of political parties in the election of interest.
Another measure is the **Least squares index (lsq)**, which measures the
disproportionality produced by the election. Specifically, by the disparity
between the distribution of votes and seats allocation.

Recently, Grigorii Golosov proposed a new method for computing the effective
number of parties in which both larger and smaller parties are not attributed unrealistic scores as those resulted by using the Laakso/Taagepera index.I will
call this as (**Golosov**) and is given by the following
formula:

*N = ∑_{i=1}^{n}\frac{p_{i}}{p_{i}+p_{i}^{2}-p_{i}^{2}}*

### Author(s)

Daniel Marcelino, dmarcelino@live.com.

### References

Gallagher, Michael and Paul Mitchell (2005) *The Politics of Electoral Systems.* Oxford University Press.

Golosov, Grigorii (2010) The Effective Number of Parties: A New Approach, *Party Politics,* **16:** 171-192.

Laakso, Markku and Rein Taagepera (1979) Effective Number of Parties: A Measure with Application to West Europe, *Comparative Political Studies,* **12:** 3-27.

Nicolau, Jairo (2008) *Sistemas Eleitorais.* Rio de Janeiro, FGV.

Taagepera, Rein and Matthew S. Shugart (1989) *Seats and Votes: The Effects and Determinants of Electoral Systems.* New Haven: Yale University Press.

### Examples

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 | ```
# Here are some examples, help yourself:
# The wikipedia examples
A <- c(.75,.25);
B <- c(.75,.10,rep(0.01,15))
C <- c(.55,.45);
# The index by "laakso/taagepera" is the default
PoliticalDiversity(A)
PoliticalDiversity(B)
# Using the method proposed by Golosov gives:
PoliticalDiversity(B, index="golosov")
PoliticalDiversity(C, index="golosov")
# The 1980 presidential election in the US (vote share):
US1980 <- c("Democratic"=0.410, "Republican"=0.507,
"Independent"=0.066, "Libertarian"=0.011, "Citizens"=0.003,
"Others"=0.003)
PoliticalDiversity(US1980)
# 2010 Brazilian legislative election
votes_2010 = c("PT"=13813587, "PMDB"=11692384, "PSDB"=9421347,
"DEM"=6932420, "PR"=7050274, "PP"=5987670, "PSB"=6553345,
"PDT"=4478736, "PTB"=3808646, "PSC"=2981714, "PV"=2886633,
"PC do B"=2545279, "PPS"=2376475, "PRB"=1659973, "PMN"=1026220,
"PT do B"=605768, "PSOL"=968475, "PHS"=719611, "PRTB"=283047,
"PRP"=232530, "PSL"=457490,"PTC"=563145)
seats_2010 = c("PT"=88, "PMDB"=79, "PSDB"=53, "DEM"=43,
"PR"=41, "PP"=41, "PSB"=34, "PDT"=28, "PTB"=21, "PSC"=17,
"PV"=15, "PC do B"=15, "PPS"=12, "PRB"=8, "PMN"=4, "PT do B"=3,
"PSOL"=3, "PHS"=2, "PRTB"=2, "PRP"=2, "PSL"=1,"PTC"=1)
PoliticalDiversity(seats_2010)
PoliticalDiversity(seats_2010, index= "golosov")
``` |