### The Inverse Herfindahl–Hirschman Index as an “Effective Number of” Parties

I learned of the passing of Albert Hirschman on December 11, and while better and more instructive tributes to his life can be read elsewhere, I wanted to focus on a little piece of Hirschman’s work that I use all the time: the (inverse) Herfindahl–Hirschman Index.

The HHI is basically a measure of market concentration, but when inverted, it is an “effective number of” whatever grouping you might be interested in, such as parties. Essentially, this statistic can be interpreted as, “Individuals are distributed across groups in such a way that they are as concentrated as they would be if divided across [HHI value] groups evenly.”

This is perhaps best understood by example, and fortunately, my field of American Politics offers an interesting one. The U.S. South, between Reconstruction and the Civil Rights Act, was commonly known as the “one-party South,” due to the overwhelming dominance of the Democratic Party in Southern Politics. We can see evidence of this dominance by calculating the Effective Number of Parties-in-the-Electorate, using the HHI.

As the graph below illustrates, non-Southern states have consistently featured just over two “effective” parties (Democrats, Republicans, and some Independents/Others), while the South lagged behind in this measure up until the 1980s.

The inverse HHI is an elegant little function (the square of the sum over the sum of the squares), and plyr makes it very easy to calculate for any dataset.