The effective number of political parties is a measure of how many political parties "effectively" participate in a government or are represented in an electoral system.
It is generally defined as follows. For political parties $~$1, 2, \ldots, n$~$, let $~$p_n$~$ represent the Proportion of total support that party $~$n$~$ has in a country, which is a number from $~$0$~$ to $~$1$~$. Then, the effective number of political parties is $~$\displaystyle \frac{1}{\sum_{i=1}^n p_i^2}$~$.
Motivation for the measure
The federal government of the United States of America is generally referred to as a "two-party system", although there are over thirty federal political parties actually registered. This is because the Democratic and Republican parties tend to dominate all popular support to the point of marginalizing all the other parties. The Libertarian and Green party are examples of major parties that have comparatively little popular support.
Because of this discrepancy in the total number of parties and the number of parties likely to actually be represented, it seems useful to have a measure of the number of parties that effectively participate in a country's politics.
There are many approaches we can take to this. We could measure the effective number of parties using "the number of parties that have more than $~$x$~$% support", but this breaks down as the number of small parties with significant support increases.
For example, if we set the cutoff to a reasonable-sounding 5%, then Algeria is a degenerate case. In Algeria's most recent legislative election in 2012, the three parties who had 5% support or more only represented about 30% of the total popular vote, and there were at least 20 parties that had between 1% and 3% support, which makes it seem very unsuitable to say there are effectively only three political parties in Algeria.
Sliding the percentage around doesn't help either, because you get degenerate cases in the other direction. If you lower the cutoff to 1% to make Algeria's number make sense, then Canada in its most recent 2015 election had five parties effectively represented by popular vote, even though the Green Party and Bloc Québécois had barely no representation between them (about 3~4% apiece) compared to the big three (Conservative, Liberal and NDP).
Desirable properties of an EPP measure
What we really want is a measure that:
gets close to $~$n$~$ as the distribution of votes between $~$n$~$ parties becomes more even, regardless of how big $~$n$~$ is.
gets close to $~$k$~$ as $~$k$~$ equally matched parties approach total domination of the electoral system (including the case where $~$k = 1$~$).
changes very little with the introduction of a new political party with comparatively negligible support.
Our cutoff measure violates rule 1, because as $~$n$~$ gets larger, more and more parties fall below the cutoff.
A good place to start looking for such a measure is to find a measure of the "average proportion of support" of political parties, and then invert that to get the effective number of parties. Immediately we run into the problem that adding up all the $~$p_i$~$ (as defined above) and dividing by $~$n$~$ (the number of parties) always gives us exactly $~$1/n$~$, which is a relatively useless measure — that's the total number of political parties again, which violates both rules 2 and 3.
But what if we use a weighted average? In particular, what if we redefine the average proportion of support as the average support for a voter's political party over all voters in the electoral system, rather the average support for a party over all parties? Then, a political party with $~$p_i$~$ support also gets $~$p_i$~$ weight in the average, because $~$p_i$~$ of the voters in the electoral system are represented by that party.
This gives us a weighted average of $~$(p_1 \cdot p_1) + (p_2 \cdot p_2) + \ldots + (p_n \cdot p_n) = \sum_{i=1}^n p_i^2$~$, which we invert to get the formula defined at the top of the article. This measure satisfies all three properties we desire from our measure, which is left as an exercise.
[todo: solution to exercise]