One can measure the degree of market concentration using formulas also used for political parties and for populations of organisms:
Effective number of parties and
Diversity index and
Herfindahl–Hirschman index
One starts with market-share fractions or vote fractions or legislature-seat fractions or population fractions f
i for entity i of whatever one is working with.
There are several formulas for the effective fraction, and by extension, the reciprocal of it, the effective number. These are generally weighted averages, the weighting from the fractions of the entities:
\( <f> \ = F^{-1} \left( \sum_i f_i F(f_i) \right) \)
where F is the averaging function. For a power law, \(F(f) = f^q\) we get
\( <f> \ = \left( \sum_i (f_i)^{q+1} \right)^{1/q} \)
For q -> 0, one gets the entropy or Shannon information measure:
\( I = - \sum_i f_i \log(f_i) \)
For base-2 logarithms, that gives the number of bits needed to specify which of the entities.
For q = 1, one gets the reciprocal of what is variously called the Laakso-Taagapera number of political parties, the Herfindahl–Hirschman index of effective number of competing firms, and the Simpson index of species diversity:
\( <f> \ = \sum_i (f_i)^2 \)
For q -> +oo, one gets the maximum of the f's.
From the effective number of parties is also Grigorii Golosov's formula for the effective number:
\( \displaystyle{ <N> \ = \sum_i \frac{f_i}{f_i + (f_1)^2 - (f_i)^2} } \)
where \(f_1\) is the largest fraction.