Decomposition
If we suppose that N firms share all the market, then the index can be expressed as where N is the number of firms, as above, and V is the statistical variance of the firm shares, defined as . If all firms have equal (identical) shares (that is, if the market structure is completely symmetric, in which case si = 1/N for all i) then V is zero and H equals 1/N. If the number of firms in the market is held constant, then a higher variance due to a higher level of asymmetry between firms' shares (that is, a higher share dispersion) will result in a higher index value. See Brown and Warren-Boulton (1988), also see Warren-Boulton (1990).
Read more about this topic: Herfindahl Index