# Herfindahl-Hirschman index and entropy

Herfindahl-Hirschman index (or HHI) describing how close the market is to monopolistic market. This tool is used by organizations responsible for the supervision of fair competition (e.g., when considering corporate mergers). This index is obtained using known market shares, \( s_i \in (0,1] \), of competitors in the market:

\begin{equation} H = \sum_{i=1}^N s_i^2 . \end{equation}

If \( H \) is:

- smaller than \( 0.01 \), then the competition in market is nearly perfect.
- smaller than \( 0.15 \), then the competition is not as perfect as it could be, but no competitor dominates the market.
- smaller than \( 0.25 \), then dominant competitor is present.
- larger than \( 0.25 \), then dominant competitor might manipulate the market.

Note that \( H \in [1/N, 1] \), which may be not very convenient in certain cases, thus sometimes normalization is introduced. Normalized HHI is defined as:

\begin{equation} H^{*} = \frac{\sum\limits_{i=1}^N \left[ s_i^2\right] - \frac{1}{N}}{1-\frac{1}{N}} . \end{equation}

The general form of normalized HHI suggests that now it describes how far the market is from being equally divided. Note that \( H^{*} \) and \( H \) may be very different for small \( N \). E.g., if \( N=2 \), \( s_1 = s_2 = 0.5 \), then \( H^{*}=0 \) and \( H=0.5 \). The difference lies in an intuition that competitive markets should not only be equally divided, but also have many competitors.

In order to use this tool correctly one should also consider methodology behind \( s_i \). Most importantly the market should be properly defined. Namely the competition may persist on global scale (for multiple similar or related products (e.g., all financial services)), but on local scale (subset of the previous product set (e.g., loans and deposits)) competition may not be present. Market should be also properly defined in the geographical sense - five competitors might have equal shares on a country level (globally competitive), but be monopolists in different regions of the country (locally not competitive).

Interestingly in physics there is a similar quantity - entropy. Even its mathematical form is very similar:

\begin{equation} S = - k_B \sum_{i=1}^N p_i \ln p_i . \end{equation}

Though note that larger entropy means larger competition (more states with equal probabilities).

Both indicators might be used to solve or understand the same problems. The choice should depend on the available data and should be guided by statistical tests (e.g., for binary cases one might use ROC). As far as I am aware both indicators produce similar results, yet slight variations are present.