What is the Herfindahl Index?

The Herfindahl index.according to Wikipedia, is the sum of the squares of the shares of the largest firms of a market.  Since someone's share of the market is represented by a percentage, the sums of the squares of these decimal numbers will add up to something between zero and one.  If the number is close to zero, you have a huge number of firms with very small percentages of the market - think of those paleta salesmen ringing their little bells.  If the number is equal to one, you have one person who owns the entire market, a pure monopoly.

Here's an example.  There are five firms in a market.  Each firm controls 20 percent of the market.  20 percent is 0.2.  So we add (0.2)^2 + (0.2)^2 + (0.2)^2 + (0.2)^2 + (0.2)^2 = 0.04 + 0.04 + 0.04 + 0.04 + 0.04 = 0.2, the Herfindahl index for this particular market, which is closer to pure competition than it is to monopoly. 

It's a happy accident that the inverse of 0.2 - one divided by 0.2 - is equal to five, the total number of firms in our example.  If you think about it, you can compare the players on a WNBA team to...a market.  Call it the Atlanta Dream player minutes market, for example.  There are only so many minutes to go around.  Each Atlanta Dream player controls a share of the minutes played.  Therefore, we can perform a Herfindahl index calculation that applies to WNBA teams.

Let's take two examples:  the first example is of a starting lineup where every one of the five plays the entire forty minutes every game.  Each controls 20 percent of the market in minutes - the rest of the six players have never set foot on the floor.  Like the above example, the Herfindahl index of the "market in minutes" is 0.2 - which has an inverse of 5, the number of players.

Another example is one where each of the 11 players on the team plays exactly 200/11 minutes every game.  Each of the 11 players plays exactly the same number of minutes.  Without showing the math, the Herfindahl index for this market of minutes is 0.090909..... - which has an inverse of 11, which is the number of players on the team.

Looking at the inverse of the Herfindahl index could be interpreted as an answer to the question "how are minutes spread across the team?"  The great thing about the inverse Herfindahl is that it will be a number which ranges between 5 and somewhere around 11...maybe even greater, in some circumstances.  You could interpret the final result as "among how many players are the team's minutes really spread?"

And guess what?  I've done the calculations in Herfindahl index, so I can tell you what the numbers are - so far - in 2010.

PHO    7.70
SEA    7.74
WAS    7.77
NYL    8.07
LAS    8.50
ATL    8.68
SAS    8.74
CHI    8.90
MIN    9.29
CON    9.43
IND    9.43
TUL    13.85

Three teams - the Mercury, the Storm and the Mystics - have the most uneven distribution of minutes.  There are core groups of players on those teams that do all the heavy lifting, at least in terms of how minutes are distributed.  At the very bottom is Tulsa, where the minutes are going all over the place.

Like a lot of statistics, the (inverse) Herfindahl index provides a lot of what but not very much why.  For a team with a high index, is it because the coach runs an offense that distributes the ball around and is substitution-heavy, or is it because the coach has to play a lot of stiffs because there's no one on the team that's good?  For a team with a low index, is it because the philosophy is that "the best players get the minutes" or are there players hurt, leading to an inequitable distribution?

The value of the Herfindahl index is that at times it can give you insight into how a team uses its players and its bench.  What follows is a list of every Herfindahl index value in WNBA history.  I suspect that a man from Oklahoma is going to shatter the WNBA Herfindahl index record this year.  List is the team year, city, final Herfindahl index and winning percentage of the team.  The correlation between a (high) inverse Herfindahl index and a high team winning percentage is -0.449 - medium in the negative direction, which implies that teams with low scores on the list generally win games.


2002    NYL    7.26    0.563
2002    UTA    7.30    0.625
2000    ORL    7.37    0.500
2000    WAS    7.38    0.438
1999    ORL    7.38    0.469
1999    HOU    7.40    0.813
1998    HOU    7.54    0.900
2001    UTA    7.54    0.594
1999    SAC    7.55    0.594
2000    NYL    7.68    0.625
2003    HOU    7.70    0.588
1999    NYL    7.72    0.563
2000    HOU    7.75    0.844
2002    LAS    7.76    0.781
2001    NYL    7.78    0.656
2005    CON    7.88    0.765
2005    WAS    7.88    0.471
1997    NYL    7.88    0.607
2004    CON    7.89    0.529
2007    PHO    7.90    0.676
2003    SAS    7.92    0.353
1997    PHO    7.96    0.571
1997    SAC    7.99    0.357
1998    DET    8.01    0.567
2001    LAS    8.01    0.875
2002    CHA    8.01    0.563
2009    SEA    8.02    0.588
2005    NYL    8.04    0.529
2002    HOU    8.04    0.750
2006    IND    8.05    0.618
2008    SAS    8.11    0.706
2006    DET    8.11    0.676
2004    SEA    8.12    0.588
2000    SAC    8.13    0.656
2003    LAS    8.14    0.706
1997    CHA    8.15    0.536
1997    HOU    8.16    0.643
1998    NYL    8.16    0.600
2007    SAS    8.17    0.588
2003    NYL    8.21    0.471
1999    WAS    8.21    0.375
2004    DET    8.21    0.500
2003    MIN    8.21    0.529
2001    HOU    8.22    0.594
2003    DET    8.24    0.735
2001    CHA    8.25    0.563
2007    NYL    8.25    0.471
2001    ORL    8.26    0.406
2000    LAS    8.26    0.875
1998    CHA    8.29    0.600
1999    CHA    8.29    0.469
1999    MIN    8.31    0.469
2004    CHA    8.31    0.471
2002    CLE    8.34    0.313
2000    MIN    8.34    0.469
1997    CLE    8.38    0.536
2009    WAS    8.38    0.471
1997    UTA    8.41    0.250
2007    DET    8.41    0.706
2001    WAS    8.46    0.313
2001    MIA    8.47    0.625
2005    HOU    8.48    0.559
1999    PHO    8.50    0.469
2004    PHO    8.52    0.500
2001    SAC    8.54    0.625
2003    CON    8.54    0.529
1998    PHO    8.55    0.633
2005    IND    8.56    0.618
2003    SAC    8.58    0.559
2004    LAS    8.60    0.735
1998    CLE    8.61    0.667
1999    CLE    8.61    0.219
2004    IND    8.62    0.441
2005    SEA    8.63    0.588
2008    IND    8.65    0.500
2000    IND    8.65    0.281
2007    WAS    8.66    0.471
2003    CHA    8.70    0.529
2003    WAS    8.70    0.265
2004    NYL    8.71    0.529
2009    IND    8.71    0.647
1999    UTA    8.72    0.469
2002    IND    8.72    0.500
2002    SAC    8.72    0.438
1998    SAC    8.74    0.267
1997    LAS    8.76    0.500
2000    CHA    8.76    0.250
2000    POR    8.78    0.313
2006    WAS    8.78    0.529
2009    LAS    8.80    0.529
2003    SEA    8.83    0.529
2004    HOU    8.83    0.382
2005    LAS    8.84    0.500
2009    DET    8.87    0.529
2006    PHO    8.87    0.529
2000    PHO    8.90    0.625
2008    PHO    8.90    0.471
2009    PHO    8.92    0.676
2007    SEA    8.95    0.500
2003    IND    8.98    0.471
2003    CLE    8.98    0.500
2001    POR    9.00    0.344
2002    MIN    9.01    0.313
2006    MIN    9.02    0.294
2001    CLE    9.05    0.688
2008    CHI    9.07    0.353
1998    WAS    9.14    0.100
2004    SAC    9.14    0.529
2007    CON    9.19    0.529
2009    NYL    9.19    0.382
2006    NYL    9.20    0.324
2000    UTA    9.21    0.563
2002    MIA    9.22    0.469
2002    ORL    9.25    0.500
2001    MIN    9.26    0.375
2006    CON    9.28    0.765
2005    SAS    9.28    0.206
2000    CLE    9.29    0.531
2002    WAS    9.30    0.531
2001    SEA    9.31    0.313
2006    HOU    9.35    0.529
2007    IND    9.35    0.618
1998    LAS    9.38    0.400
2006    SAS    9.38    0.382
2006    CHA    9.39    0.324
2005    PHO    9.41    0.471
2007    MIN    9.47    0.294
2004    WAS    9.48    0.500
2003    PHO    9.48    0.235
2008    SEA    9.51    0.647
2005    SAC    9.52    0.735
2007    CHI    9.53    0.412
1999    LAS    9.53    0.625
2008    LAS    9.53    0.588
2008    WAS    9.54    0.294
2006    SEA    9.54    0.529
2008    DET    9.55    0.647
2009    SAS    9.55    0.441
2009    MIN    9.56    0.412
2007    SAC    9.56    0.559
2007    HOU    9.58    0.382
2000    DET    9.60    0.438
2002    SEA    9.64    0.531
2002    PHO    9.66    0.344
2008    MIN    9.73    0.471
2005    CHA    9.79    0.176
2000    MIA    9.82    0.406
2004    MIN    9.86    0.529
2009    CHI    9.88    0.471
2005    MIN    9.89    0.412
2002    DET    9.94    0.281
2002    POR    9.94    0.500
2006    LAS    9.95    0.735
2009    ATL    9.97    0.529
2008    HOU    9.97    0.500
2001    PHO    9.98    0.500
2008    SAC    10.05    0.529
2001    IND    10.09    0.313
2008    CON    10.10    0.618
2004    SAS    10.13    0.265
1998    UTA    10.15    0.267
2005    DET    10.17    0.471
2006    SAC    10.18    0.618
2008    NYL    10.18    0.559
2001    DET    10.21    0.313
2009    SAC    10.29    0.353
1999    DET    10.29    0.469
2009    CON    10.62    0.471
2007    LAS    10.70    0.294
2006    CHI    10.87    0.147
2008    ATL    10.93    0.118
2000    SEA    11.35    0.188