A mathematical power ranking of all 30 NBA teams

I have read almost all of Steve Perrin's ruminations on the Clippers for years--even before this site was a part of SBNation.  Having enjoyed his writings for so long, I figured I should contribute to the site as a small gesture of appreciation for the time Steve (and others) have dedicated to Clipsnation.

Over the years I noticed Steve's writings had the hallmark analytical reasoning of a mathematical mind.  So, it was not a complete surprise to read (if I recall correctly) that Steve was a mathematics major in college and a future mathematics teacher.  I am a mathematics professor at a large university in California and also a Clippers fan (and now that I have tenure I can freely post this online without fear of a Lakers fan on my tenure committee).  In this post I will describe a mathematical way to rank all 30 NBA teams. 

This power ranking will be based only on the outcomes of games played and not any touchy feel-ly emotions or intuition.  The only idea I will use is this: a team should be ranked high if the team has a good chance of beating highly ranked teams.  This idea alone can give a mathematically precise power ranking.

I was thinking about this problem because the Clippers have a 15-25 record but have beaten good Hornets, Spurs, Heat, and Lakers teams.  Are the Clippers better than their record indicates?  How can this be measured?  Between the *'s below I will describe exactly how I calculated the rankings (this part may be skipped if you have an aversion to mathematics):


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Write down the 30 NBA teams in alphabetical order and let r_1 be the ranking of team 1, r_2 be the ranking of team 2, etc.  Let p_{i,j} be the probability that team i will beat team j based on games played this season.  The power ranking we are looking for satisfies

r_i = k ( p_{i,1} r_1 + p_{i,2} r_2 + ... + p_{i,30} r_30 )

for some positive constant k.  In words, the power ranking of a team is proportional to the rankings of the teams it can beat. 

Let P be the 30 by 30 matrix with row i and column j entry given by p_{i,j} and let r be the column vector with i^th entry r_i.  Writing our power ranking condition as a matrix multiplication, we have r = k P r.  Rewriting this, we have P r = (1/k) r.  Solving the matrix equation P r = (1/k) r for the constant (1/k) and the vector r is called the eigenvalue/eigenvector problem.  Techniques to solve this problem are given in sophomore matrix algebra courses taken by mathematics/engineering students. 

A similar ranking scheme was used in the infancy of google.com to determine the order in which to display search results (since then, google has spent millions of man hours refining their ranking system).

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After doing all of the calculations (and scaling the rankings to give the top team a ranking of 1), here are the power rankings of all NBA teams as of Jan. 17.  The closer a team's ranking is to 1, the closer they are to being the best team in the NBA.

Rankings using games played up to 2011-01-17:
1.000 SAS
0.896 BOS
0.857 DAL
0.800 CHI
0.784 OKC
0.780 MIA
0.770 UTA
0.767 LAL
0.761 NOH
0.733 ORL
0.724 DEN
0.656 NYK
0.633 ATL
0.633 PHO
0.565 PHI
0.553 MEM
0.540 HOU
0.524 POR
0.521 IND
0.502 LAC
0.481 GSW
0.471 MIL
0.450 TOR
0.447 CHA
0.436 DET
0.413 WAS
0.317 MIN
0.312 NJN
0.300 SAC
0.273 CLE

In this power ranking, the Clippers are ranked above the Warriors, even though the Clippers are 15-25 and the Warriors are 17-23.  This is because the Clippers have better quality wins than Golden State.  

If you're interested in more details or if you're interested in seeing how these rankings have evolved over the course of this NBA season, please let me know!