NBA Eigenvalues

As a basketball fan who is also interested in math, the lack of symmetry of the NBA regular season schedule has always been a source of small annoyance. While the league has 30 teams, each team plays 82 games (ignoring the recent in season tournament). This combination makes it impossible to play each of the other teams an equal number of times. Instead the games are allocated as follows:

A team will play teams in the opposing conference 2 times per season.

A team will play teams within their own division 4 times per season.

A team will play the remaining teams within their own conference either 3 or 4 times depending on the season.

This imbalance means that in a given season a team may have an effectively harder or easier schedule depending on the strength of their conference and division. As a fan of the Denver Nuggets, a member of the consistently superior Western Conference (and perhaps surprisingly consistent Northwest Division), this is rather annoying. Why should the Boston Celtics receive the same benefit (in terms of wins) by defeating the hapless Brooklyn Nets as the Nuggets get by winning a bloodbath with the Timberwolves?

Fun Fact! Each division winner receives a unique trophy. The Northwest Division winner receives the “Sam Jones Trophy”.

This got me thinking, wouldn’t it be nice if the value of a win was somehow proportional the quality of the teams record. We can make the value of a win against each team proportional to their record but now that results in a new record. So the question becomes, can we make the value of winning against each team such that a win value is exactly proportional to the resulting win counts of each team?

It turns out this is exactly the structure of an eigenvector problem. If we construct a matrix of the season’s results \(A\), such that the \(i, j^{th}\) entry is the number of times the \(i^{th}\) team beat the \(j^{th}\) team. Then we seek the eigenvector \(\boldsymbol{u}\):

\[A\boldsymbol{u} = \lambda \boldsymbol{u}\]

This means that when the win matrix \(A\) is multiplied by the win value vector \(\boldsymbol{u}\), we get a vector of win counts exactly proportional to the win values. We can solve this system easily with any eigenvector solver (I used NumPy). The scale of \(\boldsymbol{u}\) we get from the solver won’t make much sense in comparison to the value of a win we are used to (aka 1), so we scale things so the total number of wins is what it would be in a normal season (\(15 \cdot 82 = 1230\)). You can see the adjusted win values in the below table. You can also see the effective win total using the new win values (an “unskewed” schedule result so to speak) and how that would have affected the location of the team in the overall standings.

What are the big takeaways for the 2023-2024 NBA season?

(Note, these values don’t exactly reflect the tiebreaker scenarios the NBA employs for teams with the same records. So it is possible the ranking changes can be a little off depending on how tied teams got randomly sorted.)

The best teams are still the best. Boston stays at the top of the standings. Despite being from the weaker conference Boston simply won too many games to fall in the standings.
The worst teams are still the worst. But this approach gives a good way to quantify just how much worse. In this scenario, beating the lowly Detroit Pistons would be worth less than a third of a win - beating Boston is worth nearly 2 wins! i find it pretty amusing to imagine a league that actually operated this way, with teams going all out to beat the Boston for the extra wins while Detroit is utterly ignored (even more than usual).
While the top and bottom stayed stable, there is substantial movement in the middle. Sacramento is the biggest climber and moved up 8 spots due to their 4 additional wins. Seems like they were beating good teams. Meanwhile, New York was the biggest faller by ranking - it would appear a lot of their wins came from poor competition. Interestingly the Mavericks also had a big fall despite their eventual appearance in the conference finals. But their win count didn’t move much, so that change is mostly due to the tight race in the middle of the West standings and getting hopped by the Suns, Pelicans, and Kings.
The West definitely fares better in this exercise - in total the west moved up 14 spots relative to the eastern teams.
This approach only considers wins. There is no additional punishment for losing to a “bad” team.

I like that this adjustment takes place on an individual team level - as opposed to a conference wide adjustment. It would be fun if there was a particular “giant killer” one year who beat all the best teams despite an overall poor record.

Big shoutout to the nba_api library. I found it surprisingly difficult to find a convenient source of nba records.

Win Value	Team	Adjusted Win Count	Original Win Count	Ranking Change
1.77	Boston Celtics	64.67	64	0
1.68	Oklahoma City Thunder	61.22	57	1
1.62	Minnesota Timberwolves	59.01	56	1
1.62	Denver Nuggets	58.87	57	-2
1.43	LA Clippers	52.13	51	0
1.42	Phoenix Suns	51.79	49	3
1.40	New Orleans Pelicans	51.18	49	1
1.40	Milwaukee Bucks	50.91	49	2
1.38	Sacramento Kings	50.28	46	8
1.38	Dallas Mavericks	50.11	50	-4
1.37	Indiana Pacers	50.01	47	3
1.37	Los Angeles Lakers	49.81	47	1
1.30	New York Knicks	47.31	50	-6
1.27	Cleveland Cavaliers	46.26	48	-3
1.23	Golden State Warriors	44.93	46	3
1.21	Philadelphia 76ers	44.23	47	-4
1.21	Orlando Magic	44.13	47	-2
1.15	Houston Rockets	41.78	41	1
1.14	Miami Heat	41.38	46	-3
1.01	Chicago Bulls	36.89	39	0
0.99	Atlanta Hawks	36.17	36	0
0.82	Utah Jazz	29.98	31	1
0.81	Brooklyn Nets	29.52	32	-1
0.74	Memphis Grizzlies	26.85	27	0
0.66	Toronto Raptors	23.96	25	0
0.62	San Antonio Spurs	22.63	22	0
0.57	Charlotte Hornets	20.66	21	0
0.53	Portland Trail Blazers	19.38	21	0
0.34	Washington Wizards	12.41	15	0
0.32	Detroit Pistons	11.50	14	0