## Ranking NBA Championship Teams

The first thing to note is that just by looking at Basketball-Reference.com there are 62 factors that uniquely classify a team: MP FG FGA FG% 3P 3PA 3P% 2P 2PA 2P% FT FTA FT% ORB DRB TRB AST STL BLK TOV PF PTS OMP OFG OFGA OFG% O3P O3PA O3P% O2P O2PA O2P% OFT OFTA OFT% OORB ODRB OTRB OAST OSTL OBLK OTOV OPF OPTS PW PL MOV SOS SRS ORtg DRtg Pace FTr 3PAr eFG% TOV% ORB% FT/FGA eFG% TOV% DRB% FT/FGA, where OFGA indicates a given team’s opponent’s FGA per game average for a specific season.
The reason it is not meaningful to look at a specific statistic or a pair of statistics such as “three-point attempt rate” is that,

$\boxed{\frac{62!}{2! 60!} = 1891}$ possible comparisons can be made.

Because of this, what is required is a detailed statistic learning approach. I looked at the full season statistics for the last twenty NBA champions from the 1995-1996 Chicago Bulls to the 2014-2015 Golden State Warriors.

I employed principle compoent analysis (PCA) to reduce the number of dimensions to see which variables contribute most to the variance of the data set. I found that the first 7 of 20 principle compoents explained 88.52% of the variance. Therefore, we can effectively reduce the dimension of the data set from 63 to 7.  This can be seen in the scree plot below:

A visualization of the 63-variable data set is as follows:

The power of principle components analysis reduced this high-dimensional dataset to a more manageable (but, perhaps still complicated) 7-dimensional data set, visualized as follows:

Next, I computed the Euclidean distance metric to perform hierarchical clustering on these seven principle components. I obtained the following result:

We notice immediately that:

1. The 2015 Golden State Warriors were very similar to the 2014 San Antonio Spurs.
2. Not surprisingly, Phil Jackson’s 2000 and 2002 Lakers teams were very similar to each other but not to any other championship team, and similarly for his 2009 and 2010 Lakers teams.
3. Interestingly, the two teams that stand out which are truly dissimilar to any other championship team are the 2008 Boston Celtics and the 1998 Chicago Bulls.

This analysis also eliminates the notion that a team has to play a specific style, for example “modern-day play” to win a championship. In principle, there are many possible ways and styles that lead to a championship and an analysis such as this deeply probing the data shows this to be the case.

## The “Evolution” of the 3-Point Shot in The NBA

The purpose of this post is to determine whether basketball teams who choose to employ an offensive strategy that involves predominantly shooting three point shots is stable and optimal. We employ a game-theoretical approach using techniques from dynamical systems theory to show that taking more three point shots to a point where an offensive strategy is dependent on predominantly shooting threes is not necessarily optimal, and depends on a combination of payoff constraints, where one can establish conditions via the global stability of equilibrium points in addition to Nash equilibria where a predominant two-point offensive strategy would be optimal as well. We perform a detailed fixed-points analysis to establish the local stability of a given offensive strategy. We finally prove the existence of Nash equilibria via global stability techniques via the monotonicity principle. We believe that this work demonstrates that the concept that teams should attempt more three-point shots because a three-point shot is worth more than a two-point shot is therefore, a highly ambiguous statement.



### 1. Introduction

We are currently living in the age of analytics in professional sports, with a strong trend of their use developing in professional basketball. Indeed, perhaps, one of the most discussed results to come out of the analytics era thus far is the claim that teams should shoot as many three-point shots as possible, largely because, three-point shots are worth more than two-point shots, and this somehow is indicative of a very efficient offense. These ideas were mentioned for example by Alex Rucker who said “When you ask coaches what’s better between a 28 percent three-point shot and a 42 percent midrange shot, they’ll say the 42 percent shot. And that’s objectively false. It’s wrong. If LeBron James just jacked a three on every single possession, that’d be an exceptionally good offense. That’s a conversation we’ve had with our coaching staff, and let’s just say they don’t support that approach.” It was also claimed in the same article that “The analytics team is unanimous, and rather emphatic, that every team should shoot more 3s including the Raptors and even the Rockets, who are on pace to break the NBA record for most 3-point attempts in a season.” These assertions were repeated here. In an article by John Schuhmann, it was claimed that “It’s simple math. A made three is worth 1.5 times a made two. So you don’t have to be a great 3-point shooter to make those shots worth a lot more than a jumper from inside the arc. In fact, if you’re not shooting a layup, you might as well be beyond the 3-point line. Last season, the league made 39.4 percent of shots between the restricted area and the arc, for a value of 0.79 points per shot. It made 36.0 percent of threes, for a value of 1.08 points per shot.” The purpose of this paper is to determine whether basketball teams who choose to employ an offensive strategy that involves predominantly shooting three point shots is stable and optimal. We will employ a game-theoretical approach using techniques from dynamical systems theory to show that taking more three point shots to a point where an offensive strategy is dependent on predominantly shooting threes is not necessarily optimal, and depends on a combination of payoff constraints, where one can establish conditions via the global stability of equilibrium points in addition to Nash equilibria where a predominant two-point offensive strategy would be optimal as well. (Article research and other statistics provided by: Hargun Singh Kohli)

### 2. The Dynamical Equations

For our model, we consider two types of NBA teams. The first type are teams that employ two point shots as the predominant part of their offensive strategy, while the other type consists of teams that employ three-point shots as the predominant part of their offensive strategy. There are therefore two predominant strategies, which we will denote as ${s_{1}, s_{2}}$, such that we define

$\displaystyle \mathbf{S} = \left\{s_{1}, s_{2}\right\}. \ \ \ \ \ (1)$

We then let ${n_{i}}$ represent the number of teams using ${s_{i}}$, such that the total number of teams in the league is given by

$\displaystyle N = \sum_{i =1}^{k} n_{i}, \ \ \ \ \ (2)$

which implies that the proportion of teams using strategy ${s_{i}}$ is given by

$\displaystyle x_i = \frac{n_{i}}{N}. \ \ \ \ \ (3)$

The state of the population of teams is then represented by ${\mathbf{x} = (x_{1}, \ldots, x_{k})}$. It can be shown that the proportions of individuals using a certain strategy change in time according to the following dynamical system

$\displaystyle \dot{x}_{i} = x_{i}\left[\pi(s_{i}, \mathbf{x}) - \bar{\pi}(\mathbf{x})\right], \ \ \ \ \ (4)$

subject to

$\displaystyle \sum_{i =1}^{k} x_{i} = 1, \ \ \ \ \ (5)$

where we have defined the average payoff function as

$\displaystyle \bar{\pi}(\mathbf{x}) = \sum_{i=1}^{k} x_{i} \pi(s_{i}, \mathbf{x}). \ \ \ \ \ (6)$

Now, let ${x_{1}}$ represent the proportion of teams that predominantly shoot two-point shots, and let ${x_{2}}$ represent the proportion of teams that predominantly shoot three-point shots. Further, denoting the game action set to be ${A = \left\{T, Th\right\}}$, where ${T}$ represents a predominant two-point shot strategy, and ${Th}$ represents a predominant three-point shot strategy. As such, we assign the following payoffs:

$\displaystyle \pi(T,T) = \alpha, \quad \pi(T,Th) = \beta, \quad \pi(Th, T) = \gamma, \quad \pi(Th,Th) = \delta. \ \ \ \ \ (7)$

We therefore have that

$\displaystyle \pi(T,\mathbf{x}) = \alpha x_{1} + \beta x_{2}, \quad \pi(Th, \mathbf{x}) = \gamma x_{1} + \delta x_{2}. \ \ \ \ \ (8)$

From (6), we further have that

$\displaystyle \bar{\pi}(\mathbf{x}) = x_{1} \left( \alpha x_{1} + \beta x_{2}\right) + x_{2} \left(\gamma x_{1} + \delta x_{2}\right). \ \ \ \ \ (9)$

From Eq. (4) the dynamical system is then given by

$\boxed{\dot{x}_{1} = x_{1} \left\{ \left(\alpha x_{1} + \beta x_{2} \right) - x_{1} \left( \alpha x_{1} + \beta x_{2}\right) - x_{2} \left(\gamma x_{1} + \delta x_{2}\right) \right\}}$,

$\boxed{\dot{x}_{2} = x_{2} \left\{ \left( \gamma x_{1} + \delta x_{2}\right) -x_{1} \left( \alpha x_{1} + \beta x_{2}\right) - x_{2} \left(\gamma x_{1} + \delta x_{2}\right) \right\}}$,

subject to the constraint

$\displaystyle x_{1} + x_{2} = 1. \ \ \ \ \ (10)$

Indeed, because of the constraint (10), the dynamical system is actually one-dimensional, which we write in terms of ${x_{1}}$ as

$\displaystyle \boxed{\dot{x}_{1} = x_{1} \left(-1 + x_{1}\right) \left[\delta + \beta \left(-1 + x_{1}\right) - \delta x_{1} + \left(\gamma-\alpha\right)x_{1}\right]}. \ \ \ \ \ (11)$

From Eq. (11), we immediately notice some things of importance. First, we are able to deduce just from the form of the equation what the invariant sets are. We note that for a dynamical system ${\mathbf{x}' = \mathbf{f(x)} \in \mathbf{R^{n}}}$ with flow ${\phi_{t}}$, if we define a ${C^{1}}$ function ${Z: \mathbf{R}^{n} \rightarrow \mathbf{R}}$ such that ${Z' = \alpha Z}$, where ${\alpha: \mathbf{R}^{n} \rightarrow \mathbf{R}}$, then, the subsets of ${\mathbf{R}^{n}}$ defined by ${Z > 0, Z = 0}$, and ${Z < 0}$ are invariant sets of the flow ${\phi_{t}}$. Applying this notion to Eq. (11), one immediately sees that ${x_1 > 0}$, ${x_1 = 0}$, and ${x_1 < 0}$ are invariant sets of the corresponding flow. Further, there also exists a symmetry such that ${x_{1} \rightarrow -x_{1}}$, which implies that without loss of generality, we can restrict our attention to ${x_{1} \geq 0}$.

### 3. Fixed-Points Analysis

With the dynamical system in hand, we are now in a position to perform a fixed-points analysis. There are precisely three fixed points, which are invariant manifolds and are given by:

$\displaystyle P_{1}: x_{1}^{*} = 0, \quad P_{2}: x_{1}^{*} = 1, \quad P_{3}: x_{1}^{*} = \frac{\beta - \delta}{-\alpha + \beta - \delta + \gamma}. \ \ \ \ \ (12)$

Note that, ${P_{3}}$ actually contains ${P_{1}}$ and ${P_{2}}$ as special cases. Namely, when ${\beta = \delta}$, ${P_{3} = 0 = P_{1}}$, and when ${\alpha = \gamma}$, ${P_{3} = 1 = P_{2}}$. We will therefore just analyze, the stability of ${P_{3}}$. ${P_{3} = 0}$ represents a state of the population where all teams predominantly shoot three-point shots. Similarly, ${P_{3} = 1}$ represents a state of the population where all teams predominantly shoot two-point shots, We additionally restrict

$\displaystyle 0 \leq P_{3} \leq 1 \Rightarrow 0 \leq \frac{\beta - \delta}{-\alpha + \beta - \delta + \gamma} \leq 1, \ \ \ \ \ (13)$

which implies the following conditions on the payoffs:

$\displaystyle \left[\delta < \beta \cap \gamma \leq \alpha \right] \cup \left[\delta = \beta \cap \left(\gamma < \alpha \cup \gamma > \alpha \right) \right] \cup \left[\delta > \beta \cap \gamma \leq \alpha \right]. \ \ \ \ \ (14)$

With respect to a stability analysis of ${P_{3}}$, we note the following. The point ${P_{3}}$ is a: • Local sink if: ${\{\delta < \beta\} \cap \{\gamma > \alpha\}}$, • Source if: ${\{\delta > \beta\} \cap \{\gamma < \alpha\}}$, • Saddle: if: ${\{\delta = \beta \} \cap (\gamma < \alpha -\beta + \delta \cup \gamma > \alpha - \beta + \delta)}$, or ${(\{\delta < \beta\} \cup \{\delta > \beta\}) \cap \gamma = \frac{\alpha \delta - \alpha \beta}{\delta - \beta}}$.

What this last calculation shows is that the condition $\delta = \beta$ which always corresponds to the point $x_{1}^* = 0$, which corresponds to a dominant 3-point strategy always exists as a saddle point! That is, there will NEVER be a league that dominantly adopts a three-point strategy, at best, some teams will go towards a 3-point strategy, and others will not irrespective of what the analytics people say. This also shows that a team's basketball strategy really should depend on its respective payoffs, and not current "trends". This behaviour is displayed in the following plot.



Further, the system exhibits some bifurcations as well. In the neigbourhood of ${P_{3} = 0}$, the linearized system takes the form

$\displaystyle x_{1}' = \beta - \delta. \ \ \ \ \ (15)$

Therefore, ${P_{3} = 0}$ destabilizes the system at ${\beta = \delta}$. Similarly, ${P_{3} = 1}$ destabilizes the system at ${\gamma = \alpha}$. Therefore, bifurcations of the system occur on the lines ${\gamma = \alpha}$ and ${\beta = \delta}$ in the four-dimensional parameter space.

### 4. Global Stability and The Existence of Nash Equilibria

With the preceding fixed-points analysis completed, we are now interested in determining global stability conditions. The main motivation is to determine the existence of any Nash equilibria that occur for this game via the following theorem: If ${\mathbf{x}^{*}}$ is an asymptotically stable fixed point, then the symmetric strategy pair ${[\sigma^{*}, \sigma^{*}]}$, with ${\sigma^{*} = \mathbf{x}^*}$ is a Nash equilibrium. We will primarily make use of the monotonicity principle, which says let ${\phi_{t}}$ be a flow on ${\mathbb{R}^{n}}$ with ${S}$ an invariant set. Let ${Z: S \rightarrow \mathbb{R}}$ be a ${C^{1}}$ function whose range is the interval ${(a,b)}$, where ${a \in \mathbb{R} \cup \{-\infty\}, b \in \mathbb{R} \cup \{\infty\}}$, and ${a < b}$. If ${Z}$ is decreasing on orbits in ${S}$, then for all ${\mathbf{x} \in S}$,

$\boxed{\omega(\mathbf{x}) \subseteq \left\{\mathbf{s} \in \partial S | \lim_{\mathbf{y} \rightarrow \mathbf{s}} Z(\mathbf{y}) \neq \mathbf{b}\right\}}$,

$\boxed{ \alpha(\mathbf{x}) \subseteq \left\{\mathbf{s} \in \partial S | \lim_{\mathbf{y} \rightarrow \mathbf{s}} Z(\mathbf{y}) \neq \mathbf{a}\right\}}$.

Consider the function

$\displaystyle Z_{1} = \log \left(-1 + x_{1}\right). \ \ \ \ \ (16)$

Then, we have that

$\displaystyle \dot{Z}_{1}= x_{1} \left[\delta + \beta \left(-1 + x_{1}\right) - \delta x_{1} + x_{1} \left(\gamma - \alpha\right)\right]. \ \ \ \ \ (17)$

For the invariant set ${S_1 = \{0 < x_{1} < 1\}}$, we have that ${\partial S_{1} = \{x_{1} = 0\} \cup \{x_{1} = 1\}}$. One can then immediately see that in ${S_{1}}$,

$\displaystyle \dot{Z}_{1} < 0 \Leftrightarrow \left\{\beta > \delta\right\} \cap \left\{\alpha \geq \gamma\right\}. \ \ \ \ \ (18)$

Therefore, by the monotonicity principle,

$\displaystyle \omega(\mathbf{x}) \subseteq \left\{\mathbf{x}: x_{1} = 1 \right\}. \ \ \ \ \ (19)$

Note that the conditions ${\beta > \delta}$ and ${\alpha \geq \gamma}$ correspond to ${P_{3}}$ above. In particular, for ${\alpha = \gamma}$, ${P_{3} = 1}$, which implies that ${x_{1}^{*} = 1}$ is globally stable. Therefore, under these conditions, the symmetric strategy ${[1,1]}$ is a Nash equilibrium. Now, consider the function

$\displaystyle Z_{2} = \log \left(x_{1}\right). \ \ \ \ \ (20)$

We can therefore see that

$\displaystyle \dot{Z}_{2} = \left[-1 + x_{1}\right] \left[\delta + \beta\left(-1+x_{1}\right) - \delta x_{1} + \left(-\alpha + \gamma\right) x_{1}\right]. \ \ \ \ \ (21)$

Clearly, ${\dot{Z}_{2} < 0}$ in ${S_{1}}$ if for example ${\beta = \delta}$ and ${\alpha < \gamma}$. Then, by the monotonicity principle, we obtain that

$\displaystyle \omega(\mathbf{x}) \subseteq \left\{\mathbf{x}: x_{1} = 0 \right\}. \ \ \ \ \ (22)$

Note that the conditions ${\beta = \delta}$ and ${\alpha < \gamma}$ correspond to ${P_{3}}$ above. In particular, for ${\beta = \delta}$, ${P_{3} = 0}$, which implies that ${x_{1}^{*} = 0}$ is globally stable. Therefore, under these conditions, the symmetric strategy ${[0,0]}$ is a Nash equilibrium. In summary, we have just shown that for the specific case where ${\beta > \delta}$ and ${\alpha = \gamma}$, the strategy ${[1,1]}$ is a Nash equilibrium. On the other hand, for the specific case where ${\beta = \delta}$ and ${\alpha < \gamma}$, the strategy ${[0,0]}$ is a Nash equilibrium. 5. Discussion In the previous section which describes global results, we first concluded that for the case where ${\beta > \delta}$ and ${\alpha = \gamma}$, the strategy ${[1,1]}$ is a Nash equilibrium. The relevance of this is as follows. The condition on the payoffs thus requires that

$\displaystyle \pi(T,T) = \pi(Th,T), \quad \pi(T,Th) > \pi(Th,Th). \ \ \ \ \ (23)$

That is, given the strategy adopted by the other team, neither team could increase their payoff by adopting another strategy if and only if the condition in (23) is satisfied. Given these conditions, if one team has a predominant two-point strategy, it would be the other team’s best response to also use a predominant two-point strategy. We also concluded that for the case where ${\beta = \delta}$ and ${\alpha < \gamma}$, the strategy ${[0,0]}$ is a Nash equilibrium. The relevance of this is as follows. The condition on the payoffs thus requires that

$\displaystyle \pi(T,Th) = \pi(Th,Th), \quad \pi(T,T) < \pi(Th,T). \ \ \ \ \ (24)$

That is, given the strategy adopted by the other team, neither team could increase their payoff by adopting another strategy if and only if the condition in (24) is satisfied. Given these conditions, if one team has a predominant three-point strategy, it would be the other team’s best response to also use a predominant three-point strategy. Further, we also showed that ${x_{1} = 1}$ is globally stable under the conditions in (23). That is, if these conditions hold, every team in the NBA will eventually adopt an offensive strategy predominantly consisting of two-point shots. The conditions in (24) were shown to imply that the point ${x_{1} = 0}$ is globally stable. This means that if these conditions now hold, every team in the NBA will eventually adopt an offensive strategy predominantly consisting of three-point shots. We also provided through a careful stability analysis of the fixed points criteria for the local stability of strategies. For example, we showed that a predominant three-point strategy is locally stable if ${\pi(T,Th) - \pi(Th,Th) < 0}$, while it is unstable if ${\pi(T,Th) - \pi(Th,Th) \geq 0}$. In addition, a predominant two-point strategy was found to be locally stable when ${\pi(Th,T) - \pi(T,T) < 0}$, and unstable when ${\pi(Th,T) - \pi(T,T) \geq 0}$. There is also they key point of which one of these strategies has the highest probability of being executed. We know that

$\displaystyle \pi(\sigma,\mathbf{x}) = \sum_{s \in \mathbf{S}} \sum_{s' \in \mathbf{S}} p(s) x(s') \pi(s,s'). \ \ \ \ \ (25)$

That is, the payoff to a team using strategy ${\sigma}$ in a league with profile ${\mathbf{x}}$ is proportional to the probability of this team using strategy ${s \in \mathbf{S}}$. We therefore see that a team’s optimal strategy would be that for which they could maximize their payoff, that is, for which ${p(s)}$ is a maximum, while keeping in mind the strategy of the other team, hence, the existence of Nash equilibria. Hopefully, this work also shows that the concept that teams should attempt more three-point shots because a three-point shot is worth more than a two-point shot is a highly ambiguous statement. In actuality, one needs to analyze what offensive strategy is optimal which is constrained by a particular set of payoffs.

## Some Thoughts On Howard Beck’s Bleacher Report Article

Howard Beck had an interesting article today on Bleacher Report, basically suggesting that the NBA finals, in particular, the current style of play embodied by The Golden State Warriors is somehow a vindication of D’Antoni’s basketball philosophies: “Shoot a lot of threes”, “Shoot in 7 seconds or less”, “Play small lineups”, etc…

While the Warriors have certainly embodied some of these philosophies, my personal opinion is that D’Antoni’s style of play can only be vindicated if there is a clear trend in championship teams that reflect these philosophies. As I show below, this is simply not the case.

I looked at the last 15 NBA Champions (from 2000-2014), and tried to see if there were any clear patterns in common between the teams. This is essentially what I found:

Two things that are immediately clear are:

1. There is very little that championship teams have in common!

2. The overwhelming thing that they do have in common is that 14 of the last 15 NBA champions have all been ranked in the Top 10 for Defensive Rating, something that Mike D’Antoni’s coaching philosophy has never really included throughout his years in Phoenix, New York, and Los Angeles.

This, I believe is the grand point that no one seems to be interested in making, perhaps, because according to the “mainstream”, defensive-oriented basketball, which, by definition is “less-flashy” still is the overwhelming common characteristic amongst championship-winning teams.

Perhaps, the Warriors will win this year, but as I said above, I do not believe that one year is anywhere near enough to establish a trend and a vindication of D’Antoni’s basketball philosophies.

Further, there were some other things in Beck’s article that I found to be a bit concerning:

He claimed Today, coaches speak enthusiastically about “positionless” basketball—whereas 10 years ago, D’Antoni had to sell Marion and Stoudemire on the concept.”

This is not actually true. The triangle offense is the de facto example of “positionless” basketball, and has been around since the 1940s when Sam Barry introduced it at USC. Phil Jackson and Tex Winter’s Bulls and Lakers teams embodied the concept of positionless basketball. In fact, as can be seen from the diagram below (taken from http://khamel83.tripod.com/intro.htm), players don’t have set positions in the triangle offense. Rather, there are regions based on optimality and spacing:

Many examples can be found from teams playing in the triangle offense system of guards posting up, big men coming out to shoot threes, etc…