## How to Beat the Golden State Warriors

The Golden State Warriors have posed quite the conundrum for opposing teams. They are quick, have a spectacular ability to move the ball, and play suffocating defense. Given their play in the playoffs thus far, all of these points have been exemplified even more to the point where it seems that they are unbeatable.

I wanted to take somewhat of a simplified approach and see if opposing teams are missing something. That is, is their some weakness in their play that opposing teams can exploit, a “weakness in Helm’s deep”?

The most obvious place to start from a data science point-of-view seemed to me to look at every single shot the Warriors took as a team this season in each game and compile a grand ensemble shot chart. Using the data from Basketball-reference.com and some data scraping scripts I wrote in R, I obtained the following:

Certainly, on the surface, it seems that there is no discernible pattern between made shots and missed shots. This is where the machine learning comes in!

From here, I now extracted the x and y coordinates of each shot and recorded a response variable of “made” or “missed” in a table, such that the coordinates were now predictor variables and the shot classification (made/missed) was the response variable. Altogether, we had 7104 observations. Splitting this dataset up into a 70% training dataset and a 30% test data set, I tried the following algorithms, recording the % of correctly classified observations:

 Algorithm % of Correctly Predicted Observations Logistic Regression 56.43 Gradient Boosted Decision Trees 62.62 Random Forests 58.54 Neural Networks with Entropy Fitting 62.47 Naive Bayes Classification with Kernel Density Estimation 57.32

One sees that that gradient boosted decision trees had the best performance correctly classifying 62.62% of the test observations. Given how noisy the data is, this is not bad, and much better than expected. I should also mention that these numbers were obtained after tuning these models using cross-validation for optimal parameters.

Using the gradient boosted decision tree model, we made a set of predictions for a vast number of (x,y)-coordinates for basketball court. We obtained the following contour plot:

Overlaying this on top of the basketball court diagram, we got:

The contour plot levels denote the probabilities that the GSW will make a shot from a given (x,y) location on the court. As a sanity check, the lowest probabilities seem to be close to the 1/2-court line and beyond the three-point line. The highest probabilities are surprisingly along very specific areas on the court: very close the basket, the line from the basket to the left corner, extending up slightly, and a very narrow line extending from the basket to the right corner. Interestingly, the probabilities are low on the right side of the basket, specifically:

A map showing the probabilities more explicitly is as follows (although, upon uploading it, I realized it is a bit harder to read, I will re-upload a clearer version soon!)

In conclusion, it seems that, at least according to a first look at the data, the Warriors do indeed have several “weak spots” in their offense that opponents should certainly look to exploit by designing defensive schemes that force them to take shots in the aforementioned low-probability zones. As for future improvements, I think it would be interesting to add as predictor variables things like geographic location, crowd sizes, team opponent strengths, etc… I will look into making these improvements in the near future.

## Game 2 of CLE vs GSW Breakdown

As usual, here is the post-game breakdown of Game 2 of the NBA Finals between Cleveland and Golden State. Using my live-tracking app to track the relevant factors (as explained in previous posts) here are the live-captured time series:

Computing the correlations between each time series above and the Golden State Warriors point difference, we obtain:

One sees once again that the most relevant factors to GSW’s point difference in the game was CLE’s personal fouls during the game, GSW’s personal fouls during the game, and not far behind, GSW 3-point percentage during the game. What is interesting is that one can see the importance of these variables played out in real time matching the two graphs above.

In fact, looking at the personal fouls vs. GSW point difference in real time (essentially taking a subset of the time series graph above), we obtain:

## Analyzing Stephen Curry’s Play

As a long-time Golden State Warriors fan (go Tim Hardaway and Chris Mullin!), I have been watching the Warriors this season with great interest.

Stephen Curry has been getting a lot of attention. It is somewhat of a foregone conclusion that he will be the MVP this season, but, I am not completely convinced, in the sense that watching his play, he gets many open looks throughout the process of a game.

I was therefore interested in analyzing his FG% has a function of his shot distance from the basket and the distance of the closest defender on the court.

The NBA has made completing such an analysis somewhat easy with all of its new analytics tools like Shot Tracking but analyzing this question has proven difficult, because the trackers have not measured FG% as a function of two variables, rather, they have produced this statistic as function of each individual variable. One therefore ends up with a table of data as follows:

 FG% Distance from Basket (> 10 ft) Closest Defender Distance 1 56.5 10 NA 2 39.0 15 NA 3 46.9 20 NA 4 46.0 25 NA 5 60.0 30 NA 6 50.0 35 NA 7 36.4 40 NA 8 32.5 NA 0 9 42.4 NA 2 10 50.6 NA 4 11 47.8 NA 6

The “NA” values are the missing values as a result of not having the complete 3D set of data available.

The only way I could see to alleviate this problem was to perform some type of interpolation .

This way, I was able to perform the following surface regression:

This regression to the interpolated data points had an R^2 value of: 0.99, so the fit actually was very good.

The actual function for this surface was found to be:

where $d$ denotes the closest defender distance, and $y$ denotes the distance from the basket for shots greater than 10 feet.

Using this function and tools from multivariable calculus, we are able to conclude that:

Min FG% = 38.164% at d = 1, y = 15

That is, Stephen Curry is expected to have his lowest field goal percentage with the closest defender within 1 foot of him while being within 15 feet of the basket. Certainly, looking at the plot above, we see that his FG% increases as defenders are further and further away.

This can be also seen from the following contour plot obtained from computing the gradient of $FG(d,y)$ above:

What about trends? Well, computing the gradient of $FG(d,y)$, we find that:

$\nabla FG = (-6.813-0.6808d+1.0284d^2 + 0.9175 y - 0.3068 d y)\hat{d} + (-0.6783 + 0.9175d - 0.1534d^2)\hat{y}$

The charm of this is that we can now use methods of dynamical systems theory to obtain information about the trends! The vector field $\nabla FG$ is defined on the manifold $\mathbb{R}^2$ in the sense that it is a mapping: $\mathbb{R}^2 \to T\mathbb{R}^2$ that assigns to each point $m \in \mathbb{R}^2$ a vector in $T_{m} \mathbb{R}^2$. We can also interpret this vector field as the right-hand side of a system of first-order autonomous differential equations.

Motivated by this, we see that the fixed points are thus found to be:

$(d_1,y_1) = (0.864142, 10.1679)$ and $(d_2,y_2) = (5.11695,25.4915)$

Evaluating the Jacobian matrix in a neigbourhood of $(d_1,y_1)$ we find that the eigenvalues corresponding to this point are: $\lambda_1 = -2.21508, \lambda_{2} = 0.192138$. That is, the first point is a saddle point. Similarly, the eigenvalues of the second point are found to be: $\lambda_1 = 2.21509, \lambda_{2} = -0.192137$, which implies that this point is also another saddle point.

So, in terms of trends, there certainly exist orbits where Stephen Curry tends to shoot away from defenders while also keeping a distance of more than 25 feet from the basket. There also exists orbits where he does the opposite. However, the following vector field plot is very illuminating in terms of displaying Steph Curry’s flow during the game:

One sees that there is a tendency for his shots to converge where the defender is at least three feet away at a minimum distance of 25 feet away from the basket. The saddle point behaviour is very evident in the lower left and upper right corners of the vector field plot.

## The Three-Point Shot Delusion

The vast majority of NBA analysts claim today that the NBA has changed. It has become more fast-paced, and there is a significantly greater emphasis on teams attempting more three point shots. The evidence for this is the repeated recital of the fact that over the last number of years, the average three-point attempt rate has increased. An example of such an article can be found here.

It is my hypothesis that this is all based on a very shallow analysis of what is actually going on. In particular, there are more than 60 variables on Basketball-Reference.com that classify each team’s play. It seems strange that analysts have picked out one statistic, noticed a trend, and have made conclusions ushering in the “modern-day” NBA. As I will demonstrate below, using concepts from statistical and machine learning, many things have been missed in their analyses. What is even more strange is that there have been an increasing number of articles claiming that, for example, if teams do not shoot more three point shots, they will probably not make the playoffs or win a championship. Examples of such articles can be found here, here, and here.

I will now demonstrate why all of these analyses are incomplete, and why their conclusions are wholly incorrect.

Using the great service provided by Basketball-Reference.com, I looked at the last 15 seasons of  every NBA team, looking at more than 60 predictor variables that classified each team’s performance in the season. Some of these included: MP FG FGA FG% 3P 3PA 3P% 2P 2PA 2P% FT FTA FT% ORB DRB TRB AST STL BLK TOV PF PTS PTS/G oG oMP oFG oFGA oFG% o3P o3PA o3P% o2P o2PA o2P% oFT oFTA oFT% oORB oDRB oTRB oAST oSTL oBLK oTOV oPF oPTS oPTS/G MOV SOS SRS ORtg DRtg Pace FTr 3PAr TOV% ORB% FT/FGA  TOV% DRB% FT/FGA, where a small “o” indicates a team’s opponent’s statistics.

What classifies a playoff team?

Building a classification tree, I wanted to analyze what factors specifically lead to a team making the playoffs in a given season. I found the following:

(For this classification tree, the misclassification error rate was 2.73% indicating a good fit to the data.)

At the top of the tree, we see that the distinguishing factor is the average MOV/”Margin of Victory” measured per game. Teams that on average beat their opponents by more than 2.695 points are predicted to make the playoffs, while teams that on average lose by more than 1.825 points are predicted to not make the playoffs. Further, the only factor relating to three-point shooting  in this entire classification tree is the o3PA, which is the number of opponent 3-point attempts per game. For example, suppose a team can has an average MOV of less than -0.54 but greater than -1.825. If that team’s opponent attempts more than 16.0732 3-point shots per game, the team is expected to make the playoffs. In this particular case, getting your opponent to take a lot of three point shots is indeed desirable, and leads to the expectation of a team making the playoffs.

What classifies a championship team?

The next question to analyze is what characteristics/features classify a championship team. Looking at the last 20 years of playoff data, we see that the following classification tree describes the championship criteria for a given NBA playoff team.

(The learning error rate was 1.172% indicating an excellent fit to the data). One sees that at the very top is a team opponent’s field goal percentage (OFG.). If the average per game OFG% is greater than 44.95%, that team is predicted to not win a championship. Further, there are apparently three predicted paths to a championship:

1. OFG% < 44.95 –> ORtg (Opponent Team Points Scored per 100 possessions) < 108.55 –> FT% < 73.5% –> Opponent Offensive Rebounds per game (OORB) < 30.2405 –> Personal Fouls per game (PF) < 24.1467
2. OFG% < 44.95 –> ORtg > 108.55 –> O3P% < 32.45%
3. OFG% < 44.95 –> ORtg > 108.55 –> O3P% > 32.45% –> AST > 19.9076 –> OAST < 19.0938

This shows once again that the three point shot is not at all relevant in winning a championship amongst playoff teams, in that, shooting a lot of threes, or playing as a “modern” team, does not uniquely determine a team’s success. What is tremendously important is defense, and offensive efficiency, and there are multiple ways to achieve this. One does not need to be a prolific three-point shooting team to achieve these metrics.

Conclusions

The increasing  trend of teams shooting more threes and playing at a higher pace still does not uniquely determine whether a team will make the playoffs or win a championship, which is why I have called it a “delusion”. Indeed, the common statement that “nowadays, teams that make the playoffs also have the highest number of three-point shot attempts” is a very shallow statement, and is not actually why teams make the playoffs as this analysis very clearly shows. Further, attempting more three-point shots is not at all uniquely indicative of a team’s success in winning a championship.

## 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.