As a basketball fan, I often hear commentators stressing the importance of good shot selection, which means taking shots that have a reasonably high chance of success. Additionally, crisp ball movement on the offensive side and solid defense significantly contribute to a higher winning percentage, supposedly. With this in mind, and drawing inspiration from [1], I posit a statistical network model of basketball:

• Each player is represented by a node in the graph. Each player i also has a probability of success si, which is the probability that player i scores without passing, whether via a drive to the basket or a shot.
• There are unidirectional weighted edges from the ball handler to each of his four teammates, with each weight from player i to j denoting the probability that the pass from i to j will be successful, that is, successfully received by player j without a turnover. It is important to note here that the quality of passes obviously differs depending on the passer, and the chance of a successful pass also depends on the catching ability of the recipient. For this simple model, we assume that any pass can be either successful or unsuccessful (binary).
• There exists an edge from defensive player y to offensive player z if player y is assigned to defend player z. To allow for the possibility of double teams (and even triple teams), we can make this a many-to-one correspondence, so that anywhere from 0-5 defensive players can be defending an offensive player at any given time. However, each defensive player can defend only one offensive player at a time (again, not entirely realistic, but simplified for this model’s sake). Note that when the defensive team decides to double team an offensive player, in this model, one offensive player must necessarily be undefended.
• Each defensive player i is assigned a defensive rating di, and this di affects the probability of success s of the offensive player that i is guarding. While it’s true that a player’s defensive ability changes sometimes dramatically based on context (post defense, perimeter defense, even clutchness of situation), we can glean enough insight from the model with a static defensive rating.

Taken from [1]

Now that we’ve defined the model, we can see that striving for good shot selection means optimizing the probability of success si for the offensive player who actually attempts to score during a possession, given the defensive matchups. To make the concepts of probability of success (si) and probability of a successful pass more concrete, consider the following situations:

• Assume that we have a Lakers vs. Thunder game. The Lakers have possession, and the current ball handler is their point guard Steve Nash. If the Lakers run a pick-and-roll, then Steve Nash, Dwight Howard, Russell Westbrook, and Kendrick Perkins will mainly be involved in the play (we’ll consider other players negligible). Assume further that Perkins and Westbrook pressure the ball handler Steve Nash, so that Nash is basically being double-teamed, leaving Howard open near the basket. The probability of a successful pass from Nash to Howard would be relatively low, since Nash is being double teamed, with the Thunder center no less. However, the probability of success for Dwight Howard’s shot attempt would be extremely high, almost at 1.

Low probability of a successful pass because Steve Nash is being double-teamed.

• Keeping with the same teams, now imagine that Nash has penetrated the paint, attracting once again Perkins and Westbrook. To clear the area, Howard has, for some reason, rolled to the three-point line. To be clear, Howard is a great finisher near the basket, but a terrible three-point shooter. The graph now looks like

Now, Dwight Howard is wide open, but since he’s a terrible three-point shooter, taking a shot from there is an example of bad shot selection.

Conclusion

This post has focused on the offensive side of things, stressing the importance of crisp ball movement and good shot selection, which are represented by the probability of a successful pass and the shooter’s probability of success on the shot attempt. This model also explains some time-tested offensive schemes such as the triangle offense [2], made popular again by Phil Jackson. In this set, the center is down at the low post, the forward at the wing, and the guard at the corner, making a triangular shape. The other two players man the weak side. Basically, this creates good spacing and allows each player to pass to the other four. In the terms of our model, we can say that the triangle offense increases, or optimizes, the probabilities of a successful pass, and since it creates good spacing, the probability of a made shot also increases.

-dwc92

Sources

[2] https://en.wikipedia.org/wiki/Triangle_offense

### 3 Responses to “ Networks Analysis of Basketball ”

• Will

Being a basketball fan, I was really intrigued by your post. While there are far to many factors in an actual basketball game to model, I feel like your analysis would have a lot of relevance to basketball video games, such as NBA 2K13.

While I am not familiar with the algorithm used in basketball video games, as a player, I have noticed they take that things they take account pretty much the exact same things as you. However, they also factor in things such as pressure in tense situations, such as close games with little time remaining, and also hot and cold streaks. I also notice that NBA 2k13 factors in a player receiving a pass “hands”, or how good they are at actually catching a thrown pass, whether good or bad. Seeing your post makes me reflect and wonder just how complex the algorithms in the video games must be.

• yl582

I think that this analysis seems sound, and one of the things that is involved in basketball iq is the ability to recognize all of the possible situations that exist. Perhaps this is why coaches, especially in college, drill a scheme so hard into their players. If you allowed all of the players to just do whatever they want, the possible situations are just too numerous for any player to assess well. Restricting the number of options, while still keeping all of the available options good seems to be what most coaches strive for in a good offense. A good defense, on the other hand, is the reaction to the expectation of what the options on offense will be. There is always going to be some guesswork involved, but I agree with your analysis for the most part.

• dwc92@cornell.edu

I commented on the posts “The Friendship Paradox” and “Global Super-Entity”.