Holes in the Defense: A Geometric Currency for Basketball Advantage
This is the web version of a longer write-up. The model and the numbers are the same. The PDF with every table is in the repo.
The second the box score misses
I wanted to put a number on the space a pick-and-roll opens, frame by frame, and check whether that number behaves the way basketball says it should.
Watch a pick-and-roll from behind the play. A defender is a step late. The big steps up to show and leaves the paint empty behind him. For about a second the strong side belongs to the offense. Then help arrives, the corner defender sinks, and the space closes. Sometimes the ball gets there first. Usually it doesn't.
Coaches call that second advantage. Offensive actions try to create it and defensive rotations try to repair it, but the play-by-play only records how the possession ended. Can you measure the space itself, every fifth of a second, and does the number move when it should?
The data is SkillCorner's open release of ten Liga ACB games from Spain's top league, tracked from the broadcast: every player and the ball at 25 Hz, each position with its own expected error, plus a marked play-by-play of picks, off-ball screens, handoffs, drives, isolations, closeouts and matchups. That is enough to build the field and, more usefully, enough to test it.
What an "empty hole" is
The model has three steps.
1. Who gets there first. For every player and every square foot of the offensive half court, I estimate an arrival time. The player keeps his current velocity through a 0.2 s reaction time, then accelerates toward the spot at up to 20 ft/s² with a top speed of 20 ft/s (both fitted from the tracking; a player running the wrong way has to stop first). The arrival time is uncertain: there is a 0.15 s floor for indecision, a term from SkillCorner's stated position error (inflated for extrapolated players), and a term that grows with distance. From that I compute the probability that the first offensive player arrives before the first defender. With Gaussian arrival times this has a closed form (Clark's recursion for the minimum of Gaussians). Against Monte Carlo on real frames the mean absolute error is 0.016 in probability.
2. What the spot is worth. Multiply by the expected points of an open shot from that spot. The map is fitted on open and lightly contested shots from the other nine games and shrunk toward a simple parametric shape where data are thin. The rim is worth about 1.35, mid-range 0.85, the corner three a bit over 1.
3. Add it up. The exploitable-space measure \(A\) is the integral of that field over the cells worth at least half a point:
\[ \begin{aligned} H(q,t) &= \underbrace{P\big(\min_{i\in O} T_i(q,t) < \min_{j\in D} T_j(q,t)\big)}_{\text{offense reaches } q \text{ first}} \\ &\quad \times \underbrace{V(q)}_{\text{open-shot value}} \end{aligned} \]
\[ A(t) = \int \max\big(0,\,H(q,t)-0.5\big)\,dq. \]
Units are ft²·points. A typical half-court frame sits around 250-300.
One split turned out to matter. Every cell that counts toward \(A\) is either within 5 ft of an offensive player or it isn't. The first kind is an open player, like a shooter in the corner with nobody near him. The second kind is an empty hole: valuable space nobody is standing in, which an attacker who cut or drove there would reach before any defender. An empty hole is what a defender out of position looks like on the floor. Different actions create the two kinds.
Two simpler models run alongside as baselines throughout: plain Euclidean Voronoi (nearest body wins, no velocity, no uncertainty) and the same kinematic arrival times with the uncertainty removed, so the faster player always wins. I wanted to know whether the probabilistic version earns its extra complexity.
You can scrub through eight possessions yourself, with the empty-holes-only view, in the interactive replay.
Does it move the way basketball says?
For every marked action in the ten games I took the mean of \(A\) over the 1.5 s after it minus the mean over the second before it. I did the same for 1,350 random moments at least a second from any marked event. \(A\) drifts up by about 11 units on its own as the shot clock runs down, so the number to read is the excess over that null.
| Action | n | ΔA over null | P(ΔA > 0) | empty-hole part | open-player part |
|---|---|---|---|---|---|
| Pick (on-ball screen) | 1,196 | +49.8 | 0.84 | +43.5 | +6.2 |
| Drive | 512 | +44.2 | 0.80 | +40.0 | +4.2 |
| Off-ball screen | 838 | +11.7 | 0.63 | +12.4 | −0.7 |
| Isolation | 146 | +9.2 | 0.56 | +6.0 | +3.2 |
| Handoff | 325 | +7.9 | 0.61 | +9.1 | −1.2 |
| Pass | 2,676 | −8.2 | 0.50 | −7.4 | −0.7 |
| Closeout (defensive) | 453 | −15.6 | 0.46 | −11.8 | −3.8 |
| No action (null) | 1,352 | 0 | 0.55 | — | — |
Picks and drives are significant at \(p<10^{-30}\); off-ball screens, closeouts and passes at \(p<10^{-3}\); handoffs and isolations are not (Mann-Whitney against the null).
None of this was tuned in. The ordering is the one a coach would give: screens and drives create advantage, off-ball actions create less, and the closeout, the one defensive action in the data, is the only thing that destroys it. Almost everything a pick or drive creates shows up as empty holes, not open players, so the measure is picking up the defense being pulled out of shape. And the average pass loses a little, because the defense moves while the ball is in the air. Passes spend advantage. They don't make it.
The baselines get the closeout backwards
This is the result I care about most. On Euclidean Voronoi the closeout scores +23 over the null. A defender sprinting at the ball handler gives up the Voronoi cells behind him, so the area measure reads a defensive recovery as an offensive gain. Voronoi also scores handoffs at −20. The ball handler's nearest-defender distance moves less than 4 ft for any action, and the offensive convex hull shrinks on drives and picks. Only the two models that know about motion recover the basketball ordering, and only the probabilistic one gets the sign of every action right.
| Measure (excess over null) | Pick | Drive | Off-ball | Handoff | Closeout | Pass |
|---|---|---|---|---|---|---|
| Probabilistic \(A\) | +49.8 | +44.2 | +11.7 | +7.9 | −15.6 | −8.2 |
| Kinematic \(A\) | +70.8 | +60.3 | +4.1 | +5.3 | −14.9 | −14.8 |
| Voronoi \(A\) | +26.9 | +24.8 | +0.6 | −20.1 | +23.2 | +1.4 |
| Handler nearest-defender distance (ft) | +0.3 | −1.0 | +0.5 | +0.7 | +4.0 | +1.9 |
| Offensive convex hull (ft²) | −12 | −63 | +36 | +52 | +4 | +27 |
How fast the defense repairs it
After the post-action peak, \(A\) falls halfway back to its pre-action level in a median of 0.6 s for picks, drives, handoffs and off-ball screens. The defense fixes most holes quickly. What differs is how often it can't: 68% of drive-created advantage is still there 3 s later, against 54% for picks, 54% for the null and 47% for closeouts. Drives are the action whose advantage most often outlives the reaction, which is why they end in shots.
One currency, every coverage
Because every action is scored on the same field, the SkillCorner coverage labels can be compared directly. They line up with coaching vocabulary, and nothing was tuned to make them.
| Context | n | ΔA over null |
|---|---|---|
| Pick, handler defended over the screen | 775 | +58.6 |
| Pick, defense switches | 168 | +35.7 |
| Pick, handler defended under | 219 | +29.4 |
| Pick, screener's man shows | 485 | +58.0 |
| Pick, screener's man drops | 504 | +46.6 |
| Pick at the step-up / middle / wing | 154 / 734 / 288 | +64.6 / +55.5 / +27.3 |
| Off-ball screen, cutter trailed | 441 | +18.6 |
| Off-ball screen, cutter whipped (defender beats him through) | 319 | +0.7 |
| Off-ball screen, screener's man shows / drops | 47 / 594 | +49.9 / +9.2 |
| Drive created by a pick / by an isolation | 260 / 62 | +49.8 / +20.3 |
| Pass under 12 ft / 20-30 ft / over 30 ft (skip) | 642 / 650 / 133 | +2.4 / −19.2 / −21.6 |
| Closeout followed by a shot / by a pass | 215 / 83 | −12.2 / −22.8 |
Switching or going under costs the offense about 25 units relative to a defender who fights over. A show leaves the paint and concedes more than a drop. A step-up screen in the middle of the floor is the most productive on-ball action in the data; a wing screen creates less than half as much. A whipped off-ball screen creates nothing.
The skip pass, which coaches love because it exploits a hole, comes out negative. This is a real limitation of a field that values space and ignores where the ball is. A skip pass moves the ball into a hole that already exists, and the defense moves during the flight. Here is one.
Does the number predict anything?
Pictures are easy to make. The real test is whether \(A\) knows something that ball location, defender distances, convex hulls and Voronoi areas don't. From every half-court frame I predicted what happens in the next 3 s of the same possession, training on nine games and testing on the tenth, ten times over. Splitting neighbouring frames at random would leak badly, so I didn't.
| Next 3 s (logistic, AUC) | rate | ball only | baseline (16 spacing / defender features) | + Voronoi | + kinematic | + probabilistic |
|---|---|---|---|---|---|---|
| Any shot | 0.24 | 0.660 | 0.722 | 0.731 | 0.741 | 0.766 |
| Open / lightly contested shot | 0.06 | 0.606 | 0.699 | 0.712 | 0.723 | 0.740 |
| Shot at the rim | 0.08 | 0.724 | 0.775 | 0.776 | 0.785 | 0.797 |
| Assist opportunity | 0.11 | 0.539 | 0.606 | 0.615 | 0.617 | 0.651 |
| Paint touch | 0.14 | 0.523 | 0.578 | 0.583 | 0.585 | 0.617 |
| Blow-by | 0.02 | 0.599 | 0.619 | 0.607 | 0.628 | 0.615 |
With a linear model, every target but blow-bys improves in step with the model: Voronoi adds a little, kinematic more, probabilistic the most. That is +0.04 AUC on shots and open shots and +0.045 on assist opportunities and paint touches. The field's own summaries, with no ball or defender information at all, predict a shot in the next 3 s at 0.69.
Gradient boosting narrows the gap, as it should, since enough trees can rebuild spatial structure from raw coordinates. The field still helps on the two targets most about space: blow-bys (0.599 to 0.614 AUC) and SkillCorner's quality grade of the next shot (\(R^2\) 0.166 to 0.200, Voronoi 0.181). Nothing gets worse. Adding the one-second change in \(A\) on top of its level adds nothing. The level carries the information.
Two results went the other way. Nothing predicts the points a possession ends with from a single frame (\(R^2 < 0.005\) for every feature set). The next three seconds are geometric; the end of the possession is not. And within an action kind, the advantage an action creates does not predict the points scored on that possession. Creating advantage and using it are different skills, and this field only measures the first.
Possessions, one at a time
I picked these examples by rule rather than by eye: clean tracking (no more than one extrapolated player), gain between the 85th and 97th percentile for the action kind, distinct possessions. Dotted circles are the 5 ft occupancy radius; the white line is the \(H = 0.5\) contour. Bright space outside every dotted circle is an empty hole.
Holes as objects
Take every connected region of \(H > 0.5\) more than 5 ft from every attacker, link the regions frame to frame, and you get 15,795 holes across the ten games. Most close quickly: median lifetime 0.6 s, mean 2.0 s. The ball enters only 8% of them, but it goes where you would hope. Holes that survive 2.4 s get used 30% of the time, holes in the top mass tercile 24%, and holes that live under 1.2 s almost never. Holes born after an off-ball screen (mean 1.5 s) or a handoff (1.4 s) outlast those born after a drive (0.7 s). Off-ball actions open space that waits. Drives open space that is already closing.
What it took to break it
For 1,010 event windows I recomputed everything under 24 alternative configurations. I resampled every position from its own stated tracking error and varied every constant in the movement model, the threshold and the grid. Per-event changes stayed correlated above 0.96 with the reference, and the action ranking stayed at or near 1.0. Only two choices matter, and they are the ones that should: the value map (parametric vs. empirical, correlation 0.86) and the control model. With kinematic control the correlation drops to 0.80. With Voronoi it drops to 0.39 and the closeout flips sign.
What it does not do
- It values space, not the ball. This is the skip-pass problem. A ball-conditioned variant behaves the same way. A real fix needs a passing model, and ten games won't support one.
- Creation is not conversion. Nothing here says who scores from the advantage.
- Three seconds, not a possession. It describes the current state of the floor.
- Defensive intent is invisible. A defender sagging off a non-shooter on purpose registers as a hole. The empirical value map corrects for this on average, not by player.
- One movement model for everyone. No size, no fatigue, no slowdown while dribbling. The conclusions don't depend on the constants, but a player-specific model would sharpen the field.
- Ten games. The per-game spread of the validation AUCs, 0.01-0.04, is the honest size of the error bars.
Reproduce it
Everything is in a fork of SkillCorner's release under analysis/: the library, the eight pipeline stages, the results tables, the paper, and the replay. Stage 1 takes about four minutes a game and the rest takes minutes in total. Courts are drawn with mplbasketball, whose FIBA half court matches the tracking frame to within 0.05 ft.