First of a short series occasioned by the WNBA Finals. The Dream forced a Game 7, the first in Finals history, and it tips Saturday night in San Francisco. I have been wearing a Reese jersey to lecture. I also have a small model, which is the only kind I trust.
The Dream have the ball. Rhyne Howard has it, open, in her spot. Angel Reese is in the lane, not involved in the play, and the Valkyries have a choice to make before the shot goes up: spend their help on Howard, or spend it on keeping Reese off the glass if Howard misses.
Howard leads the league in made threes, became the fastest player in WNBA history to 400 of them this season, and opened Game 2 of these Finals with one from 27 feet. Reese broke the single-game rebounding record this year, broke the single-season record, and is the Defensive Player of the Year. So the question is which of two historically good things to worry about, and it’s a question the Valkyries try their best to answer, one way or the other, every time these two are on the floor together.
I wanted to know what a model says about it, and specifically whether Reese changes the answer, or just the margin. Unsurprisingly to me at least, she ignores the margin, and simply changes the answer. Words are important, but models make them more so. Let’s go.
The possession
The defense has one unit of extra attention, \(a \in [0,1]\), to split between contesting Howard (\(a\)) and boxing out the Dream’s rebounder (\(1-a\)).
Howard makes the shot with probability \(p(a) = q(1 – \theta a)\), where \(q\) is her percentage from that spot when open and \(\theta\) is the proportional cost of a full contest. If she misses, the rebounder gets the offensive board with probability \(r(a) = r_0 + \rho a\): \(r_1 = r_0 + \rho\) single-covered, \(r_0\) when boxed out, so \(\rho\) is what the box-out buys. An offensive rebound is worth \(v\) points; a defensive one is worth zero by normalization.
Suppose this is the last possession of the game, so that a rebound is worth only what it produces right now: the putback, and nothing after. Then expected Dream points are
\[E(a) = 3\,p(a) + \big(1 – p(a)\big)\, r(a)\, v,\]
and the defense minimizes this. Earlier in the game a rebound is also worth the possession it keeps alive, which makes \(v\) larger. I come back to that below; it changes the numbers, not the shape of the answer.
Two facts about \(E\) do all the work. First, comparing the corners, the defense guards Howard rather than helping on the rebounder when
\[\frac{q}{1-q} > K \equiv \frac{\rho v}{\theta\,(3 – v r_1)}.\]
The numerator is the boards a box-out removes, times their value; the denominator is the threes a contest removes, times theirs. Second, \(E”(a) = 2vq\theta\rho > 0\). Contesting the shot creates misses, and misses make the rebound worth more, so the defense’s two jobs are complements, and for some shooters the defense prefers to do part of each rather than all of one. That is an interior solution, and it looks like something you’ve seen: the defender runs at Howard but not all the way, a short closeout, a hand up rather than a body, and turns to find Reese as the shot leaves Howard’s hand. The defense appropriately scrambling, which is what a possession looks like when both sides are playing it right. The shooters who get that treatment are the ones in the middle: good enough that running at them is worth something, not so good that running at them is worth everything. How wide that middle is depends on who’s waiting for the rebound. Sometimes it’s empty, and the defense’s only choices are all or nothing.
An ordinary rebounder
Start with a Dream possession where the player in the lane is an ordinary big, not Reese. Suppose she gets one offensive rebound in five when she’s single-covered, and one in seven when she’s boxed out. Suppose a putback is worth a point. And suppose a hard closeout takes a quarter off Howard’s percentage, which for a 40% shooter means she’s a 30% shooter with a hand in her face.
Now price the two things the defense can buy. Contesting Howard turns ten percent of her attempts from makes into misses, and a made three is three points, so the contest is worth about three-tenths of a point a possession, less a little for the rebounds those extra misses create. Boxing out the big takes six points of rebound probability off the sixty percent of attempts Howard misses anyway, at a point each: about four-hundredths of a point. The contest is worth roughly seven times the box-out. The defense closes out, and it isn’t close.
Make the shooter worse and the contest is worth less while the box-out is worth more, since more shots miss. The two cross at about an 8% shooter. The middle, where the defense would rather do some of each, runs from 7.7% to 8.1%. There are no basketball players in that range.
A three is worth three points and a rebound is worth one, so a few percentage points of shooting outweigh a few percentage points of rebounding.
Angel Reese
Now put Reese off the ball. Three parameters move, and the model says which ones.
She rebounds more: say 40% single-covered. The box-out removes more in levels, because there’s more to remove: say she drops to 22% boxed out, so \(\rho = .18\). And a board in her hands is worth more than a point even on the last possession. She’s finished at about 67% in the restricted area since mid-August, which is 1.34 points per putback before counting fouls. Call it \(v = 1.4\). Keep \(\theta = .25\), so the only thing that has changed is who is standing in the lane.
| ordinary big | Reese | |
|---|---|---|
| \(r_1\) (single-covered) | .20 | .40 |
| \(r_0\) (boxed out) | .14 | .22 |
| \(\rho\) | .06 | .18 |
| \(v\) | 1.0 | 1.4 |
| threshold shooter \(q^*\) | 8% | 29% |
| shooters in the middle (short closeout) | 7.7% to 8.1% | 27% to 32% |
| Howard at 40% | close out, 7× margin | close out, 1.6× margin |
The threshold is now about a 29% shooter from that spot. The defense still closes out on Howard, but the margin has gone from seven-to-one to under two-to-one, and the players inside the band are now real: a 30% shooter from her spot is a role player in the corner, a guard a step past her range, anyone you’d describe as “can shoot a little.” Against them the optimal defense is neither corner. It’s the short closeout, a hand up and then a turn toward Reese, and the model has located which shooters to do it against. If the Valkyries look like they’re scrambling on those possessions, they’re supposed to be. Against an ordinary rebounder the model says to stand still and close out; against Reese it says to move.
And Howard’s margin is thin enough to be sensitive to \(\theta\). A quarter off is a reasonable contest effect for an average shooter. Howard is 6-foot-2 and shoots from 27 feet, and a closeout plausibly does less to her. If it takes a fifth off instead (\(\theta = .20\)), the middle shifts to 32% through 37%, and Howard, at her actual 36% from three, is inside it. The best shooter on the floor gets the short closeout, because of who is standing next to her.
Nothing about the inequality changed. The corner was never a rule; it was where the calculation landed for every rebounder a defense usually faces. Reese is the case where it lands somewhere else.
And that was the last possession. Earlier in the game, Howard shooting with twenty minutes left, a rebound in Reese’s hands is worth the putback plus a fresh possession on a team that leads the league in using them. That is a larger \(v\), and \(v\) enters \(K\) in the numerator and the band through the curvature, so both move the same way: the threshold climbs further toward Howard and the band widens. The last-possession case is the conservative one. The qualitative story is the same from the opening tip; it’s only the margins that change.
What the answer runs on
It’s worth being precise about which of Reese’s numbers did this, because it isn’t the one you’d guess.
Her rebounding rate \(r_1\) appears in the denominator of \(K\), where it slightly favors guarding Howard: a made three is a shot nobody rebounds, so a great rebounder on the other side makes the contest worth more. The case for helping on her runs through \(\rho\) and \(v\). And \(\rho\) is a product: how much she rebounds, times the fraction a box-out takes away. For a great offensive rebounder those two pull against each other. She has more boards to take, and she’s harder to take them from, since getting boards through traffic is close to what being a great offensive rebounder means. The TSN profile of Reese this month describes her tracking the ball’s rotation from the moment it leaves the shooter’s hand, and her own account of how she does it is “I just know.” That is a player with a low fraction and a high level.
I set the level to win above. I don’t know that it does. The model’s contribution is to say that the question “should you worry about Reese when she doesn’t have the ball” is really the question “how much does a box-out actually take from her,” which is measurable, and is not the same as how much she rebounds.
There is no playbook for this
Reese’s presence reshapes the defense’s problem: it moves the threshold, it opens the interior region, it turns Howard from a mandatory assignment into a judgment call. A natural question is what the Dream do with that. The answer is nothing, and the model says why that’s fine.
There is no commitment in a basketball game. Nobody can bind herself to crash or to get back; the Valkyries can’t bind themselves to a coverage. Reese decides when the shot goes up, with whatever she can see of the defense, and the defense decides when the ball is in Howard’s hands, with whatever it expects of Reese. Each side is best-responding to its guess about the other, and the guess is correct in equilibrium. That is the whole structure.
In a zero-sum game, this doesn’t cost anyone anything. The possession has the same value whether Reese sees the coverage before she decides, decides at the release without seeing it, or, in some impossible world, the Valkyries saw her choice first. In the model, \(\min_a \max_b E(a,b)\) and \(\max_b \min_a E(a,b)\) are the same number once each side can mix, and that is the minimax theorem. Order of moves is irrelevant. What the Dream get from Reese on these possessions is exactly what she can do, priced by a defense that knows it, and no announcement, scheme, or reputation adds to it. The threat is credible because crashing is what she’d do anyway, at the coverages where it matters.
There is one commitment device in the game, and it’s the substitution. A coach can’t bind a player to an action, but she can bind the team to a player, and the player’s tendencies come with her. “Go with Reese” is a commitment to a rebounder who will crash, because crashing is what Reese does, and the Valkyries price that the moment she checks in. That is commitment by delegation, in Schelling’s sense: you commit by handing the decision to someone whose incentives everyone can see. It’s coarse, it’s public, it’s reversible at the next dead ball, and the other bench has the same device. So it doesn’t escape the logic above. The lineup commits a capability, the defense responds to the capability, and the value of the commitment is exactly what the model computes for the possession.
That’s the formal content of “on any given Sunday.” Nothing is settled in advance because nothing can be, except who is on the floor. The game is decided by what each side can do in the moment, and the defense’s problem above is what that looks like for one possession.
The model, with sliders. The two buttons set the rebounder; the top chart is the possession, the bottom one is the rule across every shooter.