Second in a series occasioned by the WNBA Finals. The Valkyries won Game 7, 68–63, on Saturday night in San Francisco. Angel Reese had 8 points and 12 rebounds. Rhyne Howard went 3-of-14, and 0-of-7 from three. The jersey is in the wash.
Don’t hate the player. Hate the game. Both halves of that are literally true here, and the second half is the subject of this post.
The first post held everything fixed except the defense. Howard was going to shoot from her spot; Reese was going to be in the lane; the only question was where the Valkyries put their attention. That’s partial equilibrium, and it gave a clean answer. This post asks what Reese does to everyone else’s problem, including the Valkyries’, and the answer is that her indirect effects are larger than her direct ones, which is a thing worth knowing about anyone whose job is to be stood next to.
I’m still holding the players’ choices fixed. Nobody in this post decides when or where to shoot. That’s a stopping problem, it’s a good one, and it’s a different series.
What a rebound is worth
The first post priced an offensive rebound at a number, \(v\), and set it to the putback: about 1.4 points in Reese’s hands, since she finishes two-thirds of them. That was the last-possession case, where a board is worth only what it scores right now.
Earlier in the game a board is worth more than that, and the reason is the whole post. An offensive rebound doesn’t end the possession; it extends it. The rule, since 2019 in the WNBA, is that the shot clock resets to 14 seconds when the offense rebounds its own miss, rather than the 24 a fresh possession gets, and if more than 14 were already showing it simply keeps running.1 So the Dream get another chance to score, on a shorter clock, with another chance to miss and rebound again. Call the extended possession worth a fraction \(\lambda\) of a fresh one: less than one, because fourteen seconds is less than twenty-four.
So the value of a possession depends on the rebound rate, and the value of a rebound depends on the value of a possession. Three numbers describe the offense: \(p\), the chance its shot goes in; \(s\), what the shot is worth when it does; and \(r\), the chance a miss comes back. Call the value of a possession \(V\). Then
\[V = s\,p + (1-p)\,r\,\lambda V,\]
and solving,2
\[V = \frac{s\,p}{1 – (1-p)\,r\,\lambda}.\]
The denominator is the fixed point doing its work. Each miss that comes back is a possession that didn’t end, and the series \(1 + m + m^2 + \cdots\) with \(m = (1-p)r\lambda\) is the expected number of times you get to try. That series converges as long as \(m < 1\), which it always is, since you can't rebound more than every miss. A great offensive rebounder makes \(m\) larger, which makes the series longer, which raises the value of every possession, including the ones she never touches.
Notice what the formula multiplies. The numerator, \(s\,p\), is points per attempt, and it’s where the rest of the roster lives: a team of corner shooters has \(s = 3\) and a modest \(p\), a team that finishes at the rim has \(s = 2\) and a better one, and Smesko’s Dream, who take threes and layups and nothing else, have an \(s\) near the top of the league and a \(p\) that reflects it. None of that is Reese. Her number is \(r\), and it sits in the denominator, where it scales whatever the numerator is. The same rebounder on a team that can’t shoot extends possessions that aren’t worth extending. On a team built to shoot, every board she pulls is a fraction of a three. That is the sense in which her value is indirect: it’s a multiplier, and a multiplier is worth whatever it multiplies.
Here are two teams. Both shoot 35%, and to keep it simple every shot is a three, so \(s = 3\). One rebounds 20% of its misses, which is roughly what a normal WNBA team does. The other rebounds 30%, which is roughly what Reese’s Dream did on their way to the best offensive rebounding margin in league history. Take \(\lambda = 1\).
| ordinary team | Reese’s team | |
|---|---|---|
| shooting, \(p\) | .35 | .35 |
| rebounding, \(r\) | .20 | .30 |
| tries per possession, \(1/(1-m)\) | 1.15 | 1.24 |
| points per possession, \(V\) | 1.21 | 1.30 |
Nine hundredths of a point per possession, from rebounding alone, with shooting held equal. Over the 80 or so possessions in a game that’s seven points. The putback, the direct effect, is a slice of that: the rest is possessions that didn’t end, converted by shooters who are not Reese.
The possession, with sliders. Set the shooters and the rebounder, and watch where the points come from.
Direct and indirect
So split her value in two. The direct part is what she scores off her own boards. The indirect part is everything her boards do for everyone else: the extra tries, the shots those tries produce, and, from the first post, the attention she pulls from the defense.
The indirect part is bigger. It’s bigger in the model, because the putback is one term in a series and the rest of the series is other people. And it’s bigger in the data: in the semifinal, the Liberty shot a true shooting percentage of 60.6, a figure that would tie the most efficient regular-season offense in league history, and were swept. The Dream took eleven more shots a game. A team that is more efficient on every shot can lose to a team that gets more shots, and the first post’s \(v\) was a way of hiding that inside one number.
That’s the inference problem. Measure Reese by what she scores and you get a modest number, because her job is to produce possessions that other people score. Measure the Dream by their shooting percentage and you get a modest number too, since they’re taking shots they’d pass up on a team that didn’t get the ball back. Measure the Liberty by their shooting percentage and you’d conclude they played well, and they did, and they got swept. All three statistics are correct and all three point the wrong way, because each of them records a direct effect in a system that runs on indirect ones.
The other end
The same price runs the other way. A Valkyries miss hands the ball to a team that keeps possessions alive and leads the league in scoring off turnovers. That makes their misses more expensive, which raises the quality of shot they should settle for, which lowers their volume and raises their percentage. Gabby Williams scored 28 points in the second half of Game 7, a postseason record for a half, on a night the Valkyries took five fewer shots than the Dream and made 25% of their threes. That’s not a contradiction of the model; it’s the model from the other side. Facing Reese, you take the shots you can make, and when you have a Finals MVP, that’s enough.
Game 7
Which brings us to Saturday. The Dream took 69 shots to the Valkyries’ 64. Both teams got ten offensive rebounds. The Dream shot 6-of-30 from three, and Howard shot 0-of-7.
Every indirect effect was in place. Howard got her shots; the Valkyries had to defend her; the extra tries were there, five of them. What failed was the draw. \(V\) is an expectation, and an expectation is a statement about a series of games, not a game. The possession battle is a long-run edge, and on a night when the shooters miss, a long-run edge is a thing you can describe afterward.
This is where the joke stops being a joke. The player is Howard, and she took the shots the Dream’s whole design exists to produce, and seven of them didn’t go in. The game is the equilibrium, and the equilibrium was right. There is nothing in the first post or this one that says the Dream should have done anything differently on Saturday, which is an uncomfortable thing for a model to tell you about a loss, and the reason I trust it.
Reese’s line afterward was “delayed, not denied.” Read as a sentence about expected value, it’s correct.
Next: if the indirect effects dominate, then what the defense believes about you matters more than what you can do, and the belief and the ability can come apart. Reese has one skill where they can’t and one where they can.
1. The NBA made the same change a season earlier, in 2018–19, and both leagues did it to speed the game up: a 24-second reset after an offensive rebound let teams run the clock down twice on one trip. The rule is why the extension is worth \(\lambda \le 1\) of a fresh possession rather than all of it, and \(\lambda \le 1\) is also, conveniently, the condition under which the series below converges without any further assumption. ↩
2. This is a Bellman equation, the kind of thing that runs under every dynamic model in economics and under most of reinforcement learning. The value of being somewhere equals what you get now plus the value of where you end up next, and because “where you end up next” can be the same place, the unknown appears on both sides. There are two ways to solve it. The one in the text is algebra, because \(V\) appears linearly. The other is to guess a value, plug it into the right-hand side, take what comes out as the new guess, and repeat; the map is a contraction whenever \(m < 1\), so the guesses converge to the same answer at rate \(m\). Convergence is slower when \(m\) is larger, which is a small joke: a better offensive rebounder makes the fixed point harder to compute, because her possessions take longer to end. The usual Bellman equation also has a decision in it, a max over what to do now; this one doesn't, because I've held everyone's choices fixed. That's the thing being deferred to the stopping-problem series. ↩
With that, I leave you with this.