Nylon Calculus: Game theory and the deep 3

For years, the Houston Rockets’ most defining trait, aside from James Harden’s beard, has been their willingness to stretch the theoretical limits of mathematical application on a basketball court, eschewing almost all semblance of a mid-range strategy on offense and pushing their shot selection more and more into two distinct categories — the 3-pointer and shots at the rim.

They boasted a 3-point attempt rate of 0.502 last season, regularly producing showings where they gleefully launched more than 50 3s at a time. Some have called it audacious; others have called it an affront to basketball itself. But it’s indisputable that the Rockets’ style has been one of the primary offensive innovations of the 21^st century. With analytics proving the 3-point shot to create such an outsized amount of value, it’s hard not to let your mind drift to the theoretical extreme: why ever take a shot that isn’t from behind the arc?

The answer should be strategically obvious. Of course, if you only ever take one type of shot, defenses will key 100 percent to stop that shot and thereby dramatically reduce your effectiveness. Ah, but I know that, you’ll respond in return. If the defense sells out fully at the arc, then we can just blow by for a layup!  And so on the cycle will continue. And so, perhaps accidentally, you’ve now ventured into the realm of game theory.

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Brian Burke, now of ESPN, published a read that is well worth your time illustrating this exact phenomenon, but used for Run/Pass decisions in football.  As Brian notes, game theory concepts such as this work particularly well in sports, as sports are a zero-sum game. Anything that happens to the offense’s credit is always the direct inverse of the defense’s utility (and vice versa). If the one team scores seven points, the opposing team has “lost” seven points. However, given the scoring and gameplay structure of football versus other sports, one could argue that basketball is even better suited for game theory since every play/possession has an immediate scoring outcome.

Kostas Pelechrinis, among others, has highlighted how such game theory could be used in deciding between taking a 3 or taking a 2 in a come-from-behind end-game situation. But I wanted to go a slightly different direction.

Think for a moment on how many times over the last few seasons you’ve seen Steph Curry pull up from right in front of the logo for a deep 3, only to splash it in front of a throng of incredulous fans and viewers. A direct consequence of the exodus away from the mid-range and towards the arc has been that the functional geometry of a basketball court has irrevocably changed. Not just Steph, but players like the Rockets’ own James Harden and Eric Gordon are fearless bombing away from way behind the 3-point line. Gordon is especially proficient on deep 3s, converting at an eye-opening 36 percent clip last season on 308 shots between 26 and 40 feet, per Basketball-Reference.

The game’s best players are becoming better at shooting from farther and farther out. And as this trend continues, it’s fair to expect the deep 3 to become a more and more critical part of every elite shooter’s toolkit. And we can turn again to game theory to illuminate why that is.

Let’s set up a simplified simulation with some important assumptions. For illustrative purposes, let’s just file away any rebuttals about things like team defense and rotations or passing the ball or clock situations. We are going to purposefully distill an offensive possession into it’s most basic tenets and matchup. On offense, we will have three choices: take a deep 3, take a regular 3, or go to the rim. On defense, we will also have three choices: press all the way up at half-court (intended to counter the deep 3), stick a balanced defense around the arc (to counter the regular 3), or pack the mid-range and the paint (to counter any drives to the rim). With three different defensive strategies designed to fundamentally counter one specific “level” of the court’s geometry, we assume that a given defense can either produce a tightly contested shot, a modestly contested or average one, or leave space for a wide open one.  Thus, the matrix of strategies looks like the following, with red indicating a tightly contested shot, yellow an average one, and green a wide open one:

You may already be starting to pick up on a pattern from the above matrix, but let’s now fill it in with examples from some players. We’re going to use base assumptions of the shooting percentages from last season at 0-3 feet, 22-25 feet (3s only), and 26-40 feet pulled from Basketball-Reference’s shot finder. We’re also going to borrow Sandy Weil’s findings about tight defense from his 2011 Sloan Paper on using player tracking data to analyze NBA field goal shooting. He showed that tight defense reduces a player’s shooting efficiency by roughly 12 percentage points. And bouncing off that, we’ll assume that a wide open shot experiences the exact opposite effect, with efficiency jumping up 12 percentage points. We can then use these derived shooting percentages and multiply them by the value of the shot in order to get the expected points outcome.

Now, I’ll reiterate again that these are highly simplified assumptions, attempting to condense a highly complex set of interactions to a very basic level for illustrative purposes. I’ve also decided for now to ignore the effects of shooting fouls in the expected points calculations. While they changed the underlying details, they didn’t materially change the outcomes or the takeaways for the following examples, so I decided to exclude them for simplicity’s sake. With all that out of the way, let’s look at the matrices for Steph Curry and James Harden, two of the premier offensive forces in the NBA.

In the above matrices, I’ve both written out the base shooting percentages from each range as well as highlighted the highest value offensive strategy against each defense. Clearly, the value of the deep 3 is much different for Curry than it is for Harden, given their stratospheric difference in efficiency from that range. However, look again at the middle rows in each payoff matrix. Regardless of defensive tactic, there is always a higher-value play than the regular 3-point shot. This is what is known in game theory as a dominated strategy.

“A strategy is dominated if, regardless of what any other players do, the strategy earns a smaller payoff than some other strategy. Hence, a strategy is dominated if it is always better to play some other strategy, regardless of what opponents may do.”

We can confirm this again by solving for the Nash equilibrium of the given “games” to find the optimal distribution of offensive strategies indifferent to the defense. In order to solve for the equilibrium, we will set up a system of equations as such, with P₁, P₂, and P₃ representing the payoffs from the three offensive plays and Y₁, Y₂, and Y₃ representing the three defensive columns:

By the definition of equilibrium, Y₁, Y₂, and Y₃ need to be set equal to each other. Once those equations are set up, we can plug them into any system-of-equations solver (Wolfram-Alpha has a good one!) and get our equilibria distribution of strategies. For Steph Curry, the equilibrium works out to be 40 percent deep 3s and 60 percent rim-attacks. For the Bearded One, the equilibrium works out to be roughly 38 percent deep 3s and 62 percent rim-attacks. As expected, the dominated strategy got completely erased in both cases. Even despite the difference in Harden and Curry’s percentages on deep 3s, their overall equilibrium was not too divergent, which speaks to the power of the value from extended range.

Of course it would be ill-advised for LaMarcus Aldridge to start jacking up 30-footers only to miss them, but for the league’s best shooters, there’s a case to be made that if they’re going to shoot a t3 anyway, it’s usually preferable to take a few steps back and stretch the defense even further. And not just for game theory reasons!

As Justin Willard has written about previously, even deep 3s up to 30 feet out have a comparable points-per-attempt efficiency to a mid-range jumper. Thus, it should come as no surprise that the most prominent preachers of efficiency, the Rockets, boast a cadre of players with an affinity for the deep 3: Eric Gordon, Ryan Anderson, James Harden, and the list goes on. Moreyball is not just about 2s versus 3s. It’s about fundamentally reshaping how we understand the geometry of a basketball court and the value chain of each spot of available real estate on offense.

In the 1800’s, settlers started at the coast and expanded westwards. In basketball, teams have started moving from the midrange to the three, but their “westward expansion” farther and farther behind the arc is still a largely untapped opportunity. Forcing a defense to respect you from half-court onwards has massive implications. Heck, even just positioning shooters a few feet behind the arc can considerably distort the defense towards their breaking point. Justin has covered this topic as well in his indispensable Week-In-Review series. As he illustrated, in a four-out offense, just moving one shooter at the wing back by 3.5 feet increases the offense’s functional area by greater than 70 square feet, which can make all the difference in how a defense decides to send a double team or rotate on a drive.

In a league where spacing is so crucial to offensive success, that’s the kind of value chain which needs to be accounted for when analyzing how to best stretch a defense. Trying to push the 2-point-vs-3-point ratio to its theoretical extremes is old hat. The best offenses and the best offensive players in the NBA have already moved onto the next frontier — pushing geometry itself to its theoretical extremes.