5 Ways the Power Law Shapes Sports

Discover the secret force that shapes the sports we love.

We live in a power law world.

Power law thinking isn’t easy. But, it’s essential to understanding why a small few succeed and most others lag behind.

One of the most obvious places to see power laws is in the business world. Take search engines, for example. Google owns 92.06% of the global search engine market. Bing is #2 with 2.61% market share. None of the other “competitors” own even 2% of the market. That’s the power law.

But the power law doesn’t solely exist in business. 

As Peter Thiel points out in Zero to One, the power law is everywhere. Thiel: 

“This extraordinarily stark pattern, in which a small few radically outstrip all rivals, surrounds us everywhere in the natural and social world. The most destructive earthquakes are many times more powerful than all smaller earthquakes combined. The biggest cities dwarf all mere towns put together. And monopoly businesses capture more value than millions of undifferentiated competitors.”

Sports are not an exception. Once you know what a power law is and how to identify them, you’ll start to see them everywhere.

The power law has been the force behind the success of Ted Williams, World Cup goal scorers, James Harden’s rise to superstardom, and the two most successful franchises in the history of the NBA.

Oh yeah. It also governs how coaches like you build and maintain team culture. Surprise surprise…

Winner-take-all effects are all over. All one must do is look.

Ted Williams and the Power Law of Batting .400

In The Science of Hitting, Ted Williams shared a graphic on how he thought about the strike zone.

Over his career, Williams realized there were a small number of factors that were the main drivers of his success. One of them was his approach to managing the strike zone.

As Williams wrote:

My first rule of hitting was to get a good ball to hit. I learned down to percentage points where those good balls were. The box shows my particular preferences, from what I considered to be my “happy zone”–where I could hit .400 or better–to the low outside corner–where the most I could hope to bat was .230. Only when the situation demands it should a hitter go for the low-percentage pitch.

Williams understood that his batting results were not evenly distributed. Instead, a small group of pitch locations would give him the best chance of achieving the results he was after. In the graphic, these locations are colored red, orange, and yellow.

By waiting to swing at a pitch in one of these spots, he would increase his chance of having a productive at-bat. Over the course of a season, Williams believed, would lead to the success he was after.

This approach was key to him becoming Teddy Ballgame, arguably the greatest hitter who ever lived.

Ted Williams trusted the power law.

The Beautiful Game and the Power Law

Players took 1,200 shot attempts in the group stage of the 2014 World Cup.

Perhaps unsurprisingly, some shots were more likely to lead to goals than others.

Kirk Goldsberry points out that 41% of shots that came from within a small slice of the penalty box produced a goal 46% of the time–25% more than any other area on the pitch.

From Sprawlball by Kirk Goldsberry

This observation points to an obvious winning tactic of soccer: get the ball as close to the net as possible before you shoot.

Everyone likes to see a firecracker of a goal like this from Giovanni Van Bronckhorst, but relying on goals like this isn’t a winning strategy. You want to take shots where you’re most likely to get the best results.

But there’s one thing I haven’t mentioned. 

Since 1966, 81% of World Cup penalty kicks have been converted into goals.

Which means that on average, a penalty kick is worth 0.35 points more per attempt than a shot in even the highest-yield areas on the pitch.

So, the best strategy to rack up goals is not to shoot the ball as close as possible to the net. Instead, players should seek to earn a penalty kick.

Now, we can sit here and point out some of the ridiculous lengths soccer players go to in an attempt to draw fouls, but that’s beside the point.

The whole point is that in a power law world, soccer players know where they can go to get the most goals: as close to the net as possible. Or even better, 11 meters away from goal as the team’s best penalty taker prepares to send the ball into the back of the net.

And come to think of it, this all sounds an awful lot like what the best NBA players have figured out as well.

What Makes James Harden so special?

In Sprawlball, Kirk Goldsberry graphed the relationship between shot distance and points per shot in the NBA. 

Remember the general shape of a power law curve:

Now look back at the shooting graph again. You’ll see that exact curve replicated until the 22 feet mark. Then, the line breaks altogether. That’s where a shot moves from being worth two points to three.

In the NBA, shot values follow a power law curve. The most valuable shots on average fall in the range of a dunk to a 3-foot bunny. Then, 3 pointers become just as valuable as a 4 footer and are more valuable than any shot between 5 and 21 feet away from the basket.

As we can clearly see, NBA scoring follows a power law distribution. Shots at the rim and beyond the 3-point line are worth disproportionately more than all other shots.

But, the NBA also has the free throw. And that is what makes James Harden one of the game’s best players.

As of the 2020 coronavirus layoff, James Harden is an 85.8% career free throw shooter. So let’s say Harden gets fouled on a layup attempt. He goes to the charity stripe for 2 shots.

Statistically speaking, we can estimate that for every two free throws Harden takes, the Rockets will add 1.7 points to their total (.858*2). Of course, that’s not actually what happens. They get 0, 1, or 2 points. But played out over the course of a game or an entire season, we can confidently say that each free throw Harden takes is worth ~0.85 points.

But even that doesn’t fully explain why James Harden is so special.

During the 2016-17 season, Harden drew one hundred and sixteen 3-point shooting fouls. That’s 62 more than the second-ranked player, Lou Williams.

And, applying the same math we did a couple paragraphs earlier, we could expect these trips to the charity stripe to be worth just over 2.5 points. Over the course of a season, that adds up to 298 additional points (or over 3.5 per game).

So, the important thing to note here in terms of the power law is that there are actually three types of shots that are ideal in the NBA: bunnies, threes, and free throws. But as I wrote in The NBA & Match Quality, the important thing is to match your shot selection to your abilities.

Perhaps no player does this better–and benefits from the power law more–than James Harden.

For more insight into this, check out The Midrange Debate by Ben Falk at Cleaning the Glass.

NBA Championships and the Power Law

Okay, this is laughable.

Of 73 total NBA titles, two teams completely dominate all the others.

The Boston Celtics and Los Angeles Lakers have combined for 33 total championships. That’s equal to 45% of all championships.

You get the point. NBA championships follow a power law.

Team Culture and the Power Law

In Zero to One, Peter Thiel remarked that VC investors have a tendency to spend the most time with failing companies. Nobody likes their investment to fail. Stubborn investors who don’t recognize the power law’s effect on their returns poorly allocate their time amongst the companies in their portfolio. They spend the most time with the worst companies.

Personally, I see an application to coaching.

Whether you’ve coached in the professional arena or not, you likely tend to spend a significant amount of your time working with a small selection of your team.

If you want to follow the Thiel Approach to time allocation, you’re better off spending the majority of your time working with prospects–the few players on your team who seem to have the brightest future. Roughly speaking, you’d want to invest as much (or more) time and energy on them as you would on all other players combined.

While this might be good for getting maximal returns out of your most promising players, this can lead to negative psychological effects on players and will likely lead to cultural issues in your team.

The Roberts Approach is a different way to approach time allocation as a coach. I learned about it from Dave Roberts, manager of the Los Angeles Dodgers, during his appearance on the Flying Coach podcast. Here’s what he said:

It’s interesting and I think that we’ve all lived this where you get into coaching because you love the game, there’s a passion you want, you have the desire to teach. But as a major league manager, your scope obviously broadens. And there’s front office, there’s media, there’s ownership. And obviously, the coaches, training staff and the players and the players are always most important, but your time is certainly more limited. So it’s ironic that as you kind of work your way up the ladder, you get less and less time with what got you initially into coaching and what you love to do. And that’s connect with players. So, to that question, I try to make a point every single day to touch base with every single player and fortunately, I don’t have to worry about, you know, 50-60 players like Pete [Carroll] does, but I think that I talk to, touch, check in on every single day. And I think for the most part it’s kind of the bench players or the role players or the relief pitchers.

As manager, Roberts knows that it’s crucial to take the pulse of his players every day. That’s why he actually spends a disproportionate amount of his time talking with his bench players and relief pitchers compared to his starters.

By doing this, he’s tackling a problem that Phil Jackson wrote about in Eleven Rings: that good team chemistry was often more about investing in the role players than the superstars.

As Jackson wrote:

One of the hardest jobs of a coach is keeping the role players from undermining team chemistry…In basketball, the guys who hate you are usually the ones who aren’t getting as much playing time as they think they deserve.”

Your role and circumstance will call for different strategies when it comes to time allocation. It’s not as simple as maximizing returns because coaches aren’t economists. But as we’ve alluded to, most time allocation strategies won’t be evenly distributed. They will follow the power law.

The key to doing it well is to recognize it and not feel bad about it. The truth is that we live in a power law world.

But even in a power law world, there’s room for others.

Where the Power Law Isn’t…

Of course, it’d be wrong of us to assume that everything follows a power law. 

Every team needs role players. These players aren’t responsible for the majority of the team’s success, but they’re vital to winning championships. Remember that John Paxson hit the winning shot in ‘93. Steve Kerr did the same in ‘97. They weren’t stars. But they did have an important role on the team.

For a snippet of time, as the ball exited the fingers of Pax and Kerr, they became the most important player on the court. Is that a power law in and of itself? I don’t know. But I do know that moments like that are what make sports beautiful.

Thank you, Zack Jones, for proofreading this essay.