That shot accuracy is decreasing in distance from the hoop is a fairly obvious observation to anyone who has played (or watched) basketball. For example, here's the percentage and number of shots made by distance in the 2020-2021 season so far:

The first thing to notice is that, apart from a slight improvement in shooting accuracy between 5 and 15 feet, the further a player is from the hoop the less likely he is to make a shot. For 2-point attempts, each additional foot, on average, is associated with a 1.6 percentage point drop in accuracy. Past the three-point line, the per-foot penalty drops to 1.0 percentage points.

The second thing to notice is this decrease is highly nonlinear. The effects of distance are almost solely concentrated within the paint and beyond the three point line. There is scant evidence that, in equilibrium, distance matters within deep twos (5 to 20ft from basket).

The final thing to notice is that the accuracy on a deep two and a roughly equally-deep three are remarkably similar. I say "remarkably" because given the wide array of factors associated with shot selection, the cutoff is hardly arbitrary.

However, what matters in basketball is not baskets made but points made. As many analysts have pointed out, the 3 point "revolution" in basketball is largely driven by the recognition that when adjusting field goal percentages by their point value, 3-point attempts (3PA) offers substantially higher returns. Taking the previous figure, for example, and multiplying it by the points value of each shot, we can compute the expected value of taking a shot each distance. Now, unlike raw shooting percentages, we observe a discrete jump in the expected value of a shot around the three point line:

This local divergence is responsible for the increase in three-point shooting over the past several decades:

3-point shots have clearly increased in popularity, with much of the increase coming in the past several years. What's interesting about the above graphs is that they suggest the increase in expected value has basically come “for free”. As long as 3 point shots offer a higher expected return to 2 point shots, we would expect players to attempt them more. Taking more and harder 3PA would drive down the marginal benefit — as harder shots have lower accuracy — until the expected value of a 2PA and 3PA are the same. Nor does the change need to occur purely through changes in offense: the defense should shift toward defending three-point shooters to reduce the excess expected value they are achieving.

By way of analogy, if one asset pays higher risk-adjusted returns than another asset, we would expect investors to shift holdings into the more profitable asset until the point at which the returns of the two assets offer equal returns. This is a basic equilibrium condition.

The reason this additional EV appears to be free is that, over time, its not clear that the excess return of 3PAs has decreased, even as three-point shooting has rapidly increased. That is, players have taken more three-point shots without having to sacrifice anything in the way of accuracy and hence expected value. This is perhaps clearest by plotting the evolution of accuracy over time. As we see, there has been little change in accuracy around the three point line: