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The rise of the three-pointer

The other day I was wondering about whether or not teams act rationally when making their shooting decisions, and I have some ideas about how to determine this, but I’ll post on that later. For now, I’d like to present a something really interesting I discovered as I was going through the data.

First of all, over the Modern era (1979-80 through today), points per three-point attempt have almost exactly equaled points per two-point attempt, at approximately 0.973 (which is, interestingly, almost exactly one point per fga). This lead me to believe, at first, that it would be a good measure of shooting-decision rationality to compare a team’s or a player’s points-per-shot from inside and outside the arc: if pts/3fga, for example, is substantially higher than pts/2fga, it would seem to indicate that too many bad two-pointers are being attempted, and that some of those should be passed out for three-point attempts (but more on this in another post).

What I found in doing a little EDA, however, indicated that if this is how we conceive of rationality, the league as a whole has not been rational on a year-by-year basis (which sort of undermines my claim). What I did find, instead, was that pts/3fga are now higher than pts/2fga, leaguewide. However, this has not always been the case:

Adoption of the three-pointer


It appears as though when the three-pointer was first introduced (the season which I mark as the beginning of the “modern” era) defenses were at first unprepared to handle it, but then the quickly adapted, and the ratio of pp3/pp2 was fairly low for a while. However, from the mid-80s onward, that ratio steadily increased, until today, when the league seems to have achieved some sort of equilibrium at which pp3 is noticeably greater than pp2. Not being a basketball historian, this is just the best explanation I have come up with, and I would be very interested in hearing someone more knowledgeable give me a better story to go with the data.

NBA Game Theory

It is common basketball folk wisdom that teams which share the ball unselfishly, passing around until someone has an open shot, should be successful. Such a concept would appear to make sense, if we accept (a) increased passing leads to increased likelihood of a relatively uncontested shot for any given possession, (b) relatively uncontested shots have a higher likelihood of going in than contested shots, and c) teams with higher field goal percentages, all else equal, are more successful than teams with low shooting percentages.First, I would like to illustrate (although it’s not proof) the likely validity of these three claims; first with correlation coefficients, and then with a scatterplot. Since I don’t have statistics for the number of uncontested vs. contested shots, we’ll combine (a) and (b) and assume that higher field goal percentages result from more uncontested shooting. For lack of an easier way to display them, the following is a list of relevant correlations.

  • cor( as/fga , fg% ) = 0.687
  • cor( fg% , win% ) = 0.429
  • cor( as/fga , win% ) = 0.462

These are all high, and the correlation between assist ratio and shooting percentage is very high. At this point, I should note that we cannot here make any causal inferences for certain. Correlation is not causation, and neither are the easily seen trends in the scatterplot below. For example, it is almost certain that while better passing to the open man leads to more wins, teams with more wins likely have better players who are better passers and shooters in the first place, so we have an endogeneity problem, at the very least.

Nevertheless, I present visual evidence of the relationship; assist ratio is on the x-axis, increasing from left to right, win percentage is on the y-axis, increasing from bottom to top, team field goal percentage is coded by color, increasing in reverse-rainbow order (red is best, purple is worst). Note that the trend is obviously up and to the right, indicating a positive association of the two statistics. Also notice that the top right is more red-orange-yellow than the bottom left.

Assist Ratio by Win% by FG%

(Click to enlarge)

Finally, I’d like to develop some theory concerning why, if passing to the open man leads to wins (which we have some evidence for, but haven’t technically proved), some teams just don’t pass as much as others? To answer this, we may turn to game theory for a classic game with a similar idea.

I argue that the question each player faces about whether to take a contested shot, or pass to another man, is in some sense a Prisoner’s Dilemma. If we assume that players are financially motivated, and that contracts are based more on individual statistics than on overall team success (evidence for which assumption awaits another article), the payoffs to being a high scorer on a bad team are probably better than being a low-scorer on a better team.

The Prisoner’s Dilemma for a two-player team thus looks like this:

  Look to pass to open man Be a chucker
Look to pass to open man winwin lose muchwin much
Be a chucker win muchlose much loselose

(Note that we’re also assuming that both players decide simultaneously before the game or season which kind of player they’ll be.) The Nash Equilibrium for this game is that both players will choose to shoot first, pass second. This can be extended to apply to five-player teams as well, with similar results. In fact, the more players there are, the more likely one would be to defect, and since the win-win outcome requires complete cooperation, increasing the number of players increases the likelihood of all defecting to Chucker. To the outide observer, this is disheartening, because the win-win outcome is a team with a good record, the lose much/win much outcomes at least might result in a scoring title, and the lose-lose outcome results in the New York Knicks. Yet, all of the individual incentives pull toward the lose-lose outcome. I can offer no better evidence for my thesis than the following table of teams thus far in the ’07-08 season, ordered by assist ratio:

Team as/fga fg% win%
PhoenixSuns 0.320 0.490 0.703
UtahJazz 0.317 0.490 0.564
NJNets 0.303 0.433 0.486
BostonCeltics 0.293 0.469 0.833
LALakers 0.287 0.475 0.703
DetroitPistons 0.286 0.460 0.737
PortlandTrailBlazers 0.281 0.461 0.622
SanAntonioSpurs 0.281 0.458 0.694
DenverNuggets 0.279 0.458 0.611
TorontoRaptors 0.268 0.451 0.541
CharlotteBobcats 0.267 0.449 0.378
DallasMavericks 0.266 0.470 0.684
LAClippers 0.265 0.424 0.303
MilwaukeeBucks 0.265 0.453 0.395
IndianaPacers 0.265 0.443 0.436
AtlantaHawks 0.265 0.445 0.500
OrlandoMagic 0.260 0.463 0.590
MiamiHeat 0.259 0.460 0.216
MemphisGrizzlies 0.257 0.459 0.270
HoustonRockets 0.256 0.444 0.526
ChicagoBulls 0.256 0.417 0.400
GSWarriors 0.255 0.451 0.579
NOrleansHornets 0.248 0.448 0.676
SeattleSupersonics 0.244 0.436 0.243
Philadelphia76ers 0.242 0.444 0.368
WashingtonWizards 0.240 0.448 0.556
ClevelandCavaliers 0.232 0.430 0.514
SacramentoKings 0.228 0.456 0.417
MinnesotaT-wolves 0.224 0.436 0.139
NYKnicks 0.217 0.435 0.278

I have tried to make the case that more passing leads to better shooting leads to better records. However, from a rational choice perspective, each individual player has an incentive to shoot (unless you genuinely believe that they’re “in it for the love” and not the money). These conclusions make it all the more easy to admire teams like the modern Phoenix Suns and the Lakers and Celtics of the mid-late ’80s–teams with players who overlooked their own personal monetary incentives, and chose to assist each other to victory.

Thanks to basketball-reference.com and dougstats.com for the statistics. If you enjoyed this article, you might also enjoy the statistics and visualizations found at Envisioning the NBA.