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 and for the statistics. If you enjoyed this article, you might also enjoy the statistics and visualizations found at Envisioning the NBA.

11 responses to “NBA Game Theory

  1. Pingback: NBA similarity networks « The Arbitrarian

  2. Pingback: Game theory for team games « Green-tinted Glasses

  3. Great Job! Keep up the good work.

    I really enjoyed this article and I think that it either highly emphasizes why having a great point guard on your team is vital to the teams success or overlooks the fact that certain players are smart enough to see that helping a teammate out (either screening, huslting, passing, etc.) will actually elevate his own chances of succeeding. Obviously the later is impossible to quantify but I think it is something that must be considered, effective pick and roll’s and pick and pop’s do not work without a pick and pull up jumper or drive to the basket to set up the other move.

    Very interesting to see the Nets low win % and Kidd’s high assist ratio, probably why Mavs traded for him. Kidd, Nash and Deron Williams team’s are the three top teams on assist ratio (I’m suprised to see New Orleans/Chis Paul so low).

    Think that you are right on about how the best teams are the ones with a good combination of scorers, hustlers, and passers, with the very best teams having those special players that figure out how they can best help their team.

    I think my only complaint is with some missing factors to the prisoners dilemma. If a player shoots too much and is not good at it his coach will not be happy with him and may sit him out of the game. It is also true that if a player does not run a play the correct way (i.e. missing a screen or shooting when they should be passing) the coach will once again have some incentive to sit the player down. One way to boost individual stats is to avoid being benched by the coach and increase minutes played (except when pace adjusted). At a certain point pride has to become a psychological factor in a players performance, I don’t think that you can base everything off of having an incentive to shoot if the player knows he is better off not shooting and staying in the game rather than shooting and possibly coming out of the game and decreasing his personal stats.

  4. Great post.

    The one reaction I would have is adding onto what Viduka said above.

    The problem is that on bad teams, there may not be enough good shooters to go around. You note above that, “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.” The problem with the game theory application as you outline it is that it assumes the two players are of equal talent.

    However, in many scenarios, the better shooter taking a contested shot is actually still a higher percentage shot than the poor shooter taking an open shot. On a poor team, the better player may not have a better alternative than themselves shooting a contested shot. That said, the team is still not successful because the other team’s defense is focused on them. Even if that focus pushes it to the point that the other players’ shots are marginally better, the team is unsuccessful, because they don’t have enough good players.

    Where that marginal switch occurs between it being better for the good player to take a shot or the bad player who’s largely undefended relies on the player and the coach making the right decisions about this. Unfortunately coaches aren’t always perfect and probably don’t put their optimal lineup on the floor. And for sure players have a hard time in the heat of the moment feeling that moment at which a relatively poor teammate becomes a better option than themselves. Likewise, it’s probably hard for them to adjust their style when they do move to a good team to one of sharing the ball from one of dominating the ball from a past that likely included that as the optimal success option in high school, college, and their prior experience on bad NBA teams.

    All this said, your conclusion that championship teams tend to exhibit less selfishness in working the ball around for the best shot seems valid. Unfortunately, the American public does not seem to care for that style of basketball as much as one that is superstar driven as evidenced by the media ratings after Pistons and Spurs championships.

  5. Is FG% here raw FG% or effective FG% adjusted for 3 pt shooting? This might make a difference.

  6. This is brilliant stuff, Arbatrarian. I love shit like this and gave you the respect you deserved on my little Milwaukee Bucks blog. Keep it up!


  7. rapidadverbssuck

    Crow: this is just regular-type FG%. You’re right, it might make a difference, but probably not enough to eliminate the positive correlation.

  8. I wonder if there are longer term financial benefits to being on a winning and especially championship team, even if individual stats have suffered along the way. Players who go on to do broadcasting, coaching and management are almost always from this pool. Extended careers, even as a so-called “veteran bench presence” are available to team-player types that have rings.

  9. Nice article.

    I saw you used ass/fga for your correlations. My first thought was whether the results would get better or worse for ass/fgm, as well as the TS% adjustment mentioned above.

    You’ve got some nice graphics going here. What are you using to plot your data?

  10. I like the material, but the game theory stuff is just really, well, rough. I’ll try to provide some insight.

    Overall, I think the main problem is with the assumptions.

    Sure, If the NBA game only had one skill – shooting – and that your scoring total was the only thing you had to maximize, then, a prisoner’s dilemna framework might be acceptable.

    Unfortunately. it sounds too much like pick-up at the YMCA (or rec league for 12 years olds) and less like the guys who play in the NBA.

    Instead, the way I thought about it was to frame the problem as an optimization exercise.

    Given the options open to a player, what would he wish to do? What does he want to maximize?

    In my mind, the average NBA player maximizes his lifetime income (just like all other agents). In turn, I find it hard to argue that they way he does this is by maximizing the number of shots (unless shooting is your core competency: kapono, peja).

    On the contrary, I believe it’s a lot more likely that a player wants to maximize playing time (especailly for contract year guys) Minutes are a primary driver in the levels of most statistics. I think this holds for majority, especially for guys not on the rookie scale and in the middle of their careers.

    For a sizable percentage, however, I think they may max a combo of other things, maybe team winning percentage or a special category (think: 3 point shooting: barry, shot blocking: camby, ratliff, mutumbo). These tend to be older guys. Role players.

    Given these objective functions – and the construction of two types of players – where can we go?

    1) Well, we can go back to game theory. You might be able to build a zero-sum game model for teammates who are fighting to maximize playing time. If you want, you could build in kindness functions and multiple types (with tools from uncertainty and behavioral economics). You might get interesting results. It might however lead you far afield from the original question about passing and teamwork.

    2) Or you might get traction on something built on Evolutionary Stable Strategies (ESS). A more complex model, sure. You’d have an NBA team composed of “agents” with a mix of objective functions and see who performs best through simulation. Complexity studies do this well (Scott Page). Been used to study diversity. Might be useful lens for unselfishness or teamwork.

    3) While both 1 and 2 take you down roads somewhat far afield from the original assertion that more assists lead to better shots and therefore, better performance, you might be able to attack the original question with better data and technqiues.

    My first goal would be trying my darnest to get rid of some the endogenity problems.

    The main sticking pont for me (and I may be missing something here):
    – assists can only be registered on made baskets
    – field goals are not the only way – but a sizable way – in which teams score points
    – winning teams score more than their opponents (by construction)

    thus, how can we accurately assess the relationship between passing and performance – if the two naturally move together?


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