# Tag Archives: gametheory

## 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.

(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 win-win lose much-win much win much-lose much lose-lose

(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.