While I’m on a flight between Albuquerque to Oakland, let’s take a quick glance at another advanced analytic: Game Score. Game score is a metric that was developed by John Hollinger (one of the Godfathers of basketball analytics) to quickly give a rough estimate of a player’s contribution to a game. If a player scores 10, then the player had an average day. But if a player scored 40, then they had a superstar night. Think of this as an equivalent metric to Bill James’ (Godfather of Godfathers) game score metric for baseball. So in this post, let’s break down what it takes to be average.

Game score is given by the function

Points + 0.4FGM  – 0.7FGA – 0.4(FTA – FTM) + 0.7OREB + 0.3DREB + STL + 0.7AST + 0.7BLK – 0.4PF – TO

This equation may seem quite ridiculous, so let’s understand its utility. Here, we count the number of points scored by a player; the single most important value in any game. Similarly, we subtract the number of turnovers; which makes complete sense as a turnover eliminates a possession and therefore erases potential points off the board. The remainder of this equation becomes more muddled.

Here, offensive rebounds are valued more than defensive rebounds. Offensive rebounds are indeed tougher to obtain, and indeed prolong a possession; however, defensive rebounds terminate possessions for opponents. In this case, offensive rebounds are valued at 2.3 times more important than defensive rebounds. A steal, however, takes away a possession outright; therefore eliminating a possession for an opponent, which can be viewed as gaining an extra possession for their team.

So the weights somewhat make sense to a degree. However, we have not statistically tested them. So let’s allow them to stand on their merit.

Here, we are interested in identifying how well a shooter must perform in order to increase their game score. To do this, we must partition off the shooting elements of game score and see what they truly mean. In doing this, we will obtain break-even shooting percentages necessary to improve game score.

So the question is… how good of a shooter must you be in order to improve your game score?

Partitioning the Shooting Element

In the game score equation, the points scoring elements are given by

PTS + 0.4FGM – 0.7FGA – 0.4(FTA – FTM).

We can rewrite the number of points as

PTS = 3*3PM + 2*2PM + FTM.

Similarly, we can write the number of field goals made and attempted as

FGM = 3PM + 2PM,

FGA = 3PA + 2PA.

This allows us to write the scoring element of game score as

3*3PM + 2*2PM + FTM + 0.4*3PM + 0.4*3PM – 0.7*3PA – 0.7*2PA – 0.4*FTA + 0.4*FTM.

Arranging the three different types of shooting attempts, we obtain the nicely structured equation

All we have to do now to break even in game score is to set this equation to zero and treat each component as a vector space. In this case, we note that the number of attempts a player takes cannot be negative. Since we are interested in maximizing each field goal and free throw percentage, the coefficients cannot be zero. This therefore allows us to write each coefficient equal to zero. However, if we do this…

How’s My Shooting, Chief?

Setting the coefficient for three point percentages to zero, we obtain a three point percentage of 0.7/3.4; or 20.5882 percent. Similarly, if we look at the coefficient for two-point field goals, we break even at a percentage of 29.1667 percent. Yikes.

For free throws, all a player needs to shoot in order to break even in game score is 28.5714 percent. Shaq would be proud.

So let’s get this straight. If a player shoots 21 percent for three’s, 30 percent for two’s and 30 percent for free throws, then all they have to do to improve game score is shoot a ton? The answer is YES.

Comparison to Bazemore

Let’s consider a side-by-side comparison of this terrible shooter and Kent Bazemore of the Atlanta Hawks. Bazemore, over the 2017 NBA season, was a 34.6% three-point shooter (92 for 266) and a 44.6% two-point shooter (203 for 455). Bazemore had a mediocre free throw shooting season at 70.8% (119 for 168). Over a total of 73 total games, this equates to an average shooting-only game score of 5.4068 per game.

Let’s compare Bazemore to Shooty McShootsalot, who shoots 30% from three’s, 40% for two’s, and… let’s be generous here… 50% from the foul line. If this player has the exact same rate of shooting as Bazemore, then Shooty will make 80 three’s, 182 two’s, and 84 free throws. This results in a shooting only game score of 3.4863 per game. This means that a shooter who shoots worse than Kent Bazemore with the same load as Kent Bazemore will have a worse game score when compared to Kent Bazemore. That’s a good sign.

Now, what if this player’s usage is increased? Let’s say up from the 10.42% workload that Bazemore sees: 721 field goal attempts against the team’s 6918  (It’s even lower if you consider free throws) to 1150 field goal attempts; with the same ratios as Bazemore broken out. This means the player takes on 16.2% of the team’s offense.

In this case, the player is only taking 5 – 6 more shots per game than Bazemore would. If this carries out, this player now has an average game score of 5.51370. This identifies the player to be better than Bazemore, just by taking 5-6 more field goal attempts per game.

Let’s understand this… By taking 5-6 more field goals per game, with the percentages that the shooter has, this player will average 15.04 points per game but require an extra six to seven possessions to do this. This means that despite the game score being higher than Bazemore’s, the points per shooting possession goes down from 1.1107 (Bazemore) to 0.9547 (Shooty). Extrapolating to per 100 possessions and we have that the Hawks with Bazemore will gain roughly five points per game than the Hawks with Shooty McShootsalot.  This is an important note to make despite game score favoring the shooter.

How Do We Improve Game Score?

Game score can be improved the same way that PER gets scaled. However, instead of scaling at the high level, after shooting has been contributed, we scale based off the data; making game score a relative measure.

Let’s start by looking at the distribution of game score. Let’s take a look at Bazemore’s distribution of shooting only game score.

We find that the shooting portion of Bazemore’s game score actually lands at a mean of 4.2986. This is the average of scores, as opposed to the average of averages we performed above. Due to the skewness of this distribution, these values are not the same.

We’d like to standardize, however, there is one glaring problem if we simply standardize.

Game Scores are Not Just Shooting…

For every missed field goal attempt, a rebound can be made. A miss is equivalent to earning negative 0.7 points. However, a rebound is equivalent to either 0.7 points or 0.3 points. Good News: an offensive rebound effectively wipes out a player’s miss; however, if the player does not collect their own rebound, they are penalized despite having no actual negative impact on the play.

If we are to stick to our guns about this rebounding relationship, then we can indeed impose standardization. In this case, we identify that the league shot 23,748 for 66,422 from beyond the arc this past season. This resulted in a 35.7532 percent shooting. Similarly, the league shot 72,313 for 143,693 from inside the arc; a 50.3247 percent rate.

To enforce a league average for shooting, say 35% for three’s and 50% for two’s; we can reverse engineer a better multiplier or split up FGM/FGA. For a better multiplier without separation, we can form the regression problem:

In doing this, we get instead of 0.7, we obtain 1.2117008. Let’s set this to 1.2.

Now for offensive rebounds, we should keep in faith of keeping a possession alive and adjust 0.7 to 1.2 as well. However, in determining the relationship of offensive and defensive rebounds, we scale the defensive rebounds mechanism according to distributions of rebounds.

Adjusting Rebounds

Last year, there were 24,937 offensive rebounds compared to 82109 defensive rebounds. This means that 23.2956% of all rebounds were on offense. With the value of offensive rebounds being three times more desirable, we can adjust defensive rebounds to have a game score of 0.3 to 0.4.

Other Factors

We can adjust other factors according to the data as well. In this case, we can take assists, steals, turnovers and personal fouls and weight them against the number of points scored. These values are a little more muddled as the question is “why 0.7 assists per point?” in the original model. Take note… last season there were 0.2578 assists per non-free throw point scored. So any desirable weighting can be used here; provided it passes some sort of eye test. There are many possibilities, but beware; they need to make sense. 20% from three’s is enough to get a player yanked after a while; but in considered as contributing to a score.

As a side note, I definitely get ten boards, three assists, and shoot 3-of-12 for three’s in pick up ball nowadays. That gives me a nice 6.9 game score. Never mind we lost 21 – 17.

So how would you go about adjusting game score?