Marvin Williams Makes a Hypothetical Deal

Posted on December 16, 2007 by


Back in 2005 the Atlanta Hawks held the second pick in the NBA draft.  With point guards Chris Paul and Deron Williams on the board, the Hawks opted for forward Marvin Williams. 

Last year, while both Paul and D.Williams were posting above average campaigns, M. Williams was decidedly below average.  And this is true regardless of which summary statistic you like.  Per 48 minutes he posted the following:

Win Score: 5.3

NBA Efficiency: 17.7

Game Score: 11.8

To put these numbers in perspective, an average power forward would have posted the following marks per 48 minutes.

Win Score: 10.3

NBA Efficiency: 23.4

Game Score: 15.0

On Game Score

As we can see… oops, wait a minute. I probably need to explain Game Score.

John Hollinger created the Player Efficiency Rating (PER).  Game Score is Hollinger’s simplified version of his complex PER model.  It’s calculated as follows:

Game Score = PTS + 0.4*FGM + 0.7*ORB + 0.3*DRB + STL + 0.7*AST +0.7*BLK – 0.7*FGA – 0.4*Missed FTA – 0.4*PF – TO

PER, which is a per-minute measure, involves more than this simple equation, but the results are almost identical.  For the 458 players who appeared in 2006-07, Game Score per-minute and PER had a 0.99 correlation.  In essence, Game Score and PER are telling the same story.

When you look at the Game Score formulation you will note that a player receives credit for each point scored, as well as an additional credit for each field goal made.  A player is also charged for each field goal he takes. 

When you analyze these values it becomes apparent that a player who makes 30% of his two point field goal attempts will still come out ahead. Let’s quickly review the numbers.  If a player made 30 out of 100 shots from within the arc he would score 60 Game Score points, receive an additional credit of 12 for making 30 shots, and be charged 70 for his attempts.  In sum, his Game Score would rise by 2 (60+12-70).  And if he took 200 shots his Game Score would rise by 4 (120+24-140).  In other words, the more he shoots – despite the fact that by any reasonable assessment this player is not very good at shooting – the better he looks.

And the story is the same from beyond the arc.  From three point range the player only has to convert on at least 21% of his shots to come out ahead.  And again, once he passes this threshold, the more he shoots, the better he looks.

Marvin Makes a Hypothetical Deal

For those who have read the WoW Journal for awhile, you know I made this argument in November of 2006.  So this has been said before.  What I wish to do in this column is apply the lesson we learn from Game Score to the Marvin Williams we saw in 2006-07.

Williams took 706 shots from the field last season, of which 45 came from beyond the arc.  From this distance he converted on 11 shots, for a conversion rate of 24.4%.  Such a mark is below average, but still exceeds the threshold noted above.  From within the arc Williams made 44.6% of his attempts.  Again, this is below average, but this mark also exceeds the Game Score thresholds already detailed.

If we consider all that Williams did from the field – or his adjusted field goal percentage – we see a mark of 44.1%.  This means that per field goal attempt, Williams averaged 0.88 points, and again that is well below average.

Turning to rebounds, we also see problems.  The average power forward would grab per 48 minutes 3.7 offensive rebounds and 7.7 defensive rebounds.  Williams, though, only captured 1.9 offensive boards and 5.6 defensive rebounds per 48 minutes.  So on the boards, he was below average as well.

Now let’s imagine the following scenario.  The coaches for Atlanta come to Williams and note his below average performance with respect to Game Score.  And since Williams spends so much of his playing time at power forward, the Atlanta coaches ask Williams if he could boost his productivity by grabbing more rebounds, especially on the defensive end.

But Williams is smart.  He has studied Game Score and understands how this is calculated.  Consequently, he comes back to the coaches with an offer.  Why not re-design the offense so that his field goal attempts double?  As Table One illustrates, even if his below average shooting efficiency is unchanged, doubling his field goal attempts would cause his Game Score to rise to 17.4.  And this level would easily pass the average mark for a power forward. 

Table One: The Marvin Williams Experiment

Williams goes on to deflate the coaches’ focus on defensive rebounds.  Since each defensive rebound increases Game Score by only 0.3, focusing on this factor does not generate much of a return.  In fact, if Williams tripled his defensive rebounds – so that he was now averaging 13.2 rebounds per game – his Game Score would only rise to 15.2.  Yes, that would be above average.  But not nearly as good as Williams just doubling his shot attempts.

We should note that if Williams doubled his shot attempts, his per game scoring mark – even with shooting efficiency and free throws unchanged – would rise to 22.8.  In other words, he would be among the games scoring leaders.  And since scoring is the driving force behind play pay, the offer Williams is hypothetically making shouldn’t just make his coaches happy, it should also make Williams a much richer person.

Again, this story is hypothetical.  We do not know that Williams is being evaluated in terms of Game Score or PERs.  But this scenario does highlight the basic issue with the Game Score approach.  This model tries to incorporate the idea of “usage” in the evaluation of players.  Because it is believed – although in my view not systematically proven (see what Martin Schmidt said in June of 2006)  – that more shots leads to less efficiency, Game Score is constructed to give players extra credit for taking shots. Unfortunately, this model ends up teaching us that inefficient scorers can increase their value by simply taking more shots. And the increase in value dwarfs a substantial increase in rebounds.   

When we turn to NBA Efficiency (a metric that is also highly correlated with Game Score and PERs), we don’t see exactly the same story.  Yes, an inefficient scorer will increase his NBA Efficiency mark with more shots.  But because defensive rebounds are worth the same as points, tripling defensive rebounds results in a higher NBA Efficiency level than just doubling shots.

Of course, that shouldn’t make us happy with NBA Efficiency.  Think about baseball for a moment.  If a hitter had a 0.200 batting average, we would say he is below average.  We would not think that such a player would be helping his team by doubling his at-bats.  Giving more at-bats to an inefficient hitter should hurt a baseball team, not help.

When we look at Win Score, that’s the story we see.  Williams posted a per-48 minute Win Score of 5.3 last season, which was below average.  If he doubled his shot attempts, his per-48 minute mark would fall to 3.5.  In contrast, if he grabbed more rebounds he would be well above average.

Reconciling Game Score and Win Score

Earlier I noted that Game Score and PERs were highly correlated.  What of Game Score and Win Score? Here are the two metrics side-by-side.

Win Score = PTS + REB + STL + ½*BLK + ½*AST – FGA – ½*FTA – TO – ½*PF

Game Score = PTS + 0.4*FGM + 0.7*ORB + 0.3*DRB + STL + 0.7*AST +0.7*BLK – 0.7*FGA – 0.4*Missed FTA – 0.4*PF – TO

If we look at per-minute performance in Win Score and Game Score from 2006-07, we find a 0.82 correlation. 

What if we change how Game Score regards shooting efficiency?  If we drop the extra reward for field goals made (the 0.4*FGM term) and change the value of a field goal attempt from -0.7 to -1.0, the correlation between per-minute Game Score and Win Score rises to 0.93. 

What if we also change the return to defensive rebounding? If we also change the value on defensive rebounds from 0.3 to 0.7 – or the same for offensive rebounds in Game Score – the correlation between the two metrics now rises to 0.98.

And if we say that each rebound is worth a point, then the correlation rises to 0.996. 

In sum, the real difference between Game Score and Win Score is how it treats shooting efficiency and defensive rebounds.  One doesn’t have to agree that each rebound is worth a point (although my regressions say they are).  But if we agree that defensive rebounds are worth at least as much as offensive rebounds, and inefficient shooters should not be rewarded for just increasing their shot attempts, then we have essentially resolved all the substantial differences between Game Score and Win Score.

The Importance of All This

And if all that happens, what will that do for us? My answer returns to baseball. Earlier I talked about a player with a 0.200 batting average.  Batting average has been around since the 19th century, and it has been criticized as inadequate since at least the early 20th century (see Alan Schwarz’s The Numbers Game).   And yet, during every single televised baseball game, it’s batting average that the announcers note.  Remember, this is a measure that says a home run is equal to a bunt single.  Still it’s cited in every single game. 

Now when I see this, I always throw my shoe through the television set.  In fact, we keep a large supply of televisions on hand for every game I watch.  Whenever batting average is mentioned, I throw my shoe through the screen.  And then we spend a few minutes hooking up a new TV.  Yes, this takes up time.  But I don’t want to hold my anger within.  It needs to be expressed.

Okay, no one does this (I hope). Although baseball keeps citing measures that might be “wrong”, life goes on.  And I feel the same way about measures in basketball.  John Hollinger can spend the rest of his life (hopefully a long one) calculating PERs, writing about PERs, etc… And this should not make any difference to anyone.  This is just basketball.  We are not talking about global warming or the latest macroeconomic model.

We should remember, global warming and macroeconomic models matter. If we get those wrong, then very bad things happen to very real people.  If Marvin Williams gets to collect more money because he takes more inefficient shots, well that’s good for Marvin Williams and his family.

In essence, I think these stories are “interesting.”  But I don’t consider this particular issue to be “important.”  And that distinction sports fans should always keep in mind.

– DJ

Our research on the NBA was summarized HERE.

The Technical Notes at provides substantially more information on the published research behind Wins Produced and Win Score

Wins Produced, Win Score, and PAWSmin are also discussed in the following posts:

Simple Models of Player Performance

Wins Produced vs. Win Score

What Wins Produced Says and What It Does Not Say

Introducing PAWSmin — and a Defense of Box Score Statistics

Finally, A Guide to Evaluating Models contains useful hints on how to interpret and evaluate statistical models.