EZPM: Yet Another Model for Player Evaluation


Note #1: Before I get into this, I want to make it clear that many, perhaps most (hell, maybe all), of the ideas in the proposed model I am going to describe are not new. They may seem new to you, but I promise there are folks out there who have thought about these things before and heavily influenced my particular choices for all terms and coefficients in the model. Some of these folks you may have heard of before, including Dean Oliver ("Basketball on Paper"), John Hollinger (Pro Basketball Prospectus, ESPN), Dave Berri (Wages of Wins, Stumbling on Wins), and Dan Rosenbaum (believe he does stats for the Cavs for the past several years).

Note #2: Another motivator for my doing this is to help others get up to speed on (most) of the issues that need to be addressed when considering or developing player valuation models. As an engineer, I'm a big fan of rigorously defining a problem before tackling it. In developing this model and writing up my findings, it has helped me to better understand the problem (and basketball, hopefully). It has also raised many questions that need to be tackled going forward.

Note #3: This is the start of a discussion, not the end of one.

Note #4: Some will take this post as an implicit criticism of some other existing models. To some extent, that is obviously true. Let me name one, to be exact, since it's the elephant in the room: Wins Produced or WP (the metric developed principally by Dave Berri). First, let me emphasize that my understanding and appreciation of WP is what led me to start considering alternative models. My model, on the face of it, is not drastically different from WP. In fact, since both models are tied to point margin, it is really only the player valuation aspects that will be different. Of course, you will say, that is a large part. True. And if it weren't important, I wouldn't have bothered with all this. Finally, I would add that it has become clear to me that WP is not really going to change any time soon. Many of the components of my proposed model could easily be fit into the framework of WP, but I don't see that happening. If my post inspires Berri and others to re-consider their models, great. If not, that's fine, too. I don't mind rowing the canoe solo. That pretty much sums up my entire academic career. If this post inspires others to develop their own models or revise mine, that would be the greatest outcome of all. Go for it! (And let me know the results.)

First, what is a +/- model? Ok, let me back it up one step. What is +/-? At the team level, this is the number of points a team goes up or falls behind while it is on the floor. If starters played an entire game, this would simply be the point differential at the end of the game. Because players come in and out of the game, +/- typically refers to the number of points a team goes up or down while a player was in the game. In other words, +/- is assigned to individual players, but represents a team outcome. If Stephen Curry's +/- is +10 in a game, that doesn't (necessarily) mean Curry created a +10 point differential by himself. All it means is that while Curry was on the floor, the Warriors outscored their opponent by 10 points. Therefore, to be a meaningful statistic for evaluating players, what we really want to know is the individual +/-, or how a particular player contributed to the aggregate +/-. Say Curry was +5, Ellis was +6, Lee was +2, Biedrins was +1 and Dorell Wright was -4. In this example, the team +/- is +10, but Dorell Wright was actually a negative contributer. Going a step further, we can foresee a situation where some players are attributed +/- stats simply by playing with 4 other really good players.

One more thing, before I go further. Why do we care about +/-? Intuitively, it should be obvious that teams that score more points than they allow will win games. In fact, there is such a good relationship between +/- and wins, that we can actually formulate a simple equation that predicts wins based on point margin (another term for +/-). For example, Fig. 1 shows wins as a function of point margin per 100 possessions (simply offensive rating - defensive rating) for all teams for the 2009-10 NBA season.

Figure 1

Here, the "winning formula" (literally) is:

  • W = 2.54 * p.m. + 40.9

A team with a p.m. of zero, wins about 41 games (hey, that makes sense, right?). A team with a p.m. of +10 (that would be San Antonio currently), should win about 66 games. The Warriors with a p.m. currently of -6.6 are predicted to win 24.5 (ugh). Now, it should be noted that p.m. can be adjusted for strength of schedule, which makes a lage difference early in the season, and between conferences, since the East is weaker than the West (at least, I think that's still the case). Now we can move on...

So, that's what +/- is, and that's also why it should be clear that we need some sort of model for decoupling individual contributions from team +/- stats. How do we do this?

Fundamentally, it's a simple problem. What creates +/-? Points scored by the team minus points scored by the opponent. So, we can just take points scored by each player, distribute the points allowed by the team across all players evenly, and voila! That would be a very simple way to do this. Would it be "correct"? Well, technically, yes. All the points scored minus all the points allowed has to add up to +/-. But is it "right" in the sense that it properly attributes value? What about all the other things that players do besides score? What about rebounds? Assists? Steals? Turnovers? Why do we account for these stats? And how do we account for these stats? I'll get to the how in a bit, but let me address the why, right now. Points don't travel well. You can take that almost literally. If you put a bunch of high scoring point guards out on the floor, they probably won't come anywhere close to their individual +/- stats, because they will lose a ton of possessions due to poor rebounding. What if we put together a team of all centers? Well, they will probably get a ton of rebounds, but they won't be able to get the ball up the floor because they are such poor ball handlers compared to the guards. In short, a team is the sum of five positions, each of which bring their own strengths and weaknesses. To have a useful model — ideally, a predictive model — we need to account for everything. Well, everything that we can get our hands on.

The way that Dean Oliver, Berri, and others have created such models is by considering the possession as the atomic unit of basketball. A possession is usually defined by what ends one: FGM (field goal made), DRB (defensive rebound), FTM (free throw made —well, some of them, anyway), TOV (turnover). That's it. Each team essentially will have the same number of possessions during a game. Another useful fact is that the average number of points scored or allowed per possession in the NBA is 1.0. (Technically, it's a few hundredths of a point higher, but 1.0 is going to be good enough for our purposes.) With this number in mind, we can actually assign a value to the result of every possession at the team level and the player level. This is the marginal point value. Let me start with the team level:

Result Offense Defense
FG (2 PT) +1 -1
FG (3 PT) +2 -2
FG miss -1 +1
FT made +0.5 -0.5
FT miss -0.5 +0.5
And1 +1 -1
ORB team +1 -1
DRB -1 +1
Assist 0 0
Block 0 0
PF 0 0
TOV -1 +1
STL -1 +1

Note that assists, blocks, and personal fouls all have values of zero. The reason for this has to do with accounting. At the team level, all three are already accounted for by other results, either made or missed field goals (assists and blocks, respectively) or free throws (personal fouls).  There is no need to count these again. At the player level, we will account for these. So, the next step is to distribute value at the player level so that points add up to the same total at the team level. Does that make sense? Here's how I distribute marginal points across players (explanations follow):

Result Offense Defense
2PT Assisted FG (shooter) +0.7
2PT Assisted FG (passer) +0.3
3PT Assisted FG (shooter) +1.4
3PT Assisted FG (passer) +0.6
2PT Unassisted FG +1.0
3PT Unassisted FG +2.0
Any FG missed (minus BLK) -0.7 0.14 (all)
2PT FG made -0.2 (all)
3PT FG made -0.4 (all)
FT made (minus And1) 0.5 -0.5
FT missed (minus And1) -0.5 +0.5
And1 made +1 -1
And1 missed 0 0
ORB +0.7 -0.7*DRB%
DRB +0.3 -0.3*ORB%
BLK -0.7 +0.7
TOV -1 +0.2
STL +1

Let me go through the rationale for each of these assignments:

  1. Field goals: On offense, the +1 or +2 values for made shots are obvious, and to my knowledge, all marginal value metrics (Oliver, Berri) use these values. For missed field goals, we encounter the first bit of separation from WP, namely that a player is debited only a portion (0.7) of a marginal point. The reason for this is that there is a 30% chance that the offense will get an ORB and the possession will continue. I discussed this a bit a few days ago . WP subtracts a full point in this case.
  2. Assists: In my model, an assist is valued at 30% of the marginal value of the type of field goal that is made. This is one of the weakest parts of my model. The "true value" of an assist is not currently known (not that it is not knowable, though). I think everyone can agree that assists are helpful. Not all players can create their own shot. If we completely ignore the assist, then a player appears to be a more efficient scorer than he might otherwise be on a team lacking a good point guard. Although my 30% is essentially picked out of thin air, it is important to note that the total value of the assisted field goal is still +1.0 or +2.0 at the team level. This is not true for WP, which essentially double counts assists (or I should say 1.5X counts to be more accurate), since there is no distinction between assisted and unassisted FG in that model.
  3. Free throws: Fairly straightforward, except accounting for these is dramatically different if we are using box score stats or play-by-play (PBP) data. If we use PBP data, FT can be attributed easily (but analyzing PBP data is hard). If we use box score stats, we don't exactly know who to blame for giving up free throws, so we apportion blame to the players proportional to their PF, noting that 0.44 FTA is roughly one possession.
  4. And1: These are free points, since the player would have already been credited with a made field goal. You are credited value for making one, but not debited for missing.
  5. Blocks: In my model, the +0.7 coefficient for blocks has real meaning, since it gives the full value of the opponent's missed field goal to the player getting the block. To me, this makes perfect sense, since the guy getting his shot blocked missed the field goal and loses 0.7 pts. Shouldn't the blocker be credited with exactly the same amount?
  6. Let me do STL and TOV before getting to rebounds: A steal is worth +1, since you are taking away a possession from the opponent. A TOV is -1, since you are losing a possession and any chance for scoring. Team TOV should be debited or credited evenly among players (in the absence of a better model).
  7. Rebounds (yay!): This is where it gets interesting. Here's the basic logic. A missed field goal is worth -0.7 pts. The other -0.3 pts are lost, if the team does not get the ORB. On the other hand, I credit the defense +0.7 (distributed evenly among 5 players) for creating the missed field goal, and +0.3 to the player who gets the DRB. On the offense, the players on the team that "gave up" the DRB are debited by an amount that adds up to -0.3 and is proportional to the league average DRB% at their position. So, a PG loses a tiny bit of value, while the PF/C loses quite a bit more. Two things to note here. 1) This is vastly different from the WP model. 2) Everything adds up and defense is accounted for. Now, you may argue it's not right to split the defensive credit evenly. I agree! Well, I don't disagree. Truth is I'm not sure right now what the best way to debit value (maybe subtract 0.7 from the position counterpart that scored), but what I am sure of is that there needs to be this credit/debit for defense, and it needs to be of larger value than it is in the WP model. In the WP model, a DRB is worth +1. You can see now that is the equivalent of crediting the player with creating the missed field goal (i.e. the entire defensive possession). Is that a fair way to value the player? I dont' think it is.

Ideally, we would apply this scoring system to PBP (play-by-play) data. But that is hard, and I have not implemented such a system yet. Over Thanksgiving, however, I checked that everything adds up by going through the PBP data line by line for a single game (remember that Denver game?). In lieu of PBP data, we can use box score stats. I've done that for the Warriors through Friday's game against the Heat. As a point of comparison, I also ran the numbers for the Spurs. Here are the numbers (for players with > 100 minutes played):

2010-11 Warriors

Stephen Curry PG 673 76.5 5.6 -29.5 -2.2 47.0 3.4
Monta Ellis SG 918 63.1 3.4 -43.2 -2.3 19.9 1.1
Reggie Williams SG 515 44.2 4.2 -49.2 -4.7 -5.1 -0.5
Jeremy Lin PG 134 -8.9 -3.3 6.9 2.5 -2.0 -0.7
Dorell Wright SF 909 41.8 2.3 -59.5 -3.2 -17.7 -1.0
Rodney Carney SF 266 -0.9 -0.2 -24.7 -4.6 -25.6 -4.7
V. Radmanovic PF 264 12.5 2.3 -38.3 -7.1 -25.7 -4.8
David Lee PF 539 12.0 1.1 -67.4 -6.2 -55.4 -5.1
Charlie Bell SG 106 -7.0 -3.2 -7.0 -3.3 -14.0 -6.5
Andris Biedrins C 636 6.9 0.5 -93.0 -7.2 -86.0 -6.7
Jeff Adrien PF 162 4.3 1.3 -26.3 -8.0 -22.1 -6.7
Dan Gadzuric C 217 -2.7 -0.6 -33.0 -7.5 -35.7 -8.1

2010-2011 Spurs

Manu Ginobili SG 735 149.2 10.2 -10.2 -0.7 139.0 9.5
Richard Jefferson SF 729 120.2 8.3 -39.4 -2.7 80.8 5.6
James Anderson SG 106 16.8 8.0 -5.3 -2.5 11.4 5.4
George Hill PG 596 79.0 6.7 -15.3 -1.3 63.6 5.4
Tony Parker PG 758 93.2 6.2 -18.5 -1.2 74.7 5.0
Gary Neal PG 365 50.7 7.0 -20.2 -2.8 30.5 4.2
Matt Bonner PF 359 57.6 8.1 -32.3 -4.5 25.3 3.6
Tim Duncan C 662 54.5 4.2 -47.0 -3.6 7.5 0.6
DeJuan Blair PF 472 24.2 2.6 -30.5 -3.3 -6.3 -0.7
Tiago Splitter C 223 15.1 3.4 -26.4 -6.0 -11.3 -2.6
Antonio McDyess PF 380 6.3 0.8 -36.7 -4.9 -30.4 -4.0

The column you want to look at is EZPM100, which is the point margin (+/-) for the player per 100 possessions on the floor according to my model. I'm not going to go into a heavy discussion of the actual numbers now (although the rankings generally make sense, but there are probably some things that will surprise you). The model is still preliminary, as far as I'm concerned. As I said in the beginning of the post, this is the start, not the end of my work on the model. What I would really like to hear is feedback. Be as critical as you want. That's what makes this fun. If you think the coefficients are crap, say so. Besides using PBP data (which I think will have a big effect), I have tons of ideas for how to improve and test the validity of the model. Defense needs much more work. Do you have ideas for crediting/debiting value on defense? If so, I want to hear them. What about assists? How can we do that more rigorously? So many things to discuss.

Update (12/13/10):

  1. I realized that I had put the missed FTA (debits) in the defensive column, instead of the offensive column. I have updated the tables to reflect that change.
  2. I am now using the rebounding stats from basketball-value.com, so the actual number of rebounds while a player was on the floor is taken into account (as opposed to an estimate of total rebounds based on minutes played). This appears to boost second unit guys more than starters (which is actually the opposite of what I had predicted).


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I think the net value of a player's scoring, assists, and turnovers should equal the league average because those are the 3 things of value (either positive or negative) that a player can do with a possession (unless we wanted to get into hockey assists). I find that the value of all assists and turnovers come very close to netting out to zero for the league (not quite but close) depending on the value assigned to an assist. Assuming assists/turnovers net to zero, then all I am really interested in from a scoring perspective is scoring efficiency relative to the average TS%. If an individual player's assists and turnovers net out to either a positive or negative number because he is either a very good/bad ball handler or play maker then those skills are captured separately. I guess what I am saying is that I don't like the idea of punishing a player's PPP by including his turnovers without also considering his assists.

By the way, I love what you are doing.

I see, but then I'm not sure why turnovers are being discussed. I realize that some metrics add turnovers into possessions. That makes sense, but I don't see why it makes sense if you are trying to isolate the break even value of scoring alone. In my own analysis (extremely crude I admit), I compare each player's TS% to the league average TS% of around 54% and credit or deduct value relative to that. I then have a separate adjustment for usage just as you suggested that tries to capture the incremental turnovers/assists associated with handling the ball more/less often than average and the probable incremental value of associated with high/low usage scorers taking more or less tough shots away from their teammates. The problem is that I don't think there is an exact formula for this because it depends on the makeup of the team. IMO the value of the extra or lower usage is a variable.

Wayne, did you read the post on "breaking even"? There's no reason why that analysis can't be extended to take into account assists, as far as I can tell. I just neglected it at the time. But all that is a moot point if you think that "break even" point shouldn't involve anything other than shooting (which I disagree with).

If I understand the core of the argument on scoring , you are saying that higher usage scorers will tend to turn the ball over more times and get punished in most ratings systems for those extra turnovers. So to offset that, the real break even point is lower than the league average eFG% or TS%. Perhaps, I’m not explaining your position well, but I want to counter it anyway. Higher usage scorers also tend to get more assists. Perhaps for a player to really be effective at a higher usage he should be getting enough extra assists to offset the extra turnovers and we should set the break even point to the league average after all.

Wayne, this is not an argument about high vs. low usage. The entire analysis was assuming average usage (which is about 20%). Indeed, higher usage players will turn the ball over more, so in the current system they would have to shoot a higher percentage to break even. Perhaps, that is not fair. To correct for this, one possibility is simply to credit players for higher usage. I have actually played around with this idea in my model - crediting (or debiting) usage above (or below) league average - but have not settled on a good method for installing it in an update version of ezpm yet.

I quickly checked 82 games but missed that study. While charging players for not getting a shot up before 20 seconds have elapsed based on play by play would be best, you could do at season level. Adjust all players on a team by the league average or team average time period efficiency difference using the frequency (%) the clock got down to 4 or less on the team as a whole and minutes they were on the court. Then offset that with a small credit for the crunch-time shooters at season level. That would be fairly simple.

Clarification: In above I should have said: ...10-20 seconds "into" (not "on") the shot clock...

There are different degrees of complexity and accuracy. The simplest way to handle it might be to find the league average for shots taken from 10-20 seconds on the shot clock (earlier is fast break or advantaged early offense) and find the difference between that and the average for shots taken from after 20 seconds are used. Charge the per possession difference in net shooting efficiency between these two time periods evenly to all or maybe better based on their usage. It is a small charge but it helps move towards being more fair to end of clock shot creators and the others.

82games (of course) had a study several years ago: The results are not exactly what I thought. Here are the PPP for different times (elapsed): 0-10 Secs: 114.0 11-15 Secs: 105.2 16-20 Secs: 101.9 21+ Secs: 94.9 I suppose 0-10 seconds are most efficient overall, due to fastbreaks outweighing other factors.

Miss early in the clock get charged a full share for the miss; miss late in the clock the shooter gets charged less with the difference being deducted from the other players who didn't get up a shot before it was "late" as late is shown at 82games to be far less efficient time for anyone to shoot on average.

I was thinking the same thing. The numbers would be kind of arbitrary, but interesting, nonetheless. Most players shoot worse as the shot clock goes under 10 seconds (except for Monta Ellis, which I wrote a post on a while back). An "average shot" should be taken at an "average" time on the shot clock, which I'm guessing is between 10-15 seconds for the league. This will have to wait for like ezPM3.0 or something, though.

If someone shoots below average for the league of all attempts, the next questions are the %assisted and non-assisted and the FG%s on each. And the next question is the team context. It is possible that a player with a relatively good unassisted FG% is helping his team on those shots if they are attempts that are not ignoring assisted opportunities or stealing shots from an average level of assisted opportunities. You might "suggest" that if the player plays on a team that is still getting an average or higher level of assists that his nonassisted attempts were "normal" / acceptable and a player with a relatively good unassisted FG% is helping his team on those non-assisted shots in the team context where a potential assist is not always available. You might also look at shots in say the first 10-15 seconds of the shot clock. A self created dribble-shot in that time period might be viewed as more selfish than one that occurs later or certainly in the last few seconds of the clock where the threat of not getting any shot is greater. If you want to go for better modeling of reality and fairer credit I think it might be time to get involved with time of shot on the shot clock when charging for missed shots.

"One player shoots 46% efg. The other player takes no shots at all. Should the guy shooting 46% have a higher ezPM?" Yes, if you are trying to score and win a game and only these 2 players were involved. Looking at 2 players in isolation can be both helpful and misleading. But teams need to score to win. Taking no shots at all may be mathematically neutral but it is not neutral on court +/- impact to the eye and logic or in the Adjusted +/- data. I have seen charts that show that league wide extremely low usage (or in the absolute case of no usage) is correlated strongly with negative offensive Adjusted +/-. Playing "4 on 5" or sort of on offense is generally harmful at the team level even if it is not marked so by an individual metric.

Playing “4 on 5″ or sort of on offense is generally harmful at the team level even if it is not marked so by an individual metric. OK, that was a poorly chosen example, because of externalities. Evan, how about running 2 identical .484 shooting teams through EZPM and see if they end up positive offense/negative defense, and then run 2 identical "breakeven" shooting teams through EZPM and see if they end up 0 offense/o defense?

Those numbers you quote are for jump shots only, not all 2-pointers. The relevant figures for "close shots", according to that article, are 70.3% and 54.3%. Overall 2 point fg% seems to be 48.4% for this season. ()

Ah, ok. You're right. Thanks, for the correction.

Amusingly, this season it turns out that Vladimir Radmanovich is shooting exactly .440 on 2-pointers. So instead of talking about abstract players, you can ask yourself, "is Vladimir Radmanovich helping the Warriors by taking those shots?" Search your feelings :-)

Fred, have you seen this: I want to reference this finding: The average field goal percentage for assisted 2 point jump shots is 42.8% and the average field goal percentage for created 2 point jump shots is 33.7%. If these averages are true, then a player who could create his own 2pt field goals at a rate of 45.8% would appear to be positive in comparison to league averages.

Sorry, this sentence should read "And that in turn means that some FG% BELOW .488 will yield the zero ezPM point (or what Fed calls ‘break even’)."

Fred is making a simple error here. He is saying that a .488 FG% team will break even, which I assume is true in his data. He then assumes that your metric should show that .488 shooting is "break even." This is false. Nothing requires that possessions resulting in a FGA have the average 1.06 value. In fact, since TOVs result in zero points, the average value of FGA possessions MUST be higher than 1.06. And that in turn means that some FG% BELOW .488 will yield the zero PPS point (or what Fed calls 'break even'). The model already incorporates TOV, so no change is needed here. If you calculate a team with average TOV% that also shoots 44%, it will have a negative ezPM, just as it should. What Fred has discovered is that a 44% FG% team (or whatever it is) than never commits a TOV will be average offensively. This is mildly interesting, but not a problem for the model. Fred made this mistake originally, then corrected himself (I thought), but now is making it again. If he will correct himself one more time (but no more thant that!), I think you guys will be on the same page.

In fact, using the four factors regression equation, a team that has 0 TOV, only needs to shoot ~36% (eFG) to be average (assuming all other factors are average, of course).

A team that shoots .36 with no turnovers with an ORR of 26.3% (league ave last year) will average .72 points on the first attempt and get another shot .64*.263=~1/6 of the time. So they'll make ~.12 points off of second shots and ~.02 off of third shots, for a grand total of .86 points per possession. That's way below league average points per possession. Either I'm missing something important or your equation is significantly off.

Well, I'm missing fouls, obviously. .265 free throws per possession*.764 ft% = .20 ft points per possession, +.86= 1.06, right about at the league average points per possession. still doesn't address the problem that the .36-.488 shooter isn't helping his team vs an average opponent.

Guy, the problem is that you're looking at team EZPM and I'm looking at individual. In your example, the guy getting docked for the turnover is the one making the team EZPM score negative, not the guy shooting poorly, who is not docked for shooting poorly. They both deserve negative points. Why should a guy shooting below average get positive points? How is he helping his team win?

Fred, you showed above that if a team doesn't turn the ball over, they only have to shoot 36% to break even, given average foul rates and ORR. Now, it seems to me that when a player takes 20 shots, we know that on those 20 possessions, he definitely did not turn the ball over. Therefore, what I'm thinking is that essentially what the players does for the team that is positive is increase the number of possessions that don't end in a turnover. Taking this to the logical limit, if the same player takes every shot for the team, and doesn't turn the ball over, he really doesn't have to shoot 50.1% to break even. I think Guy was trying to make that point (although he can correct me if I got it wrong). I'm not 100% positive about this idea, but it seems to be consistent with the math both of us have been showing.

Have to think about this one. It seems like for everything except turnovers, being at the league average and playing average opponents will give you positive ezPM points. League average turnovers (assuming all positions had the same number of turnovers) would give you 0 ezPM points. (You'd get 20% of the value of the opposing team's turnovers, which counteracts your own 20% of your team's turnovers.)

Eh, this was incorrect: "It seems like for everything except turnovers, being at the league average and playing average opponents will give you positive ezPM points." All of these things should zero out, some in unexpected ways. (Turnovers - steals + .2*opponent turnovers=0 is pretty easy, but to zero out blocks, you need to combine it with your own shooting stats, to account for the times you were blocked.).

No, the problem is not team vs. individual, it's that you insist on evaluating part of a player's performance (such as FG%) in isolation. You keep talking about a below-average shooter, but tell us nothing about his turnovers. A below-average shooter will NOT get positive points if he is average everywhere else. Try giving us ALL the stats for a hypothetical player, and then explain why you think his expm is wrong. Otherwise, this is a waste of time.

Take two players, Guy. They have the exact same non-shooting stats. One player shoots 46% efg. The other player takes no shots at all. Should the guy shooting 46% have a higher ezPM? I say no, I say the guy who's not shooting is better for his team. Do you disagree?

Fred, you take two guys who don't shoot the ball, and I'll take my chances with two guys who shoot 46%. In fact, I'll take a guy who shoots 10% over the hypothetical guy who takes zero shots, and I'll give you points to start.

Evan: does the shooter get teammates who shoot 48.8%, and do you promise to keep guarding the non-shooter? Obviously that was an extreme case. It's more to answer Guy's suggestion that other non-shooting factors are relevant to this discussion, than to make a case for non-shooters having a place on a team.

"One player shoots 46% efg. The other player takes no shots at all. Should the guy shooting 46% have a higher ezPM? I say no, I say the guy who’s not shooting is better for his team. Do you disagree?" This is a very odd player, who never shoots yet still gets fouled. But yes, I disagree (a little). The 46% shooter delivers 1.075 PPP when making a FGA, slightly above average. So those possessions have a little positive value. At 45%, he's average. And that assumes you mean the "never shoots" guy gets some assisted FGs. If not, then the 46% guy is much more valuable.

Guy, we disagree. If you think someone who shoots below average for the league, and whose shooting average, if his entire team had it, would result in that team losing most of their games, deserves a positive evaluation *as a shooter*, I can't agree with you. You're looking at the entire possession and saying, well, if this guy never turned it over and shot every time at his low rate, he'd generate value for his team. That may be true, but if so that's because he's not turning it over; he deserves to gain points in the model for that skill. It's not because of his shooting; he deserves to lose points in the model for that skill.

Fred, we do not "disagree" -- you are mistaken. A player who shoots 46% on FGA and is otherwise average gets a negative ezPM, just as you think he should. You seem upset that his FG components are positive, but there is no reason at all to think all FGAs must equal zero on average. Do all FTAs add to zero? Of course not. All Turnovers? Obviously not. Your insistence that the subset of possessions ending in a FGA must average zero is arbitrary and illogical. Why do you keep insisting on it? Spend some time with a spreadsheet, calculating ezPM for different hypothetical players, and report back when you find players who make an average team worse yet get a positive ezPM. Until then, I don't see what there is to discuss.

This is supposed to be about net +/-. Making an average percentage of your shots does not alter your net +/-. That's why I am arguing that an average shot % should lead to 0. I guess as long as you don't break down ezpm into its component elements, the net result may (emphasis on may) end up all right. I'm still concerned about the imbalance between offense and defense and thought this might be contributing to it. (The Spurs, an above-average defensive team, are considered deficient in offense in the example at the top of the page.)

Aha. I think it does affect offense vs. defense. As I said, league average fg% is .484. If both teams play entirely average and entirely identically (identical turnover rate, identical rebound rates), and both shoot .484, which I repeat is average, then both teams will be considered by EZPM above average on offense and below average on defense (I think -- check me on this.). All the turnovers and rebounds will net to 0, and we'll be left with the raw math of makes and misses on both sides. Both teams' overall EZPM will be 0, so that will be accurate, but the breakdowns into offense and defense will not be accurate.

Fred, your own analysis showed that with free throws, the shooter would net 1.06 PPP, which is league average. Are you backing off of that now?

1.06 is a breakeven of 45.8%. getting closer to observed results, but still significantly different

A possession is still worth the same amount, so any factor involving a change of possession should keep the same value (I think). Dunno about defense.

If you adjust the value of a 2p shot to account for turnovers as I suggest [ 2- (1.045/.866) ] you get a breakeven point of .495, closer to where I think it should be. The value of free throws would also need to be adjusted similarly if you adopted this.

Wouldn't that be double counting TOV, since teams are already credited/debited for those? I'm not clear what the theoretical justification here is. In other words, I need to be able to explain it to someone else before I would consider putting it in the model.

It's a way to make the numbers match up to the league average shooter, who has to deal with turnovers as well. Thought experiment: we're playing in the Nerf Basketball Association, where there are no offensive rebounds and no fouls; all there are are turnovers and 2-point shots. Point guard dribbles up the floor and attempts a pass, if successful the target takes a shot, end of possession. The opposing league point guards on average turn it over on 20% of the possessions and opposing shooters make 50% of their shots. Therefore opposing teams average .8 points per possession. You have a point guard who never makes turnovers. Your shooters make 45% of their shots, so you average .9 points per possession, which is above the league average. Are your shooters helping your team?

Luvs me a good thought experiment! I was planning to come up with one myself, but you appear to have done the work for me. Let me think about it some more. I am wondering how it would change the value of the other parts of the model. Would I then need a similar correction on missed shots, for example? Also, by making the value of a made shot decrease, the value of a defensive stop goes down. Thanks, Fred. You've given me much food for thought the next few days.

You've designed a scenario where every basket gets an assist or none do. That is not the real NBA and so effectively you skirt the assisted FG value vs assisted FG value issue that was being debated.

Clearly, these factors are important. Hey, glad they're built into the model! Who would've thought it? It will be important to use the correct factors for comparing players in different seasons/eras.

With 1.06 as the PPP, a player shooting 41/100 gets -7.7 pts.

Actually, I may need to change again. I've been using Hoopdata values. Basketball Reference says the league average OFF is about 1.062. That will impact the results further.

Hoopdata and Basketball Reference are at least 2% different in their count of possessions.


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