Ben's Proposed Basketball Player Rating System

Brian Haeffner's Tiger Basketball Player Ratings site has stats and computations of this rating system for the Missouri Tigers and all Big 12 teams.

Okay, the goal is to create a rating system which assigns credit to individual players as accurately as possible given the limitations of the stats available. As a means of achieving this goal, I try to stick to a normalization of game points. Roughly speaking, if team A wins by 15, and the player ratings from team A add up to 80, then the player ratings from team B add up to around 65. (This latter goal could by perfectly achieved by using points scored as a rating system, but then we've lost completely the original goal.)

So, the method is to tie the value of various stats to their actual point scoring impact, and to keep an eye on the balance sheets.

Possessions

   Possessions = FGA + TO - OR + 0.45*FTA

Teams average around 1 point per possession (PPP). I will use this as a benchmark for now but this could be modified in face of stats or a good argument.

Field Goals Attempted/Made

Now let's examine a system which awards 2 or 3 points (accordingly) for each field goal made. What 2PT percentage would correspond to the break-even point? Take

   2*p - 0.7*(1-p) = 0  (break even)

   3*p - 0.7*(1-p) = 0  (break even)

1. the field goal shooter is getting credit for a lot of his teammates' work - they're all running the offense to create the higher percentage shot (setting screens, making assist-worthy passes,...)

2. the made field goal ends the possession! The team as a whole would average 1 point on that possession, so the 2PT field goal doesn't lift you 2 points above expected points but only 1.

We can cure both of these problems with one swoop. For a successful 2PT shot, we credit the scorer with 2-1=1 points (or 3-1=2 points for 3PT shots), and divvy up the other point to the teammates. Points credited for assists are taken from this team point, and the remainder are divided among the five players on the court, including the shooter.

First, what impact does this have on shooting percentages? Break even is now given by

    2P:  1*p - 0.7*(1-p) = 0      3P: 2*p - 0.7*(1-p) = 0

Now, the impact of awarding points to the team is that everyone will be getting points for how many possessions they play. While it's not possible to keep track of the number of possessions each player was in the game for, this could be well-approximated by the number of possessions in the game times the fraction of the time the player played. So basically points are awarded for minutes, in absence of more precise information about how useful each player was on each scoring possession. Assists modify this a little - below. The impact of this, in comparison with the existing NEP or Ratings system, is that the regular players will all be closer together in rating per game, which I think is probably accurate. If one looked instead at possession-normalized stats, the impact is only a uniform shift in everyone's stats.

A final comment: awarding 1 for 2PT made, 2 for 3PT made, and -0.7 for FG missed can be restated as

    -0.7*FGA + 1.7*2PT-made + 2.7*3PT-made.

     -0.7*FGA + 1.7*FG-made + 1*3PT-made

Assists

By the way, SBA are perfectly well defined, just not recorded. That's a shame. Another useful stat would be 3PT assists versus 2PT assists. Some opportunities for rational stats at the introduction of the 3PT shot were definitely squandered.

It's a guess how much shooting % goes up, but it's a guess constrained by knowing 2PT%, 3PT% and that 55% FG made are assisted. I can try to write some of my thoughts on this up eventually, but I came out with rough guesses of an increase +30% for 2PT shooting percentage and +20% for 3PT shooting, so the SBA added a value of 0.6 points in both cases.

So how to translate SBA back to assists? Well, it turns out SBA'd FG are around 55% accurate, so roughly 3 assists are awarded for every 5 SBA. We can't know how many SBA a player has, but we can make the assumption that all players have an SBA/assist ratio of 5/3, and just multiply this factor times their assists. The final, extremely approximate result:

   assist = (5/3  SBA per assist)*(0.6 pts/SBA) = 1 pt

Points per Possession

   Poss Pts = 0.15*(minutes/40+any OT)*(T-FGA + T-TO + 0.45*T-FTA - T-OR)

Now, I know it seems strange to award points for nothing, but here's the argument. First, there are a lot of intangibles that go into good offense and there are no stats for them. Therefore we can choose not to award points for them, but that's not really accurate. Those intangibles are earning points for the team. The best we can do is to assume the coach is pretty well tuned into these and is doling out playing time appropriately. So our best guess of who is "doing the right things", which is better than assuming no one is doing them, is to base it on PT.

A final comment about these "possession points": if they are determined game by game from the box score - the best method - and then also determined from the season stats, there might be some discrepancy. For example, suppose player A only plays 10 minutes per game in fast paced games, but plays 25 minutes per game in slow games. If the tempo of the games varies quite a bit, then using team-averaged possession/game and player A's averaged min/game would give a misleading picture, making A appear to have more possessions than he really did. Personally, I doubt this is much of an effect, and if it is, then I'd try to use the game-by-game numbers.

Free Throws

1. the "and-one" on a 3-point play: 0 possessions and 1 FTA
2. a 1-and-1: 1 possession and 1.7 FTA (70% chance of making first)
3. a 2 shot foul: 1 possession and 2 FTA
4. a 3 shot foul: 1 possession and 3 FTA
5. a technical: 0 possessions and 2 FTA

None of these are particularly common, though. The 1-and-1 acts to increase the number of possession per FTA, but by very little (since majority of 1-1 become 2 shots anyway). Numbers 1,4, and 5 have a stronger impact per rate of occurance.

1-1 situations are actually fairly rare also these days, since only 6 are possible in a game and some number of those get replaced by 2 shot fouls. All in all, the total number of times 1, 4 and 5 happen is roughly about equal to the number of 1-1 in a game, and since they have a stronger impact I'd go with possessions = 0.45*FTA. It's possible that 0.4*FTA would be more accurate - I'll think more about this.

This can be checked roughly by computing possessions (from the top) for each team in a game, particularly a game with a disparity in FTA. The possessions always come out within 1 or 2 of each other, indicating that this is pretty accurate.

Okay, now that we have that established, each FTA costs 0.45 possessions so with the 1 PPP benchmark, it costs -0.45 points (note I'm assuming no chance for the rebound). Each free throw made is simply worth 1 (we've already "paid" the team their 1 PPP), so the break even FT% is simply 1*p - 0.45 = 0 or 45%.

Wow! That sounds really low...but it isn't. What this reflects is that (1) you've only got to hit 1 of your two shots to reach your average PPP and (2) some number of those shots are possession freebies: the "and-one" or the 3-point shot, for example.

Since (nearly) everyone can shoot better than 45%, everyone benefits from more FTA. This gets right at what Brian was advocating based on intuition: the rules are set up so that if you get to the foul line, you are helping your team. We don't have to award any extra points for this, because we're already favoring FTA. I think it's nice how these intuitive results (also shooting %) come out of quantitative arguments.

Rebounds

Now that might sound low - after all, if you don't grab the defensive board, it could become an offensive board. But the fact is defensive rebounds ARE easier to come by. The same people are trying at both ends and 70% of the boards go to the defense. Since it's more a matter of blocking out and taking what comes to you, it's worth less. I've got more arguments I can make on this point, but maybe it's unnecessary - I'll see what the response is.

By the way, here is where some additional stats could be useful: is the offensive board simply a result of some defender missing their blocking assignment? If so, that defender should be penalized (a "missed block" stat) and then in the balance sheet the offensive board would be worth less. Oh well, can't have it.

Turnovers and Steals

So I thought, until I had insomnia one night. Actually, much of this is a product of insomnia, which makes me wonder if it's a SOURCE of insomnia. I'm not going to pursue that thought.

Here's the problem: it's double dipping. Every steal is also counted as a turnover to the other team, so if you award +1 to the stealer and -1 to the stealee, you've awarded a two point swing where in fact only a 1 point swing has occured. How do we deal with this?

Pretty nearly half the TO are also steals. Considering those half, how much of the 1 point do with give the stealer, and how much do we take away from the chump who lost the ball. I'm just going on my game sense here when I say I think the stealer deserves the majority of the credit. I would give them at least 0.7, and maybe 0.8. Let's say +0.7 for a steal.

Now, half the turnovers are a full -1 because no steal was involved, and the other half are -0.3, so the net effect is -0.65 per TO (since we don't know how many of a given player's turnovers were steals). Since the steal/TO ratio was an estimate anyway, I'd make a TO -0.7 for simplicity.

Funny how simple things are all complicated... (now I've got the Who in my head)

Blocks

Here's my guess: A block takes a FGA with an average scoring potential of just over 1 point (>50% 2PT since usually inside) and makes it a miss. Let's give the block 0.2 for this. Now what fraction of blocks result in a change of possession? I think it's less than half. That would be a nice stat to have....Let's put it at 30%, so at 1 PPP that adds an average value of 0.3 per block, giving them a total worth of 0.5. This is very debatable.

Fouls

What this says is that many fouls are non-shooting fouls before the bonus, and have almost no impact on the game (as for bringing a team closer to the bonus - read on). To calculate the average value of a foul, consider how many points typically result from it minus the cost of possessions (there is a reason for fouling at the end of the game!) I'm not explaining this well, maybe the formula will make the point: assuming a 70% FT shooter, a foul gives 1.15*0.70 = 0.805 points, but costs (-1 pt/poss)*(0.45 poss/FTA)* (1.15 FTA/foul) = 0.518 points. So at this level of accuracy, I'd make a foul worth -0.3 points. My number of 1.07 would result in a foul value of -0.27. I leave this minor point to be settled by more research.

Now that's really an average. Some of those fouls were pre-bonus and had no impact, but some were after the bonus and had more impact. But under the assumption that players don't show a tendancy to foul only before the bonus or only after the bonus, then divvying up the total damage done by a per foul basis is fair.

Defensive Adjustment

The single most important job on defense is to make the opponent take a low percentage shot!

Nobody gets credit for this. And it's a team effort. And it can be the difference between a W and an L, so it should show up in our stats. And it shouldn't double count steals or defensive rebounds, which do help the defensive effort, but those points are already awarded. So we want to look at points scored per shooting possession (PPSP)

    PPSP = Points/(FGA + 0.45*FTA)

  DP = (0.2)*( 1.1*(OFGA + 0.45*OFTA) - OPTS )*(minutes/40)

Rating System

   Team stats: Poss = FGA + TO + 0.45*FTA - OR

 1.7*FGmade + 1.0*3PTmade - 0.7*FGA + 1*FTmade - 0.45*FTA + 0.7*OR + 0.3*DR

 + 1*AST + 0.7*STL - 0.7*TO + 0.5*BLK - 0.3*Foul

 + 0.15*(min/40+)*Poss  + 0.2*(1.1*(OFGA + 0.45*OFTA) - OPTS )*(minutes/40+)