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### Calculating Player Value: An Introduction to HOOPWAR

## March 8, 2012 | By Benjamin Miraski

Over at Hustle Belt, I published some data on player valuation.

It is an overly complicated formula, designed to try and come up with a measure similar to Wins Above Replacement for baseball.

I promised a breakdown of the formula so here goes.

The basic formula is: (Points Saved + Points Earned) * Average Pace/Team Pace - Replacement level / 30.

We will take each piece separately to try and explain the methodology

Points Saved = (Defensive Rebounds + Steals + Blocks - Fouls) * Team defensive points per possession

I use the standard definition of a possession. I just calculate it for the total stas of the opponents a team faced during the year.

Here is the reasoning behind each of the pieces:

- DR: Defensive rebounds stop the other team from having the ball.
- Steals: Steals stop the other team from having the ball
- Blocks: Blocks stop a shot. It doesn't necessarily change possession but it does stop a potential scoring shot.
- Fouls: I figure this will be controversial. I could weight fouls based on how many fouls a team has end up being shooting fouls, and then multiply that by the free throw percentage of the opponents, and there are offensive fouls, and fouls that don't actually result in points. I get that. But here is why I use just straight fouls: impact. We have all watched a game where a player makes a stupid foul, or players play differently with a certain foul total than without. The point is that fouls affect your defensive play. In some cases they turn directly into points. In other circumstances, they result in your being less effective. The end result is that I feel fouls have to be counted in defense, and I count the full amount.
- Team defensive points per possession: I could have used the league average, but defense is a team effort. I felt that if your team was good on defense, you are getting some help on the defensive front. You might make better stats than if you were the lone player in a bad defense. This was my way of making sure that the points a player saved its team were counted, at the average rate the team gave up points.

To use Kentucky's Anthony Davis (through 29 games) as an example, which is admittedly taking the most ridiculous example possible:

Points Saved = (198 + 43 + 138 - 58) * 0.8807 = 285.07

Points Earned = FG * (2 * (1-TeamAstPct)) + 3PT * (1-(Team3ptShotPCT*TeamAstPct))+ FT + Off Reb + Ast * TeamAstPct * ( 2+Team3ptshotPct) - ( TO * Team offensive Pts per Poss)

Terms explained:

- FG: Number of field goals that a player makes (both 2-pointers and 3-pointers). Simple: Makes shot, get credit... but how much credit?
- 2*(1-TeamAstPct): Since I am giving some of the credit for the shot to the person making the assist, you only get partial credit. Basically, you get credit for the percentage of the points that your team on average didn't assist on.
- 3PT: I do the same adjustment for three pointers. You get the extra point in a ratio based on assist percentage and the percentage of threes that your team makes.
- FT: You get credit for all of your free throws made. I could take off for free throws missed, which is something I might think about because you should be above a league average FT shooter, but for now, full credit
- Offensive rebounds: You have to get some credit for keeping the possession alive.
- Assists: You get credit for your assists multiplied by the team assist factor times 2 plus your team's three point percentage. In essence, this is trying to distribute the points not earned by the actual scorer to the guy making the assist.
- Turnovers: You lost the ball. You cost your team the equivalent of the points per possession that your team makes on offense. That is taken away from your total.

Anthony Davis again.

Points Earned = 158 * ( 2 * (1-.485)) + 0 * ( 1 - (163/802) * 0.485) + 99 + 87 + 26 * ( 0.485 * ( 2 + (163/802))) - 27 * 1.16 = 345.109

Adding those two numbers, you get: 630.179

That is Davis' raw score.

Now, divide that by the total number of possessions that Kentucky had this year, both offense and defense. This gives you the contribution of Davis per possession.

630.179 / 3859.5 = 0.1632

Again, Davis is a rare case. Most players are not worth 0.1 points per possession, no matter how slow their team plays.

Multiply that per possession number by the number of possessions in a game for an average team in NCAA basketball and then again by the number of games a team played. Some teams will play more games, and their players may be worth more as a result. But they also contributed to more wins. At this point, we could go to a per game basis, but let's continue.

0.1632 * 136.2 * 29 = 645.27

Now we have to subtract out the contribution of a replacement player playing in Davis' minutes.

How did we arrive at the replacement player? Well, the thought was that a team of replacements would somehow manage to win nine games, or at least be worth nine games worth of wins.

Teams that won nine games this year had about 1920 points of offense. If you look at the formula for Points Earned, you aren't going to recoup all of the points you scored. Some of that will be made up on defense.

I am assuming that a team of 12 replacement players would be worth 1920 points overall. It might be a bad assumption (see my thoughts below). It might actually be too high.

It is what it is. Twelve replacement-level players, all playing 500 minutes over 30 games would therefore be worth 160 points.

To get Davis' value over replacement, take his total score and subtract out 160 points for every 500 minutes he played.

645.27 - ( 160 * ( 911/500 )) = 353.748

You can see, Davis is still worth a heck of a lot to his team. But how does that turn into wins?

Using the Pythagorean Formula for basketball, you can approximate a team's winning percentage using points scored and points against. The exponent in the formula varies, according to posts I have seen. I used 10.45. At that exponent, it take a marginal change in points of about 30 to "earn" an extra win.

At the upper levels of the formula, it gets hard to move that needle, but that is why it is very difficult to win 100 percent of the games.

So much like 10 in baseball, I divided the total by 30 to get what I am calling HOOPWAR

Davis: 353.75 / 30 = 11.8

The man is a god on the basketball court. I have found no other player worth that much.

I have run Thomas Robinson, I have run Draymond Green; I have run the entire MAC conference.

If Davis isn't the player of the year in college basketball, I don't know what team that guy plays for, because 11.8 is turning out to be a big number.

Now, there are some things that I am continuing to think on:

- Replacement level might still be off, but I can't lower replacement value without inflating Davis' number into a region that I don't think any player could possibly achieve. Maybe that is OK. I am not comfortable with it right now.
- The end result of No. 1 is you have a lot of guys who play AT REPLACEMENT LEVEL. Maybe it is just the teams I have run, and if I ran more of the teams from the top half of the country, there would be more impact players. But right now, I have a glut of players who come in at right around 0 for their worth. Am I OK with that? I would need a better sample size to answer that.
- If you run a team, the combined HOOPWAR value plus 9 (the value of the replacements playing) does not equal the number of wins for the team. What can I attribute this to? Some of it is the normalization of the number to an average pace. Some of that might be getting more from your players than the actual sum of the parts. Some of that might be a contribution from the losing team (if you play a really bad team, you could almost be guaranteed a win based on their ... crappiness). If you look at baseball, there is a similar phenomenon, but I think the data is closer. In some cases, I have 20 win teams that were worth 15 wins.
- Is the defensive rating right? I know usage on defense is a big deal and centers and forwards do have a better chance to compile defensive numbers. With more data, I might see a pattern in how the big men are leading the way with the numbers, but in a league like the MAC, where I have run all the numbers, there were still a few guards scoring well on defense, and overall (perhaps because of the assist addition). Just something to think on.

Feel free to email me at bmiraski at mrisports.com if you want to discuss.

Posted March 8, 2012 10:09 PM

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