# Calculating Win Shares

### I. Introduction

Stealing a page from baseball's Bill James, I decided to attempt to calculate college basketball Win Shares. This article will describe how I came up with the Win Shares system for college basketball (please see this article for the pro version of the system). If you believe that any attempt to attribute team success to individual players is an abomination, then read no further, as this article will be of no interest to you.

### II. What is a Win Share?

Bill James developed his system such that one win is equivalent to three Win Shares. My system deviates from James's system in three key ways:

1. In James's system, one win is equivalent to three Win Shares. In my system, one win is equivalent to one Win Share.

2. James made team Win Shares directly proportional to team wins. In his system, a baseball team that wins 80 games will have exactly 240 Win Shares, a baseball team that wins 90 games will have exactly 270 Win Shares, etc. In my system, a basketball team that wins 25 games will have about 25 Win Shares, give or take.

3. James did not allow for the possibility of negative Win Shares. In his system, the fewest number of Win Shares a player can have is zero. In my system, a player can have negative Win Shares. I justify this by thinking about it in the following way: a player with negative Win Shares was so poor that he essentially took away wins that his teammates had generated.

### A. 2009-10 to present

Offensive Win Shares are credited to players based on Dean Oliver's points produced and offensive possessions. The process for crediting Offensive Win Shares is outlined below (using Jared Sullinger of the 2011-12 Ohio State Buckeyes as an example):

1. Calculate points produced for each player. In 2011-12, Sullinger had an estimated 609.7 points produced.

2. Calculate offensive possessions for each player. Sullinger had an estimated 500.3 offensive possessions in 2011-12.

3. Calculate marginal offense for each player. Marginal offense is equal to (points produced) - 0.875 * (division points per possession) * (offensive possessions). For Sullinger this is 609.7 - 0.875 * 1.0170 * 500.3 = 164.5. Note that this formula may produce a negative result for some players.

4. Calculate marginal points per win. Marginal points per win reduces to 0.5 * (division points per game) * ((team pace) / (division pace)). For the 2011-12 Buckeyes this is 0.5 * 68.0 * (67.0 / 66.3) = 34.36.

5. Credit Offensive Win Shares to the players. Offensive Win Shares are credited using the following formula: (marginal offense) / (marginal points per win). Sullinger gets credit for 164.5 / 34.36 = 4.8 Offensive Win Shares.

### B. 1998-99 to 2008-09

Because the number of statistics we have available to us in our college basketball database is limited prior to 2009-01 (no offensive rebounds, no turnovers, etc.), we will not use methods based on Dean Oliver's work, although the basic framework will remain the same. Here is the process for crediting Offensive Win Shares (using Shane Battier of the 2000-01 Duke Blue Devils as an example):

1. Calculate the player's modified points. The formula is:
```  2.0 * (field goals) * (1 - ((team assists) / (team field goals)))
+ 1.5 * (field goals) * ((team assists) / (team field goals))
+ 1.0 * (three-point field goals)
+ 1.0 * (free throws)
+ 0.5 * (assists)
```
Plugging Battier's statistics into the formula above we get 741.71 modified points.
2. Calculate the player's modified shot attempts. The formula is:
```  1.000 * (field goals) * (1 - ((team assists) / (team field goals)))
+ 0.500 * (field goals) * ((team assists) / (team field goals))
+ 1.000 * ((field goal attempts) - (field goals))
+ 0.475 * (free throw attempts)
+ 0.500 * (assists)
```
Plugging Battier's statistics into the formula above we get 587.44 modified shot attempts.
3. Calculate Division I points per shot attempt. Division I points per shot attempt is equal to (division points) / (division field goal attempts + 0.475 * (division free throw attempts)). For 2000-01 this is 1.0600.

4. Calculate marginal offense for each player. Marginal offense is equal to (modified points) - 0.875 * (division points per shot attempt) * (modified shot attempts). For Battier this is 741.71 - 0.875 * 1.0600 * 587.44 = 196.86. Note that this formula may produce a negative result for some players.

5. Calculate marginal points per win. Marginal points per win reduces to 0.25 * (team points per game + opponent points per game). For 2001-01 Duke this is 40.31.

6. Credit Offensive Win Shares to the players. Offensive Win Shares are credited using the following formula: (marginal offense) / (marginal points per win). Battier gets credit for 196.86 / 40.31 = 4.9 Offensive Win Shares.

### A. 2009-10 to present

Crediting Defensive Win Shares to players is based on Dean Oliver's Defensive Rating, which is an estimate of the player's points allowed per 100 defensive possessions. Here is a description of the process (once again using Sullinger in 2011-12 as an example):

1. Calculate the Defensive Rating for each player. Sullinger's Defensive Rating in 2008-09 was 85.8.
2. Calculate marginal defense for each player. Marginal defense is equal to (player minutes played / team minutes played) * (team defensive possessions) * (1.125 * (division points per possession) - ((Defensive Rating) / 100)). For Sullinger this is (1123 / 7800) * 2550 * ((1.125 * 1.017) - (85.8 / 100)) = 105.0. Note that this formula may produce a negative result for some players.

3. Calculate marginal points per win. Marginal points per win reduces to 0.5 * (division points per game) * ((team pace) / (division pace)). For the 2011-12 Buckeyes this is 0.5 * 68.0 * (67.0 / 66.3) = 34.36.

4. Credit Defensive Win Shares to the players. Defensive Win Shares are credited using the following formula: (marginal defense) / (marginal points per win). Sullinger gets credit for 105.0 / 34.36 = 3.1 Defensive Win Shares.

### B. 1998-99 to 2008-09

Here is the process for crediting Defensive Win Shares (once again using Battier in 2000-01 as an example):

1. Calculate team marginal defense. Team marginal defense is equal to 1.125 * (division points per shot attempt) * (team field goal attempts + 0.475 * (team free throw attempts)) - (opponent points). For 2000-01 Duke we get 834.60.

2. Calculate the player's share of the team's marginal defense. The player's share of the team's marginal defense is equal to 0.40 * ((total rebounds) / (team total rebounds)) + 0.25 * (steals / (team steals)) + 0.25 * (blocks / (team blocks)) + 0.10 * (assists / (team assists)). How did I get those weights? Of the statstics we have available, pro Defensive Win Shares are most dependent on defensive rebounds, steals, blocks, and assists, so I regressed pro DWS on those stats and then found the relative importance of each regressor (approximately 40% for defensive rebounds, 25% for steals, 25% for blocks, and 10% for assists). Getting back to our example, Battier's share on 2000-01 Duke is equal to 0.2540

3. Calculate marginal defense for each player. Marginal defense is equal to (team marginal defense) * (player share). For Battier this is 834.60 * 0.2540 = 211.99. Note that this formula may produce a negative result for some players.

4. Calculate marginal points per win. Marginal points per win reduces to 0.25 * (team points per game + opponent points per game). For 2000-01 Duke this is 40.31.

5. Credit defensive Win Shares to the players. Defensive Win Shares are credited using the following formula: (marginal defense) / (marginal points per win). Battier gets credit for 211.99 / 40.31 = 5.3 Defensive Win Shares.

### V. Putting It All Together

The final step of the process is to add Offensive Win Shares to Defensive Win Shares. In our examples, Jared Sullinger's total in 2011-12 is 4.8 + 3.1 = 7.9 Win Shares and Shane Battier's total in 2000-01 is 4.9 + 5.3 = 10.2 Win Shares.

### VI. Does This Work?

Because this metric is designed to estimate a player's contribution in terms of wins, it makes sense to see if the sum of player Win Shares for a particular school closely matches the school's win total. For 2011-12 Ohio State the sum of player Win Shares is 35.1 while the school's win total is 31, an error of 31 - 35.3 = -4.3 wins. For 2000-01 Duke the sum of player Win Shares is 39.3 while the school's win total is 35, an error of 35 - 39.3 = -4.3 wins. These errors are actually larger than the "typical" error; looking at all Division I schools from 1998-99 (the first season we have complete player statistics) through 2011-12, the average absolute error is 1.6 wins and the root mean squared error is 2.1 wins.

### VII. Feedback

If you have any comments or questions about the Win Shares methodology, please send us some feedback.