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Calculating Win Shares

Note: This article was edited on April 26, 2006. I changed the way defensive Win Shares are credited to individual players.

Introduction

Stealing a page from Bill James, I decided to attempt to calculate basketball Win Shares. Due to data limitations the current system only works for the seasons 1977-78 forward, but at some point I hope to modify the system for seasons prior to 1977-78. This article will describe how I came up with the Win Shares system for basketball.

What is a Win Share?

A Win Share is worth one-third of a team win. Win Shares are assigned to players based on their offense, defense, and playing time. If a team wins 60 games, there are 180 Win Shares to distribute among the players. This is always true; if a team wins n games, then there are 3n Win Shares to allocate to the players.

Splitting Win Shares Between Offense and Defense

Splitting Win Shares between offense and defense is based on something called marginal points. To calculate marginal points scored, do the following (I will use the 2003-04 San Antonio Spurs as an example):

  1. Calculate league points scored and league possessions. Calculating league points scored is straightforward. In 2003-04 the league scored 222097 points. Possessions are estimated using a formula presented by Dean Oliver in Basketball on Paper (please see Oliver's book for complete details). In 2003-04 this works out to 215776.07 league possessions.
  2. Calculate the league's points per possession. In 2003-04 the league scored 222097 / 215776.07 = 1.02929 points per possession.
  3. Calculate the team's expected points scored. Expected points scored are (league points per possession) * (team offensive possessions). The Spurs had an estimated 7330.74 offensive possessions, so they were expected to score 1.02929 * 7330.74 = 7545.46 points.
  4. Calculate marginal points scored. Marginal points scored are points scored minus 0.92 times expected points scored. For the Spurs this is 7501 - (0.92 * 7545.46) = 559.18 marginal points scored.

The next step is to caclulate marginal points allowed. To do this:

  1. Calculate the team's expected points allowed. Expected points allowed are (league points per possession) * (team defensive possessions). The Spurs had an estimated 7353.42 defensive possessions, so they were expected to allow 1.02929 * 7353.42 = 7568.80 points.
  2. Calculate marginal points allowed. Marginal points allowed are 1.08 times expected points allowed minus points allowed. For the Spurs this is (1.08 * 7568.80) - 6909 = 1265.30 marginal points allowed.

Marginal points are the sum of marginal points scored and marginal points allowed. For the Spurs this is 559.18 + 1265.30 = 1824.48 marginal points. In order for the system to work, there must be a relationship between marginal points and team wins. The relationship is this: 1 win is equal to m marginal points, where m is equal to one-third of the league average points per game for a team. In other words, based on their marginal points the Spurs would have been predicted to win 1824.48 / (93.4 / 3) = 58.6 games. They actualy won 57 games, for an error of 57 - 58.6 = -1.6 wins. If marginal points are used to predict team wins for all teams since 1977-78, the root mean squared error (rmse) of the predictions is 3.86 wins.

Aside: Where did the 0.92 and 1.08 multipliers come from?

Good question. I figured a team made up of marginal players would win about 10 percent of their games, so I solved the following equation for x:

(1 - x)**14 / ((1 - x)**14 + (1 + x)**14)) = 0.10

The value of x is roughly 0.08, so 1 - x = 0.92 and 1 + x = 1.08.

The Spurs won 57 games in 2003-04, so the team has 3 * 57 = 171 Win Shares. To determine how many Win Shares to allocate to the offense and defense, divide marginal points scored (or allowed) by marginal points and multiply by Win Shares. For the Spurs offense this is (559.18 / 1824.48) * 171 = 52.4 Win Shares, while for the Spurs defense this is (1265.30 / 1824.48) * 171 = 118.6 Win Shares.

In order to prevent teams from having too few or too many Win Shares credited to their offense, at least 25% and at most 75% of a team's Win Shares must be credited to the offense. This rule did not affect the 2003-04 Spurs, as 52.4 / 171 = 30.6% of the team's Win Shares were credited to the offense.

Crediting Offensive Win Shares to Players

Offensive Win Shares are credited to players based on Dean Oliver's Offensive Rating. Offensive Rating is an estimate of the player's points produced per 100 offensive possessions. The formula is quite detailed, so I would point you to Oliver's book Basketball on Paper for complete details. The process for crediting offensive Win Shares is outlined below (using Tim Duncan of the 2003-04 Spurs as an example):

  1. Calculate the Offensive Rating for each player. In 2003-04, Tim Duncan's Offensive Rating was 107.385.
  2. Calculate offensive possessions for each player. Once again, I would point you to Oliver's book for complete details on calculating player offensive possessions. Tim Duncan had an estimated 1373.19 offensive possessions in 2003-04.
  3. Calculate marginal offense for each player. Marginal offense is equal to (player possessions) * ((player Offensive Rating) / 100) - 0.92 * (league points per possession)). For Tim Duncan this is (1373.19) * ((107.385 / 100) - 0.92 * (1.02929)) = 174.26. Note that any player with negative marginal offense gets zeroed out at this step.
  4. Calculate team marginal offense. The team total for the 2003-04 Spurs is 647.299.
  5. Credit offensive Win Shares to the players. Offensive Win Shares are credited using the following formula: ((player marginal offense) / (team marginal offense)) * (team offensive Win Shares). Tim Duncan gets credit for (174.26 / 647.299) * 52.4 = 14.1 offensive Win Shares.

Crediting Defensive Win Shares to Players

Crediting defensive Win Shares to players is based on Dean Oliver's Defensive Rating. Defensive Rating is an estimate of the player's points allowed per 100 defensive possessions. I know I'm sounding like a broken record, but please see Oliver's book for further details. Here is the process, once again using Tim Duncan as an example:

  1. Calculate the Defensive Rating for each player. Tim Duncan's Defensive Rating in 2003-04 was 88.501.
  2. Calculate marginal defense for each player. Marginal defense is equal to (player minutes played / team minutes played) * (team defensive possessions) * (1.08 * (league points per possession) - ((player Defensive Rating) / 100)). For Tim Duncan this is (2527 / 19755) * 7353.42 * ((1.08 * 1.02929) - (88.5001 / 100)) = 213.18. Note that any player with negative marginal defense gets zeroed out at this step.
  3. Calculate team marginal defense. The 2003-04 Spurs team total is 1245.79.
  4. Credit defensive Win Shares to the players. Defensive Win Shares are equal to ((player marginal defense) / (team marginal defense)) * (team defensive Win Shares). For Tim Duncan this is (213.18 / 1245.79) * 118.6 = 20.3 defensive Win Shares.

Putting It All Together

The final steps of the process combine offensive and defensive Win Shares such that the final result is an integer. They also ensure that player Win Shares sum to three times the team's win total. Here are the remaining steps:

  1. Add offensive and defensive Win Shares. Tim Duncan's total is 14.1 + 20.3 = 34.4.
  2. Separate the figure above into Win Shares and partial Win Shares Tim Duncan had 34 Win Shares and 0.4 partial Win Shares.
  3. Compute final Win Shares. Sort the team's players by descending partial Win Shares. If the sum of player Win Shares does not equal team Win Shares, add one Win Share to each player on the sorted list until the two are equal. For example, the player Win Shares for the 2003-04 Spurs sum to 164. The Spurs as a team had 171 Win Shares. Therefore, the players with the 7 highest partial Win Shares get one Win Share added to their total. This did not affect Tim Duncan, so his season total remains 34 Win Shares.

Here are the final Win Shares totals for the 2003-04 San Antonio Spurs:

Name                WS

Tim Duncan          34
Manu Ginobili       24
Tony Parker         20
Rasho Nesterovic    20
Hedo Turkoglu       18
Bruce Bowen         15
Robert Horry        12
Malik Rose          10
Jason Hart           5
Devin Brown          5
Kevin Willis         3
Ron Mercer           2
Charlie Ward         2
Anthony Carter       1
Shane Heal           0
Matt Carroll         0
Alex Garcia          0

This is my first attempt at basketball Win Shares, so this should be considered a work in progress. If you have any comments or questions, plase send me some feedback.