## Wednesday, June 26, 2013

One of the nice things about an in-game win probability model is that you can use it to see which plays (and players) had the biggest impact on the outcome of a game. Traditional box score stats, and even more advanced metrics like plus/minus, summarize player performance independent of the game situation. The box score counts Ray Allen's game-tying three pointer in game six of the Finals just the same as Mike Miller's three pointer made at 10:35 in the first quarter in the same game.

Everyone knows that Ray Allen's shot was much bigger than Mike's, even though they were worth the same amount of points. Win Probability Added is a stat that quantifies just how much bigger that play was. The basic idea is to score each play (and the player responsible) by looking at how that play changed the win probability for the team (see here and here for this concept applied to the NFL and MLB respectively).

When Mike Miller hit his three pointer in the first quarter, the Heat's win probability increased from 50.7% to 54.8%, worth a respectable +4.1% in win probability added (or WPA). However, when Ray Allen hit his three pointer, the Heat's win probability increased from 3.9% to 38.1%, a WPA of 34.2%*.

* A note on the win probabilities used in this post. They will not match those found in my win probability graphs. For the graphs, I use the probabilities that factor in team strength. For player evaluation, I use the "evenly matched teams" probabilities, so as to judge each player on an equal footing.

What I intend to do in this post is to lay out a general framework for constructing a Win Probability Added stat at the player level for the NBA.

### What Plays to Count

For the purpose of constructing a general WPA stat for each player, I am proposing counting the following types of plays:
• Missed shots
• Turnovers
The general idea is to view things within a possession framework. Possessions are the raw materials a team uses to build points and wins, and most possessions end with either a made shot, a missed shot, or a turnover. For both made shots and turnovers, the WPA calculation is straightforward. A made shot increases the point differential, advances the game time, and flips possession. A turnover just advances the game time and flips possession (my win probability model factors in possession as a variable, so this is all simple arithmetic).

### Missed Shots and Rebounds

Quantifying WPA (or WPMinus, rather) for a missed shot requires a bit more care. In general, a missed shot has two potential outcomes: an offensive rebound or a defensive rebound, and each outcome has a different win probability. A missed shot that is rebounded by the offense is less costly in win probability, but it doesn't seem right to credit the win probability generated by the offensive rebound to the player that missed the shot. So, my approach for missed shots will be to calculate an expected "pre-rebound" win probability, based on league average offensive rebound rates. Here is an example, using the offensive rebound that set up Ray Allen's game tying three:

The Heat began their final possession of regulation with 19.4 seconds left on the clock. With 7.9 seconds left, Lebron James missed a three point shot, which was rebounded by Chris Bosh with 6.3 seconds left. At the outset of the play, the Heat's win probability was 8.2%. Chris Bosh's rebound gave them possession with 6.3 seconds left, which is a 3.9% win probability. But if Chris Bosh hadn't grabbed that rebound and the Spurs gained possession with 6.3 seconds left, the Heat's win probability would have dropped to 0.9%. A missed field goal has a 36% chance of being rebounded by the offense, based on league average rates (it is 18.6% for missed free throws). So, the "pre-rebound" win probability as a result of Lebron's missed shot was 2.0% ( = 36% x 3.9% + 64% x 0.9%). Here is how the whole sequence breaks down according to win probability:

• 8.2% : starting win probability
• -6.2%: missed three pointer (Lebron James)
• +1.9%: offensive rebound (Chris Bosh)
• +34.2%: made three pointer (Ray Allen)
• =38.6%: ending win probability

Not only does this allow me to properly apportion win probability to the player that missed the shot, I can assign win probability values to rebounds as well.

### Game 6 Win Probability by Player

The table below ranks each player's Game 6 performance according to Win Probability Added (WPA). I have broken out my proposed "official" WPA metric into WPA due to made and missed shots (shWPA) and WPA due to turnovers (toWPA). In addition, I've also added WPA due to rebounds (rbWPA) and WPA due to steals (stWPA) as separate metrics, but ones that are not incorporated into the official WPA.

NBA Finals Game 6 - Win Probability Added
teamplayerWPAshWPAtoWPArbWPAstWPA
MiaRay Allen0.450.50-0.050.010.27
SAKawhi Leonard0.310.310.000.250.20
MiaMike Miller0.140.140.000.090.00
MiaShane Battier0.110.110.000.020.00
SATiago Splitter0.080.10-0.020.010.00
MiaChris Bosh0.050.07-0.020.110.13
SATim Duncan0.030.08-0.050.210.02
SABoris Diaw0.020.020.000.100.00
MiaChris Andersen0.000.000.000.050.06
MiaMario Chalmers-0.020.17-0.190.080.00
SAGary Neal-0.03-0.01-0.020.010.00
SATony Parker-0.04-0.040.000.100.23
MiaLeBron James-0.240.23-0.470.200.21
SAManu Ginobili-0.470.18-0.640.090.05

Not too surprisingly, Ray Allen was the WPA MVP of the game, with 3/4 of his WPA due to that single shot. More surprisingly, is where Lebron James landed. While generating a +23% WPA from made and missed shots, that was more than offset by some pretty costly turnovers, worth a total of -47% WPA (including one with 39 seconds left in regulation with the Heat trailing by 2 that cost -14%). Remember that turnovers are more costly than a missed shot in terms of WPA because a missed shot still has a chance of being retained by the offense.

But this approach may not be entirely fair to Lebron. His -0.24 WPA specifically excludes WPA added via rebounds or steals, categories where he clearly added value to the team, to the tune of +21% and +20% WPA. However, my current thinking is that these WPA numbers are oranges to the apples of my shots+turnover WPA and adding them together doesn't feel right, but I'm open to ideas as to how to incorporate them.

### Go to WAR?

Another way this approach may not be fair to LeBron and other high usage players is that it measures their performance against league averages, which is not always the best measure of value. Lebron James' -0.23 WPA was the result of 32 plays in which he was involved. Chris Andersen achieved a higher WPA than Lebron, but was only involved in two WPA-worthy plays. In fact, I had a higher WPA than Lebron (zero), which really doesn't seem fair.

One potential way to address this is to borrow the "above replacement" concept from advanced baseball stats. Instead of measuring win probability added relative to league averages, you measure it against what a replacement-level player would be expected to achieve. Tom Tango explains:
Replacement is defined very specifically for my purposes: it’s the talent level for which you would pay the minimum salary on the open market, or for which you can obtain at minimal cost in a trade.
More specifically, Fangraphs and Baseball Reference define replacement value as the level of play that would result in a 0.292 winning percentage. Dave Cameron looked at the WPA for players acquired through minor league free agency or waivers (i.e. replacements) and his results were consistent with the 0.292 value.

Defining replacement value for NBA players seems a few orders of magnitude more challenging, owing to the difficulty of quantifying individual contributions. So, I am considering the following alternative approach to defining replacement value, one that is more straightforward to calculate:

Wins Above Replacement would be measured relative to the "bare minimum" one would expect from a team and/or player with possession, where bare minimum is defined as "holding onto the ball until the shot clock runs out".

So instead of measuring win probability added relative to the play before, you measure it relative to what the win probability would have been if the team had just run out the shot clock.

This is easy to calculate and would address the Birdman > Lebron issue in the Win Probability Added table above. In the end, I think both views of a player's win probability contributions have value and it makes sense to track both, which I intend to do.

### Next Steps

I would like to calculate a Finals MVP, based on their WPA contributions throughout the series. After that, I hope to extend these metrics to the entire 2012-2013 season. Suggestions and comments are welcome.