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Tuesday, September 18, 2007
Unleveraging Win Probability
More about WPA/LI (linear weights by game state, with the leverage aspect deflated). Not for the faint of heart:
I’m having a conversation at Baseball Fever, and below is a copy of post 26:
If we stick with the simpler “walk or hit wins, out sends to extra innings” situation:
- assuming a league average OBP of .333 (my definition, including all events), the expected winning probability is easy enough to calculate (1/3 wins the game, and 2/3 sends to extra innings with a 50% chance to win) as .667.
- the LI is equally easy to calculate (.333 gain in wins happens 1/3 of the time, and .167 wins downward happens 2/3 of the time, for an average absolute movement in some direction of .2222, compared to the random of .0346, or an LI of .2222/.0346 = 6.4)
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Now, say you have an Ichiro-type, where his OBP, overall, is .400. In this case, when Ichiro is at the plate, the Mariners will have a 70% chance of winning, if he does his Ichiro thing. (40% wins the game, and 60% sends it to extra innings where they have 50% chance of winning.) However, suppose in clutch situations, Ichiro has an OBP of .460. That means, the Mariners will win .730 with Ichiro as a clutch hitter, as opposed with .700 with Ichiro as his usual self. So, he adds +.030 wins with his clutch play, per PA (in this situation), over and above his own great self. This is the method you are describing.
Let’s look at the other way. His WPA/LI would be (.730-.667)/6.4= .012. His WPA is .730-.667= .063.
That leaves his WPA minus WPA/LI as +.051 wins.
Is one way better than the other?
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Let’s consider another example. Let’s assume that Ichiro hits .400 regardless of situation, in a league that hits .333. In this case, your method would say that Ichiro has no clutch skill at all. The other method would do the following:
WPA/LI = (.700-.667)/6.4= .005. His WPA is .700-.667=.033. This leaves him with WPA minus WPA/LI of +.028 clutch wins.
But also note that in low leverage situations, he’ll end up with a minus. For example, if the LI was 0.2, and the then his WPA/LI for some blowout situation would be something like (.110-.109)/.2= .005, and his WPA would be a tiny change (.110-.109=.001), giving him a WPA minus WPA/LI of -.004 wins.
In this simplified example, if you had 87% of the games with an LI of .2, and 13% with an LI of 6.4 (overall LI of 1.0), then Ichiro’s clutch would be:
+.028 * 13%
-.004 * 87%
= .000 (rounding error)
As you can see, this method, and your method, both agree that Ichiro, overall, gets a clutch rating of 0.
And, WPA minus WPA/LI is superfast to calculate, and doesn’t require prior knowledge of the hitter’s overall stats and how that translates for each game state! Cool, right?
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As for what does WPA/LI represent, it’s his win impact, with the game situation unleveraged. For example, a hitter with an OBP of .400, in a league of .333 is worth about +.055 runs per PA, or roughly +.005 wins per PA, in a random situation, compared to the average hitter. In the game-ending situation in question, his WPA was +.033 and the LI was 6.4, making his WPA/LI as +.005 wins. So, what WPA/LI does is two fold:
1. It rebalances the walks, hits, HR relative to the out, based on how each of those events will impact the win probability
2. It deflates the leverage aspect
This metric (WPA/LI) is especially ideal for players like Ichiro, who can rebalance their game to take advantage of the new balance required for a game state (sometimes avoiding a K is necessary, sometimes getting a walk isn’t all that important, etc).