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Friday, June 11, 2010
Game-state version of wOBA
This is going to require alot of effort from you, to read and to follow. It’s a step-by-step kind of thread. And anyone out there that has ever disparaged, questioned, or just been plain flummoxed by wOBA or WPA/LI, well, this thread may be for you. I hope all your questions will have been answered after you read this. (Though it is possible that new questions you hadn’t considered will pop up!)
It took me about an hour to write this. I know I’m asking alot for your time and patience in return. The reward should be there. Let’s go…
On Base Percentage (OBP) says that each positive event counts as “1”, and you put the sum of those positive events in the numerator. The number of opportunities (plate appearances, or PA) is in the denominator. Divide the two and you get OBP.
But, the only time that a walk and a HR is indeed worth the same is in the bottom of the final inning of a tied-game with the bases loaded. Presuming it’s two outs, then the one-third of the time you reach base, you win the game, and the two-thirds of the time you make an out, you go to extra innings, where you have a 50/50 chance of winning. Your chances of winning therefore was one-third of 100% plus two-thirds of 50%, or 66.7%. A walk, hit, HR, etc, all added +.333 wins. An out subtracts .167 wins.
If you go through the exercise for every other game situation, you can guess that the value of the walk must be less than 1 each and every time, and the value of a HR is going to be more than one all the time. The only answer that is not readily apparent is: how much away from one?
If you look at it on average, over all game situations, the value of a walk is not “1” like OBP would say, but around 0.7. And the value of a HR is 2.0, not “1”. As we saw above, it does happen that a HR is the same as a walk under specific situations.
So, this is what you have to remember:
1. The average value of a positive event will always be 1.
2. The average value of a walk is centered at 0.7, with a maximum value of 1. The average value of a HR is centered at 2.0, with a minimum value of 1. (The single, double, and triple are somewhere in-between these two extreme events.)
Since players are very aware that a walk is a HR in one situation, and the batter and pitcher and fielders will approach that game situation differently, it stands to reason then that every single situation is different. And we should not treat them as if they are the same. So, rather than a value of 0.7 for a walk being a standard, that value should be dynamic, changing based on the game situation. How can we calculate that?
Let’s take another example, say with the batting team down by one run in the bottom of the 8th with 2 outs, and runner on 2B. The chance of winning is 0.2763. Here are the chances of winning following various events:
0.2998 BB (puts runners on 1B, 2B)
0.4600 1B (various possibilities)
0.5821 2B (scores runner, ties game)
0.5955 3B (scores runner, ties game)
0.8341 HR (puts team 1 run up)
0.1745 Outs (ends inning)
Suppose I tell you that the OBP is .3333. What is the chance of winning if you have a positive event (but I don’t tell you whether it’s a walk, hit, or HR)? You should be able to figure it out based on the above information.
Chance of winning
= frequency of positive events times chance of winning following positive events
+ frequency of negative events times chance of winning following negative events
0.2763
= .3333 x chance of winning following positive events
+ .6667 x 0.1745
Simple math tells us that “chance of winning following positive events” is 0.4799.
So, for a standard positive event, its win value compared to the negative event is .4799 minus .1745 equals .3054. This becomes our baseline value for a positive event.
We then compare the win expectancy following a walk (.2998) to the win value of the out, and see that the walk is worth .1253 wins. We take that value and compare it to the standard win value of a positive event (.3054), and see that the walk is 41% the value of a positive event. Repeating for each event, and we get:
0.41 BB
0.93 1B
1.33 2B
1.38 3B
2.16 HR
The average value of the positive event must be, and always is, 1. That’s one of the two rules you were supposed to remember. The math simply follows this requirement. If it’s not apparent to you that this is what always happens, read through this post again.
What we are basically doing here is creating a weighted on base average (wOBA). Every game state provides its own approach, and each outcome will have its own value based on how much it affects the chances of winning.
You can create thousands of such equation, one for each game state. Also note that you can also include the strikeout as a (negative) event in the numerator in the cases of less than two outs, and similarly a groundout or flyout (all over and above the standard out value).
In the end, we’ll end up with a game state wOBA. Each equation will tell us exactly how much win impact each event has for each game state, and it will all be along a scale where “1” is average. And if you familiarize yourself with the standard values for each event (0.7 for walk, 2.0 for HR, etc), you will start to appreciate when a walk becomes less valuable than usual (like the above illustration, where the walk value is 0.4) and when a HR becomes more valuable than usual. By knowing this, you as a batter, or pitcher, will be able to pitch to the situation to try to leverage the possibilities of those outcomes.
Now, who wants to have all those mothers of equations? Fortunately, there’s a nice shorthand, and it’s called Situational Wins, and it is calculcated by taking the WPA and dividing by the Leverage Index (LI).
Really, you are asking yourself? How does wOBA and WPA/LI relate at all? We said in the above situation, that following the standard positive event, there is a .4799 chance of winning (compared to the .2763 starting point), or +.2036 wins. And that following a negative event, the chance of winning goes down to .1745 (or -.1018). The LI in this situation is around 3.77. So a positive event would have a WPA/LI of .2036/3.77 = +.054 wins, and a negative event would be -.1018/3.77 = -.027 wins.
And -.027 wins is the standard value of an out in an average situation.
This is how WPA/LI relates to wOBA by game state: each PA is de-leveraged so that they each count the same. And the de-leveraged WPA simply becomes the linear weights for that particular game state. wOBA by game state makes it clear how things work, if you remember the two rules at the start of this post. WPA/LI is an easy calculation, especially because Fangraphs and Baseball-Reference does it for you.
Furthermore, with the currency unit set as wins, it makes the result useful.
This is why wOBA is this way it is, and uses the scale it does. And that’s how wOBA, WPA, LI, and Linear Weights are all tied-in together. For those people who want wOBA to be scaled a different way, well, it’s not going to happen, because there’s a perfect reason why it is this way, and it has to be this way.