## WE Rating System: How good was that game?

For a little while I wanted to try to use WE to quantify how "exciting" a game was.  It should be pretty simple.  In general, the more spiky the graph, the better the game and blowouts tend to be big yawnfests.

Using some data Jeff gave me (thanks), I was able to do a little number crunching and come up with some simple formulas that give you some idea of how good a game was.

The first we'll call Blowout Rating (BR).  This tells you how big of a blowout a game was (big shocker!).  It ranges from 100 to -100.  100 represents a "perfect" blowout where we win.  -100 represents the opposite.  0 tells us the WE was 50% the whole game.

The formula:  BR = 100*[2*average(WE)-1]

This basically uses the area under the WE curve to figure out whether the game was mostly hopeless or a guaranteed win.

The next two formulas are a little more difficult (not too bad) and use a little calculus (oh no!).  They are the average absolute 1st and 2nd derivatives of the WE curve.

The average absolute 1st derivative (AA1D) mathematically is the slope of the WE curve.  Absolute means if the slope is negative then we'll make it positive.  This gives us the average change in WE with each plate appearance.  In other words, if it is 10%, then whenever there was a batter up the WE changed on average 10% (in either direction).  I'd say this would signify how exciting the game is because it shows us how much the WE changes.

The average absolute 2nd derivative (AA2D) is a little bit tougher to nail down.  Its the absolute derivative of the 1st derivative (which was not absolute).  When the slope of the WE changes directions this results in a large value for the 2nd derivative.  This tries to tell us how much the slope of the WE varies and gives us a percent change in the slope of the WE.  In other words, this shows us how much the momentum varies in a game.

I tried to make a model game where we'd expect to see the WE jump around as much as is possible.  It probably isn't the theoretical maximum but its probably close enough.  I found that the AA1D(max) = 22% and AA2D(max) = 19%.  Once again, these are a little too low so maybe we'll just say that they are both 25%.

These are the games Jeff supplied me with data for.
Sep 08
Aug 15
Jul 7
May 09
Apr 21

I think the games I was given gave me a pretty good mix to work with.  So what did I get?

Blowout Rating (BR)
4_21    -72
5_9      43
7_7      69
8_15    -19
9_8     -41

So what does this tell us?  Well the April game pretty much sucked.  It sucked so much Jeff didn't write anything up for it.  On the other hand, the June game was also a pretty big blowout in our favor as the game was in hand the whole time.  The best of the bunch according to this was the August game which was back and forth a little until the last few innings where it slipped away.  The biggest shortcomings of this is that it doesn't take into account the timing of the game.  It would be kinda nice if it was weighted so that the last 1/3 of the game was more important but for now I think it is pretty decent and shows what you want in one number.  Once again, this doesn't really show us how "exciting" a game was, just how big of a blowout it was.

Absolute Average 1st Derivative (AA1D)
4_21    2.1%
5_9     2.2%
7_7     1.7%
8_15    4.4%
9_8     3.6%

Absolute Average 2nd Derivative (AA2D)
4_21    2.6%
5_9     3.2%
7_7     2.4%
8_15    6.1%
9_8     5.2%

I kinda lumped these two together because they kind of show similar things in the end.  They show that the last two games were clearly more exciting than the first three.  Looking at the WE graphs this isn't surprising.  Also, again as with the BR, there is no extra weighting for the time things occur but I think WE takes care of this since it varies much more near the end of the game which increases both metrics.  One nice tidbit you can tease out from this that isn't in the BR metric is that the September game was a more exciting game than the May game even though the BR is roughly the same magnitude for both.

One thing I didn't look into just because I didn't have the info that would be interesting is looking at the average leverage index for a game.  This would be a pretty easy way to show how many close situations there were in a game.  I'd like to look at this one and see how it'd compare to the other metrics I came up with.

Oh I also thought it'd be good to include this one that Jeff says he likes.  As a quick and dirty way he says he looks at a game to see how many plays change the WE by more than 10%.

Plays where WE changed my more than 10%
4_21    2
5_9    2
7_7    1
8_15    5
9_8    6

Appears as if Jeff's method is pretty good and goes along right along with my derivatives.  This method appears to be surprisingly good.  Once again, Jeff appears to be pretty smart.

This stuff is pretty rough but I kind of wanted to get the idea out there.  It should be pretty easy to formula a good rating system and be able to come up with a list of the most exciting games ever.

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