by Dave Studeman
December 27, 2004

http://www.hardballtimes.com/main/article/the-one-about-win-probability



I talk a lot about Win Shares, because they do something I think is really
valuable -- they estimate the contribution each player has made to his team's
wins. This is an entirely different way of thinking about players, stats and
value -- because it measures every baseball event within the context of the
ultimate goal: winning games.

But Win Shares are not the only way to skin this cat. There is another
process that goes by many names and has been "introduced" to the public many
times. In fact, a 2003 Business Week article claimed that "a novel way to
evaluate baseball players" had been invented by people who wanted to bottle,
patent and sell it. Too late. It was first introduced by the Mills brothers
in the early 1970's, and it's been done many times since.

As I said, it goes by many names: "Player Win Averages" (Mills brothers),
"Player Game Percentage" (Bennett), "Win Probability Added" (Drinen), "Win
Expectancy" (Tangotiger), "Game State Wins" (Rhoids website), "Player's Win
Value" (Ed Oswalt) and WRAP (Lonergan and Polak). I'm sure there are other
people who have done the same thing and given it a name that I've not
acknowledged here. I apologize if you're one of those people.

For purposes of this article, I'm going to use the Drinen name, "Win
Probability," because I think it's the most descriptive.

Here's the basic idea. An average team, at any point in a game, has a certain
likelihood of winning the game. For instance, if you're leading by two runs
in the ninth inning, your chances of winning the game are much greater than
if you're leading by three runs in the first inning. With each change in the
score, inning, number of outs, base situation or even pitch, there is a
change in the average team's probability of winning the game.

Christopher Shea has invented a "Win Expectancy Finder" to look up the actual
Win Probability of every base/out, inning and score combination of all Major
League games from 1979 to 1990. Chris used Retrosheet data that had been
compiled by Phil Birnbaum, and his WE Finder simply looks up the percent of
times a team in a given situation went on to win the game during those years.
Next time you watch a ballgame, use it to track the ups and downs of the
game. It will change the way you watch baseball.

Here's an example: Bottom of the ninth, score tied, runner on first, no one
out. The home team has a 71% chance of winning according to the Win
Expectancy Finder (in this situation, the home team won 1,878 of 2,631 games
between 1979 and 1990). Let's say the batter bunts the runner to second. Good
idea, right? Well, after a successful bunt, with a runner on second and one
out, the Win Probability actually decreases slightly to 70% (home team won
1300 of 1,848 games), according to the WE Finder. The bunter hasn't really
helped or hurt his team; his bunt was a neutral event.

If you're managing a team, or even following the game, you might want to know
this sort of thing. Of course, the application of actual strategy (should he
bunt or not?) depends on a lot of other factors, such as the skills of the
batter, the pitcher and the baserunner, the following batters in the order,
the game conditions and probably a number of other things. But Win
Probability sets the baseline for evaluating each event on the field.

To really have fun with this system, you can take it one step further and
track something Drinen calls "Win Probability Added" (WPA).

Once again, the concept is simple. Let's say our batter in the bottom of the
ninth hits a single to put runners on first and third with no outs. This
increases the Win Probability from 71% to 87%, for a gain of 16%. So, in a
WPA system you credit the batter +.16 and debit the pitcher/fielder -.16. If
you add up every positive and negative event from the beginning to the end of
a game, you wind up with a total for the winning team of .5, and a total for
the losing team of -.5. And the player with the most points will have
contributed the most to his team's win.

By the way, that 87% with runners on first and third in the bottom of the
ninth is on the low side for reasons I'll discuss in a minute.

If you were to track an entire season in this manner, you would have a Win
Contribution metric that is more accurate than Win Shares, because it is
based on how much each event actually contributed to the team's wins. In a
way, WPA is the ultimate baseball statistic. And in a way, it is not.

Like Win Shares, WPA is not a good predictive statistic because it's not
necessarily a good representation of a player's true talent. If a player hits
a home run in the ninth inning of a 1-0 game, he is credited with more WPA
points than if he hits a home run in the first inning of a 1-0 game. The
talent is the ability to hit the home run; when it happens in a game is
something that is pretty random. When you are thinking of acquiring a player
for your fantasy team, you should rely more on the traditional sabermetric
stats, like Linear Weights, Runs Created, DIPS, etc. etc.

Also, WPA measures the impact of an event while the game is in progress, not
after the game is over. After the game is over, the score is 1-0, and it
doesn't matter when the batter hit the home run. But during the game, it
matters a lot. Good managerial strategies, for instance, are based on an
implicit understanding of Win Probabilities. And if there is such a thing as
clutch performance, WPA might unearth it.

The most interesting and useful application of Win Probability Added -- the
one that Drinen, Tangotiger and others have spent a lot of time on -- is the
evaluation of relief pitchers and the managers who call on them. We all know
that closers are important, even though they pitch less than 100 innings a
year, right? Why? Because they pitch the most important innings.

Using the WE Finder again, if a pitcher gives up a bases-empty home run in
the first inning of a tie game, his team's Win Probability decreases about
10%. If he does the same thing in the eighth inning, it decreases about 25%,
because his team has less time to come back. In this context, the eighth
inning is about 2.5 times more important than the first inning. And if you
apply this sort of analysis to every appearance made by a relief pitcher, you
can quantify the importance of all of his innings pitched.

Tangotiger developed a system called "Leveraged Index" that measures and sums
the potential "Win Probability Added" for each pitcher's appearance. Doug
Drinen tracked a similar measure, called "P," in the Big, Bad Baseball
Annual. Though the math behind each system differs, they are both constructed
to measure the importance of relief innings. You might particularly enjoy
Tango's Crucial Situations article and chart.

Win Shares, by the way, includes an approximation of WPA for relief pitchers,
based on each pitcher's saves and holds. I've been playing with a similar
system myself, and I hope to roll out some analysis in the next few weeks.

There are a couple of reasons the Win Expectancy Finder isn't the best source
of Win Probabilities. First, it's based on the years 1979 through 1990, when
there were fewer runs scored per game than in the past few years. A one-run
lead was safer back then. Also, there are sample size issues with some of the
situations. For instance, there were only 220 games with runners on first and
third for the home team in the bottom of the ninth with the score tied.
That's not a large enough sample size. So you shouldn't take the numbers in
the WE Finder as "gospel."

The better way to develop your Win Probability table is to develop something
called "Markov Chains." I won't go into all the math here (because I doubt I
can really explain it well), but suffice to say that a proper Win Probability
table is something a good mathematician can concoct, based on the probability
of scoring a certain number of runs for each base/out situation.

And there are still a lot of Win Probability issues to be resolved. For
instance, Win Probability tables really should be altered based on the home
park. To track WPA on a regular basis, you need play-by-play data, so you
can't create it for most of baseball history. And Win Probability doesn't
solve the sticky issue of splitting credit between pitching and fielding
(something Lonergan and Park admit).

Win Probability is a complicated subject, and there's so much more I could
say. But I hope this article serves as a good introduction to a topic I plan
to return to in the future.

References and Resources
Here's an example of a game I tracked with my own Win Probability tables.
Cyril Morong has a nice review of Win Probability hitting stats on his
website. If you're a Baseball Prospectus subscriber, you can find a table of
Win Probabilities based on 2004 games in their stats section.

I recommend Alan Schwarz's "The Numbers Game" for a very nice history of the
evolution of Win Probability (including the critical role of George Lindsey).
My thanks go to Tangotiger, Doug Drinen and Jon Daly for their support and
education in this subject.