
The year 2005 marks
the first year of Hall of Fame eligibility for two of the top hitters for
average of the 1980’s. It also marks the twentieth anniversary of the 1985
season, when Wade Boggs and Willie McGee each won batting titles. In 1985,
Boggs won his second career batting title and began a stretch of four straight
years with the highest batting average in the American League. McGee won his
first of two career batting titles and won the National League MVP award. Boggs
finished with a .368 average; McGee batted .353.
Baseball’s 1985
batting champions were successful in very different ways. Willie was all about
speed—his 56 stolen bases in 1985 are more than double Boggs’s career stolen
base total. Boggs was a master at getting on base, with a career .414 on base
percentage, compared to McGee’s .333. Boggs hit from the left side of the
plate, while McGee was a switch hitter. But the biggest difference between the
two hitters is one of patience and temperament.
Willie McGee was a
free swinger—he struck out once in every six at bats during his career. Manager
Whitey Herzog once said, “Smart as Willie is, he never did learn to lay off the
sinking curveball in the dirt....The man always chases that pitch, and when he
does, you always think for a second he’s some duffer on a public course shanking
one out of bounds.” Wade Boggs, on the other hand, was one of the most
disciplined hitters in the game. He almost never swung at the first pitch, he
was great at working the count, and he almost never swung and missed in any
situation (in 1989 he had 1191 swings and missed the ball only 58 times).
So how could two
singles hitters earn batting titles with such markedly different styles? More
specifically, why on earth would someone like Boggs, who prided himself on
working the count, concede the first strike of practically every at bat? And
why wouldn’t McGee, who did not read pitches as well as Boggs, take a pitch in
order to get a better read on the pitcher? The answer lies in the value of the
information each receives from a given pitch.
Decision Theory
Decision theory is
a way of evaluating an actor’s best course of action among a number of risky
alternatives. A decision tree is a way of mapping decisions and possible
outcomes, with the tree diverging with every possible choice or event.
On the above tree,
the actor faces a choice between going to Los Angeles and going to Seattle. The
choice (which the actor has control over) is denoted by the green square before
the tree splits. Once he gets to either place, there is a chance of rain. Rain
is an event (which the actor has no control over) denoted by the red circle.
Since the outcome of the event is unknown, there is also a probability
associated with the various possibilities that shows up next to the possible
outcome. In this case, there is a 50% chance of rain in Seattle and a 20%
chance of rain in Los Angeles. Below the probability is the value associated
with an outcome. In this case, let’s say that the actor wishes to go outside,
so he places a higher value on dry weather. “True” and “false” show which
choice the actor should make.
If the actor wants
to maximize his value, he should go to Los Angeles, where he has an 80% chance
at a “10” outcome and a 20% chance at a “5” outcome, for an average value of 9
(80% of 10 plus 20% of 5). This is fairly obvious because the drier weather in
Los Angeles gives him the best shot at his most desired outcome. If, however,
he values dry weather in Seattle more than in Los Angeles, this may change the
outcome.
Hitting as a
Decision Tree
It’s very difficult
to place baseball situations into this simplified a format, so we’ll need to
make a few assumptions in order to be able to talk about what makes Wade and
Willie so different. Imagine a baseball world in which:

There are two pitches, fastball and
curveball.

There are two types of pitcher.
One type throws fastballs 80% of the time, and the other throws fastballs 40%
of the time. These types of pitcher are equally mixed throughout the
population, so overall batters will see 60% fastballs and 40% curveballs.

Batters cannot tell which type of
pitcher they are facing. So, if Willie McGee is facing Fernando Valenzuela,
he cannot tell from history what type of pitcher Fernando will be in this at
bat. The type of pitcher is determined by things like game situation and
other factors, but in any given at bat, Fernando could be more of a fastball
pitcher or a curveball pitcher.

Before offering at a pitch, batters
try to determine whether they will be seeing a fastball or a curveball. This
is often called guessing or looking for a certain pitch, but here it is simply
used to signify the fact that batters are better able to hit balls if they
have a better idea of what’s coming.

In the decision tree, the decision
will be whether to look fastball or curveball, and the event will be the pitch
that is thrown.
Hitting
Without
Information
First, let’s take a
look at a scenario in which batters have no information on the pitcher they’re
facing—there is a 60% chance of a fastball and a 40% chance of a curveball.
Wade and Willie still have to look for either a fastball or a curveball, and
depending on what they are looking for, their batting averages against certain
types of pitches will change (looking for a fastball makes you better able to
hit the fastball). Batters’ tendencies are defined by the following rules:

When Boggs is looking fastball, he
hits .420 against the fastball and .240 against the curveball.

When Boggs is looking curveball, he
hits .310 against the fastball and .380 against the curveball.

When McGee is looking fastball, he
hits .440 against the fastball and .220 against the curveball.

When McGee is looking curveball, he
hits .270 against the fastball and .320 against the curveball.
(These numbers are
oversimplified and probably somewhat exaggerated, but they’ll work.)
The following trees
show that without any information on what type of pitcher they are facing, both
players will look fastball (the word “true” appears next to the fastball
choice). Willie will hit .352, which is basically his 1985 average, and Boggs
will hit .348.
In a world without
information, both hitters are about the same. Willie’s a little better with the
fastball, but Wade can hit the curveball slightly better.
Hitting
With
Information
Suppose that these
players wish to find out what kind of pitcher they are facing. This is
reasonable and achievable, since there are multiple pitches in each at bat.
Hitters can move from a world with no information to a world with information
very easily—to find out what type of pitcher he is facing, a hitter must take
the first pitch. After the first pitch, a batter knows what type of pitcher he
is facing, and he can adjust accordingly. (Of course, you can’t really tell
what type of pitcher you’re facing based on just one pitch. Technically, seeing
a first pitch fastball would give you 67% certainty that you’re facing a
fastball pitcher, and seeing a first pitch curveball would give you 75%
certainty you’re facing a curveball pitcher. However, here we’ll assume that a
pitcher always gives himself away on the first pitch.)
Here is what
happens when Boggs and McGee enter a world where they can gather information:
After taking the
first pitch, Boggs is going to be looking fastball after seeing a fastball and
curveball after seeing a curveball. He hits .352 against curveball pitchers and
.384 against fastball pitchers, and his average is .368, his 1985 batting
average. McGee, on the other hand, is still looking fastball no matter what.
His relative inability to hit the curveball has cost him the ability to use the
information to his advantage. For Willie, nothing is gained from being in a
world with information. He will be content simply to think fastball and swing
away.
The Value of
Information
Decision theorists
define the value of information as the difference between the expected value
with information and the expected value without information. In this example
it appears that the value of information for Wade Boggs is 20 batting average
points (.368.348), or about 13 hits over the course of a year. But we really
don’t know what the value is because the model assumes that there is no cost
associated with taking a pitch. Both decision trees think of the at bat as one
event rather than a series of pitches, but the second one pulls out the first
pitch and treats it as its own event. Taking one pitch out of the at bat should
decrease the value of the at bat, but it doesn’t. Knowing the cost of the lost
pitch varies with different batters and different pitchers. For batters who are
good at making contact, the cost is relatively low: they can foul off pitches
and extend the at bat until they see a pitch they like.
For Boggs, there
clearly is a value in taking a pitch that outweighs the cost of possibly falling
behind in the count: Boggs, who can see pitches and work the count very well,
gets to acquaint himself with the pitcher’s offering and determine how to
approach the at bat. He shouldn’t be too afraid to fall behind because he can
hit the curveball, which pitchers are more likely to throw with the count in
their favor. Willie wants to jump on the first pitch because falling behind
means facing more of the curveballs that eat him up. For him, the value of the
information is outweighed by the cost of the information.
Boggs and McGee
in the Real World
This example
grossly oversimplifies the baseball world, and a few omissions are worth
mentioning.
Players learn in
ways that are much more subtle than taking pitches, and they update their
information throughout an at bat and a career. A good hitter will know what to
expect based on the game situation and his previous experience with a pitcher.
Rather than look at pitches, he can foul off pitches or study from the on deck
circle (or the de facto on deck circle, the area ten feet closer to the plate
than the on deck circle, which is more conducive to gathering information). And
while the first pitch of an at bat probably gives the most information to a
batter, part of the chess match lies in the fact that the batter and the pitcher
are constantly updating their probabilities of what the other is likely to do.
Gaining information
isn’t just a function of seeing pitches; it also depends on a player’s ability
to gather and use information. Hitters like Boggs who take an analytical
approach to hitting are likely to learn more from taking a pitch than players
like McGee, whom Whitey Herzog said didn’t ever know what the count was
(referring to his consistent behavior regardless of situation, not his ignorance
of the situation). So it’s conceivable that you could have a hitter with the
same situational averages as Boggs who would not want to take the first pitch
because he is simply not as good at using the information.
This model probably
applies best to “guess” hitters, hitters who anticipate a certain pitch and
swing accordingly. If they’re right, they look great, and if they’re wrong they
usually look silly. Not all hitters are considered guess hitters, but the logic
they employ applies in varying degrees to all hitters. Pitches simply come too
quickly for hitters to react well to everything, so it helps to have a better
idea of what’s coming.
Wade Boggs and
Willie McGee each became successful through considering numerous hitting
factors, and a decision tree that could accurately model their true approaches
to hitting would be far too complicated to be useful. Each player has his own
way of accounting for velocity, location, movement, balls and strikes, men on
base, weather, and many other variables. But what underlies all of this is a
way of updating probabilities, consciously or unconsciously, that allows a
hitter to use these variables in his favor.
