Wade Boggs, Willie McGee, and the Value of Information
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 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:
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:
(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.
© 2004 Daniel Lauve