Game Theory 3: The Mathematician Curt Schilling Or how playing the odds can make you a superstar.

April is the start of Baseball Season. Baseball is a statistics driven sport more than any other. This is probably why Baseball is the most popular sport among the math inclined. 

This is a baseball story that is also a math story. It is an interesting tale about how a good pitcher became a great pitcher simply by collecting statistics, calculating odds, and preparing for every player he was going to face.

In the winter of 1992, Curt Schilling was preparing to play for the third team in as many years. He had a good arm, and he could throw at least four different pitches well and accurately. He was a relief pitcher with an average ERA. 

It was during that winter that Curt met Yankee's ace Roger Clemens, who at the time was considered the best pitcher in all of baseball. Clemens obviously saw something in Schilling, he took him under his wing and taught him the art of preparation.

Schilling became a student of the game. He started studying the players he would face, looking for weaknesses. He studied not just the general statistics of players from the newspaper, but he studied how he pitched to players in the past, keeping track of what works and what does not. Schilling keeps all of this on a laptop computer that he takes everywhere. The tech savvy Schilling even has MPG video of his pitches to watch and study.

The preparation and study paid off big. In his first year with the Philadelphia Phillies, he went from reliever to starter, tripled the number of innings pitched, and dropped his ERA a point and a half. Schillings career stats are shown below.

This past year, Schilling was the top vote getter among pitchers in the NL All-Star team, finished with an NL high 22 wins, was Co-MVP of the World Series, and Sportsman of the Year in the both Sports Illustrated and The Sporting News. 

Yet, despite the career making season, Schilling still sees room for improvement. He allowed a career high 37 home runs last year, including three to Home Run Champion Barry Bonds. Schilling has made the lowering of that statistic a top priority this season. Only time will tell if he gets it right. Schilling is not the best pitcher in the Majors (in fact playing with Randy Johnson he is not the best pitcher on his own team).

To be fair, Curt is not the only player ever to show marked improvement through study and preparation, he is just one of the better documented cases. His is just one example. 

Many have the talent in sports, but the champions of the game also bring dedication and preparation. They are never satisfied with their performance and they are constantly looking for ways to improve. They look to exploit every advantage they can, including those buried in the numbers.

Update (August 2002):

Curt Schilling is still allowing too many home runs this season. Cardinal Jim Edmonds even hit a grand slam off of him (the first allowed by Schilling in over two years). One thing he is not allowing is walks. Schilling is on track to break the all time record for highest Strikeout to Walk ratio. His current ratio is 11.8, the all time record belongs to Bret Saberhagen who in 1994 had a K/BB ratio of 11.0 in that strike shortened season.

Baseball: A Game of Averages

As I pointed out last month, all sports have a certain amount of chance factors in them. This is more true in Baseball than in any professional sport. 

Consider the physics of baseball. You have got a round ball being hit by a round bat. The ball is going to fly tangential to the angle of contact at a speed equal to the sum of the angular momentum vector of the ball and bat at the point of contact**. If all the numbers add up right, it is a home run. If they do not, it results in either a hit, a ground out, a fly ball, a foul ball, or most likely a strike (when the ball and bat never actually make contact).

Sorry about the technospeak in the last paragraph. Basically, what I am saying is that the outcome of every pitch is determined by where the ball and bat are when they make contact. Pitchers can control where the ball will be, and batters control where the bat will be. The luck factor results from the fact that even the best pitchers are not perfect in their throws, and even the best batters are not perfect in their swings. Thus, neither pitcher or batter has complete control over the outcome.

Last year, the team with the best record (Mariners) lost 46 times. The teams with the worst records (Devil Rays and Pirates) both managed 62 wins. Chance is the great equalizer in sports. Since luck comes and goes for every one, there is not much you can do about it.

The Winning Strategy: Play the Odds

Team managers often make decisions that surprise newcomers to the game*, such as taking a right handed all star hitter out of the game and replacing him with a left handed bench player. There are usually good reasons, such as when the other team just put in a right handed reliever who struggles with left handed batters. Managers are often "playing the odds".

In a game where so much luck is involved, it is impossible to anticipate outcomes. However, it is possible to make changes that increase the odds in one's favor. Playing percentages does not always work out, managers often end up looking bad when their decisions prove costly, but while it cannot guarantee a win, it can (if done correctly and all things being equal) produce more wins.

Of course, what ultimately happens is that all baseball managers end up playing percentage strategies. There really are mathematical reasons why the fastest runner bats first, the player with the most home runs bats third, and the player with the best batting average bats fourth. There are mathematical reasons why you load your roster with right handed batters when facing left handed pitchers and vice versa. It is not just a matter of tradition, it is proven strategy.

All of this is confusing to new fans of the game, but if it weren't confusing it wouldn't be baseball.

Poker: A Game of Averages

Lets face it, most of us are never going to manage a big league team and put our statistical skills to the test. Yet, there are common games that employ similar percentage strategies for long term success. The most popular of these is Poker.

In Game Theory terms, Poker is an n-player (where n  is usually 5 or 6), random, zero sum game of imperfect knowledge. As such it is heavily dependent on luck. Like baseball, even the worst players can occasionally draw a good hand. Also like baseball, the players who employ the best strategies usually win the most hands and the most money in the long run.

The Winning Strategy: Play the Odds

Poker is not about putting money into a pile and whoever has the best hand wins. It is, like baseball, a very psychological and statistical game as well. It may sound illogical, but bluffing from time to time (betting heavily on hands that are unlikely to win) is a vital strategy to winning in the long run.

The odds against getting two pair is about 20 to 1, the odds against three of a kind is 46 to 1, yet in a five or six player poker game of five card draw with nothing wild, the most likely winning hand has either two pair or three of a kind. 

This seems counter intuitive, if five hand are dealt at random, the likelihood of one of those five hands containing two pair or better is about one in four. But, in a draw game, it is as if everyone got a second hand of five, so the odds of someone having two pair or better drops to one in two.

So half the time, nobody has two pair or higher. When this happens there are two likely outcomes: 1. No one has a hand good enough to open resulting in a re-deal, or 2. One player successfully bluffs the pot away. Either way, we never see the cards dealt.

Like baseball, whole books can be written about strategy and tactics in poker.

The moral of the story: Statistics are Important

Baseball and Poker are just games, but in a way they mirror real life strategy. Success in business and life in general is not purely about skill, but also includes a certain amount of luck. 

There are, however, ways of improving one's odds at success. They require knowing which choices in life are the most likely to result in acceptable outcomes. When statistics are correctly assembled and applied, they result in mostly good decisions. Statistics that are incorrectly collected or misapplied often result in bad decision making. 

Baseball Statistics Abbreviations and Explanations Usually Used by Statisticians and Published in Magazines and Newspapers.

HITTING:
G = Games Played
AB = At Bats
H = Hits
R = Runs
2B = Doubles
3B = Triples
HR = Home Runs
BB = Walks (Base on Balls)
HBP = Hits by Pitch
SO = Strikeouts
AVG = Batting Average = Hits/At bats = H/AB (The batting average tells you how often the batter gets a hit. Walks don't count as At Bats here.)
OBP = On Base Percentage = (Hits + Walks + Hits by Pitch)/(At Bats + Walks) = (H + BB + HBP)/ (AB+ BB) (OBP tells you how often the batter gets on base.)

SLG = Slugging Percentage = Hits + Doubles + 2 (Triples) + 3 (Home Runs)/ At Bats = (H + 2B + (2*3B) + (3*HR))/AB
[The Slugging Percentage formula tells you if the batter gets lots of big hits. It's different from Batting Average because doubles, triples, and home runs count more heavily (SLG is a weighted average).]

The most popular stat in batting is the Average. A typical average is around .250, below .200 means the batter is struggling a bit, above .300 is Great!

PITCHING

G = Games Pitched = Total number of games in which the pitcher has pitched (A game is counted even if the pitcher has only faced one batter!)
GS = Games Started = Number of times pitcher started the game G - GS = Relief appearances.
IP = Innings Pitched (Each batter that a pitcher faces who makes an out in a game represents 1/3 of an inning. Walks and hits don't count in determining this stat.)
BB = Walks Allowed = The total number of batters a pitcher allows to reach first base on a walk
SO (or K) = Strikeouts = The total number of batters a pitcher has struck out
H = Hits Allowed = The total number of hits (singles, doubles, triples, and home runs) that batters were able to get when the pitcher was pitching
R = Runs = the number of batters that a pitcher allowed to get on base who eventually scored
ER = Earned Runs Allowed = Total Runs Scored —Total Runs Scored Because of Error (Earned runs are the number of batters that a pitcher allowed to get on base who eventually scored. Runs scored because of errors don't count.)
W = Number of Games Won (This is the number of games where the pitcher's team won and the pitcher was then named the winning pitcher.)
L = Number of Games Lost (This is the total number of games in which the pitcher's team lost and he was named the losing pitcher.)
SV (or S) = Saved Games (if a relief pitcher enters the game with their team ahead by three runs or less, and the relief pitcher finishes the game without losing the lead, they are credited with a save. If their team loses, that is a "Blown Save" = BS.)

Pct. = Winning Percentage = W / (W+L) (This ratio shows the overall success of the pitcher. The percentage for a pitcher who has the same number of wins as losses is .500. A higher percentage indicates a pitcher who wins more often.)

ERA = Earned Run Average = 9 * ER / IP (This ratio gives the coach how many earned runs a pitcher gives up per nine innings pitched.)

WHIP = Walks and Hits per Inning Pitched = BB + H / IP (A popular measure in Fantasy Baseball)

SO/BB = Strikeout to Walk Ratio (This ratio compares the number of batters that a pitcher struck out to the number of batters that he allowed to get on base on walks.

The most popular stat in pitching is the ERA. A typical ERA is around 4, above 5 means the pitcher is struggling, below 3 is great!

There are also fielding stats such as fielding percentage (number of times handling the ball correctly/number of times handling the ball), and running stats such as stolen bases, stolen base percentage, and total bases reached.

*In truth, most managers make strange decisions that surprise old time fans as well.

**From a reader: You can't get a speed by adding two momentums. They measure different things! You could say "proportional to ... ." 
I have to spend a bit of time visualizing what you mean by the tangent thing. Wouldn't this situation be more like "angle of reflection = angle of incidence," like a mirror? - Bruce Bottomley
Yeah, I'm not sure I got that right either. I was thinking about adding momentum vectors in three dimensions as opposed to adding speed. As for angle of contact, I was referring to the fact that if the ball hits the top edge of the bat it pops up, the bottom edge it bounces down.
Of course this is just theory. Many batters can "pull" the ball with the right swing (keeping the bat on the ball longer than an instant, thus using the bat to shove the ball where they want it to go), then of course there is the fact that baseballs are not perfectly round, they have seams (these seams are what allows curve balls to be thrown) and when the bat hits a seam instead of a clean contact, the ball slows down due to a cushion effect and then goes off in an uncontrollable direction.

Further links:
Curt Schilling: The Sporting News Sportsman of the Year
Stathead.com - Statistics and Strategies for Baseball
Pokerinformation.com - Rules and Strategies for Poker

High Heat - The Best Baseball Simulation Program Anywhere (It is the game with Schilling on the cover)
Link to a Draw Poker Simulator I wrote for Windows

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