Koufax Kicks

You know what? He's just plain wrong.
To people with a brain, a shootout is like deciding a basketball game by having the teams play baseball. Yep, baseball. America's national blankin’ pastime. Babe Ruth. Rajah Clemons. 3000 Ks. Tony Gwynn. Greg Maddux – the great cerebrist on the mound. Did you ever consider that Brazil’s goalie might anticipate tendencies as well as Maddux does?
Here at the Jones, we could bury you with hypertext consisting of our collective musings as to how Americans can (a) find poetry in the mental and physical competition between two guys standing sixty feet, six inches apart, one wielding a bat, one armed with an applesized pelota, yet (b) dismiss as ridiculous soccer's isomorphic scenario, but we'll let you fill in the blanks yourself.
We’ve got bigger fish to fry. We want to
explain why you should get your kicks from those lifestopping kicks, and
why they reveal yet another dark side of ego in sports.
You might
be thinking, "Who cares? I don't have to watch soccer for another four
years." That's true. Point taken. But this riff deals with strategy.
Fastball or slider? Pass or run? Bunt or swing away? To understand
baseball and why Maddux keeps his ego in check, read on.
In
strategic situations, you want to keep the other guy guessing. This goes
for baseball, soccer, football, chess, you name it. If your opponent knows
your strategy, you’re done. (Unless you’re Woody Hayes and just want to
settle it like men.) If you never throw deep, the DBs creep to the line.
If you never throw to first, Rickey cruises into second.
You know, in the shootout, since the goalie is always
jumping to the left or right, guys and gals should sometimes kick to the
center. It’s wide open! But I know why they don’t. (Hint: They’re men, or
they act like them.
Matching Pennies
Before soccer there was Matching Pennies. Matching Pennies is really about baseball. Heck, everything is really about baseball.
Economics professors teach Matching Pennies in game theory courses around the world (much like soccer is played around the world). In fact, three game theorists recently received Nobel Prizes in economics.
So, even if you cannot shake that
predisposition to abhor anything soccer related, and you're counting the
days ‘til The Deuce returns to those Superfights of the 70s, stick around.
Game theory rocks.
Matching Pennies is a simplified version of
paperscissorsrock. That’s why academic types write long mathematical
papers about it. (Entire libraries are devoted to
paperscissorsrock.)
Here we go. Rajah and Clemente each have a penny they can place heads up (H) or tails up (T). Rajah wants to avoid matching. Clemente wants to match. Rajah throws fastball. Clemente thinks fastball, Clemente wins. Clemente thinks curve, Clemente loses.
So, what should Rajah do? If he always plays H, then Clemente will figure it out and also play H. Therefore, Rajah should randomize 5050 between H and T (taking the quality of the pitches out of the equation for now). If Rajah does this, it doesn’t matter what Clemente does – he’ll win half the time. If Clemente plays H, half of the time Rajah is playing H, so Clemente wins, and half of the time Rajah is playing T, so Clemente loses. If Clemente plays T, he also wins half the time and loses half the time.
Left or right
Now, let’s apply this logic to soccer. Two players, a goalie we'll call Gehrig and a kicker Koufax. In watching the World Cup (or at least the highlights), you may have noticed that Koufax seems to have two options. Left or right. Gehrig has the same two options.
Computing the probability of a goal gets a little more difficult here. It depends upon Koufax’s skill level. Koufax may be able to score with incredibly high probability, even if Gehrig guesses correctly. How? By lofting the ball into the upper corner. However, a lesser player aiming for the corner may either miss high or hit low.
We’re going to summarize all of this
information by letting Q be the probability that Koufax scores even if
Gehrig guesses correctly. Q stands for quality. (Formerly, it stood for
Quentin Dailey, but he’s long gone.) A dead guy has a Q of 0, and Pelé a Q
of 1. The rest of us are in between.
As with Rajah and Clemente
above, the equilibrium of this
matchup is for Koufax to play left half the time and right half the time.
So, if left and right are the only options, then the probability of a goal
is (1+Q)/2. If Q = 1/2, the probability of a goal equals 3/4.
You might be tempted to say that it doesn’t
take game theory to say that (a) better players score more often and (b)
the players should randomize.
True enough. But, here’s where the
game theory earns its keep. We’re going to show you that these soccer
players’ massive egos prevent them from playing an even smarter
strategy.
The question that just nags nags nags is why
the heck doesn’t Koufax kick the ball into the center of the net? If
Gehrig is jumping left or right, that means that the center is open.
Good question.
Let’s expand the game and find out what’s up.
We’ll assume that if Gehrig plays C (for center) and Koufax goes with
either L or R, Koufax scores. But, if Koufax and Gehrig both play C,
Koufax loses every time. After all, Gehrig is just standing there, like
Bill Buckner, only he stops the ball.
Now, if Gehrig always chooses
L or R, then Koufax can just play C, score for sure, and make Gehrig look
foolish. In that case, Gehrig would want to play C sometimes as well. And
Koufax should randomize so that Gehrig doesn't know whether to play L, C,
or R.
And, since Koufax would have three options instead of two, the chances that Gehrig could match would be reduced. So, why don’t we see kicks in the center of the net?
Two answers: ego and ego.
Pride goeth before a dive
First, ego. Let’s say Koufax has a quality of Q=1/2 and both players play like game theory says they should. Then, one out of five times, Koufax kicks it into the center of the net. And, one out of five times, Gehrig just stands in the center.
[Mike's comment: convince yourself why this is so]
Since their actions are independent (Gehrig
doesn’t know what Koufax will do and vice versa), the probability of
Koufax kicking into the center and Gehrig standing there waiting for it
equals 1 in 25 (that’s 1/5 x 1/5). This means that four percent of the
time, Koufax kicks the ball to a stationary Gehrig.
Imagine this
scenario. Koufax is one of the greatest athletes in the world, he’s dating
a Spice Girl, and he kicks the ball into the center of the net, where
Gehrig has luckily decided to stay. Koufax returns to England and goes on
the BBC to explain: "I know that I kicked the ball to the center of the
net to a waiting goalie and we lost the Cup, but it was an equilibrium
strategy according to game theory."
Fact is, this won't happen. As
badly as these guys want to win, they don’t want to win that badly.
Neither do the goalies. That’s why they never just stand in the center. A
German goalie who stood midgoal would be practically begging some guys
from Stuttgart to rearrange his face, if he dared show it.
Second,
ego. By not kicking into the center, players signal their quality. Who
benefits the most by using the additional strategy of kicking in the
center of the net? A player with a Q of zero increases his chances of
scoring from 50 percent to 66 percent, a 16 percent increase. But, a
player with a Q of 0.8 increases his chances of scoring from 90 to 91
percent. Hardly worth it. Therefore, by kicking into the center, not only
would a player risk humiliation, he would be announcing, "Hey, I suck!"
Would Nike pay for that?
We could go on and on with this. If the
first two players score by going left, should the third go left? Wouldn't
center be an obvious choice if the game were totally on the line?
Hey, this is actually sort of exciting. Wait,
imagine this. A limping Kirk Gibson softly leftfoots a roller toward the
gaping middle of the net, having stolen Joe Gibbs sign. Lawrence Taylor
reverses midflight, just nicking the ball, which nonetheless skips,
bounces, creeps over the line.
You spell it like this: Gibson
Outguesses Anticipatory Lawrence. GOAL.
Don't care? Go watch
Heidi again, ‘cause you'll never understand sports.