FIGURE 3-2
A simple example of skill and luck distributions
Source: Analysis by author.
To represent an activity that's completely dependent on skill, for instance, we can fill the jar that represents luck with zeros. That way, only the numbers representing skill will count. If we want to represent an activity that is completely dependent on luck, such as roulette, we fill the other jar with zeros. Most activities are some blend of skill and luck.
Here's a simple example. Let's say that the jar representing skill has only three numbers, −3, 0, and 3, and that the jar representing luck has −4, 0, and 4. We can easily list all of the possible outcomes, from −7, which reflects poor skill and bad luck, to 7, the combination of excellent skill and good luck. (See figure 3-3.) Naturally, anything real that we model would be vastly more complex than this example, but these numbers suffice to make several crucial points.
It is possible to do poorly in an activity even with good skill if the influence of luck is sufficiently strong and the number of times you draw from the jars is small. For example, if your level of skill is 3 but you draw a −4 from the jar representing luck, then bad luck trumps skill and you score −1. It's also possible to have a good outcome without being skilled. Your skill at −3 is as low as it can be, but your blind luck in choosing 4 gives you an acceptable score of 1.
FIGURE 3-3
Simple jar model
Source: Analysis by author.
Of course, this effect goes away as you increase the size of the sample. Think of it this way: Say your level of skill is always 3. You draw only from the jar representing luck. In the short run, you might pull some numbers that reflect good or bad luck, and that effect may persist for some time. But over the long haul, the expected value of the numbers representing luck is zero as your draws of balls marked 0, 4, and −4 even out. Ultimately your level of skill, represented by the number 3, will come through.6
The Paradox of Skill—More Skill Means Luck Is More Important
This idea also serves as the basis for what I call the paradox of skill. As skill improves, performance becomes more consistent, and therefore luck becomes more important. Stephen Jay Gould, a renowned paleontologist at Harvard, developed this argument to explain why no baseball player in the major leagues has had a batting average of .400 or more for a full season since Ted Williams hit .406 in 1941 while playing for the Boston Red Sox.7 Gould started by considering some common explanations. The first was that night games, distant travel, diluted talent, and better pitching had all impeded batters. While those factors may have had some influence on the results, none are sufficient to explain the failure to achieve a .400 average. Another possibility was that Williams was not only the best hitter of his era, but that he was better than any other hitter to come along since then. Gould quickly dismissed that argument by showing that in every sport where it can be measured, performance had steadily improved over time. Williams, as good as he was in his era, would certainly not stand out as much if he were to be compared with players today.
At first glance, that may seem contradictory. But the improvement in performance since 1941 may not be as apparent in baseball as it is in other sports because batting average has remained relatively stable, at around .260–.270, for decades. But the stable average masks two important developments. First, batting average reflects not individual skill but rather the interaction between pitchers and hitters. It's like an arms race. As pitchers and hitters both improve their skills on an absolute basis, their relative relationship stays static. Although both pitchers and hitters today are some of the most skillful in history, they have improved in lockstep.8 But that lockstep was not ordained entirely by nature. The overseers of Major League Baseball have had a hand in it. In the late 1960s, for example, when it appeared that pitchers were getting too good for the batters, they changed the rules by lowering the pitcher's mound by five inches and by shrinking the strike zone, allowing the hitters to do better. Thus the rough equilibrium between pitchers and hitters reflects the natural evolution of the players as well as a certain amount of intervention from league officials.
Gould argues that there are no more .400 hitters because all professional hitters have become more skillful, and therefore the difference between the best and worst has narrowed. Training has improved greatly in the last sixty years, which has certainly had an effect on this convergence of skills. In addition, the leagues began recruiting players from around the world, greatly expanding the pool of talent. Hungry kids from the Dominican Republic (Sammy Sosa) and Mexico (Fernando Valenzuela) brought a new level of skill to the game. At the same time, luck continued to play a meaningful role in determining an individual player's batting average. Once the pitcher lets go of a ball, it is still hard to predict whether the batter, however skilled, will connect with it and what will happen if he does.
The key idea, expressed in statistical terms, is that the variance of batting averages has shrunk over time, even as the skill of the hitters has improved. Figure 3-4 shows the standard deviation and coefficient of variation for batting averages by decade since the 1870s. Variance is simply standard deviation squared, so a reduction in standard deviation corresponds to a reduction in variance. The coefficient of variation is the standard deviation divided by the average of all the hitters, which provides an effective measure of how scattered the batting averages of the individual players are from the league average. The figure shows that batting averages have converged over the decades. While Gould focused on batting average, this phenomenon is observable in other relevant statistics as well. For example, the coefficient of variation for earned run average, a measure of how many earned runs a pitcher allows for every nine innings pitched, has also declined over the decades.9
FIGURE 3-4
Reduction in standard deviation in Major League Baseball batting averages
Source: Analysis by author.
This decline in variance explains why there are no more .400 hitters. Since everybody gets better, no one wins quite as dramatically. In his day, Williams was an elite hitter and the variance was large enough that he could achieve such an exalted average. Today, the variance has shrunk to the point that elite hitters have only a tiny probability of matching his average. If Williams played today and had the same level of skill that he had in 1941 relative to other players, his batting average would not come close to .400.
Hitting a baseball in the major leagues is one of the hardest tasks in all of sports. A major league pitcher throws a baseball at speeds of up to one hundred miles an hour with the added complication of sideways or downward movement as it approaches the plate. The paradox of skill says that even though baseball players are more skillful than ever, skill plays a smaller role in determining batting averages than it did in the past. That's because the difference between success and failure for the batter has come to depend on a mistake of only fractions of an inch in where he places his bat or thousandths of a second in his timing in beginning an explosive and nearly automatic swing. But because everyone is uniformly more skillful, the vagaries of luck are more important than ever.
You can readily see how the paradox of skill applies to other competitive activities. A company can improve its absolute performance, for example, but will remain at a competitive parity if its rivals do the same.10 Or if stocks are priced efficiently in the market, luck will determine whether an investor correctly anticipates the next price move up or down. When everyone in business, sports, and investing copies the best practices of others, luck plays a greater role in how well they do.
For activities where little or no luck is involved, the paradox of skill leads to a specific and testable prediction: