The Success Equation. Michael J. Mauboussin

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      Source: Analysis by author.

      In many situations we have only our observations and simply don't know what's possible.² To put it in statistical terms, we don't know what the whole distribution looks like. The greater the influence luck has on an activity, the greater our risk of using induction to draw false conclusions. To put this another way, think of an investor who trades successfully for a hundred days using a particular strategy. He will be tempted to believe that he has a fail-safe way to make money. But when the conditions of the market change, his profits will turn to losses. A small number of observations fails to reveal all of the characteristics of the market.

      We can err in the opposite direction as well, unconsciously assuming that there is some sort of cosmic justice, or a scorekeeper in the sky who will make things even out in the end. This is known as the gambler's fallacy. Say you're watching a coin being tossed. Heads comes up three times in a row. What do you think the next toss will show? Most people will say tails. It feels as if tails is overdue. But it's not. There is a 50-50 chance of both heads and tails on every toss, and one flip has no influence on any other. But if you toss the coin a million times, you will, in fact, see about half a million heads and half a million tails. Conversely, in the universe of the possible, you might see heads come up a hundred times in a row if you toss the coin long enough.

      It turns out that many things in nature do even out, which is why we have evolved to think that all things balance out. Several days of rain are likely to be followed by fair weather. But in cases near the side of the continuum where outcomes are independent of one another, or close to being so, the gambler's fallacy is alive and well. This influence casts its net well beyond naive gamblers and ensnares trained scientists, too.³

      When you're attempting to select the correct size of a sample to analyze, it's natural to assume that the more you allow time to pass, the larger your sample will be. But the relationship between the two is much more complicated than that. In some instances, a short amount of time is sufficient to gather a relatively large sample, while in other cases a lot of time can pass and the sample will remain small. You should consider time as independent from the size of the sample.

      Evaluation of competition in sports illustrates this point. In U.S. men's college basketball, a game lasts forty minutes and each team takes possession of the ball an average of about sixty-five times during the game. Since the number of times each team possesses the ball is roughly equal, possession has little to do with who wins. The team that converts possessions into the most points will win. In contrast, a men's college lacrosse game is sixty minutes long but each team takes possession of the ball only about thirty-three times. So in basketball, each team gets the ball almost twice a minute, while in lacrosse each team gets the ball only once every couple of minutes or so. The size of the sample of possessions in basketball is almost double that of lacrosse. That means that luck plays a smaller role in basketball, and skill exerts a greater influence on who wins. Because the size of the sample in lacrosse is smaller and the number of interactions on the field so large, luck has a greater influence on the final score, even though the game is longer.⁴

      The Two-Jar Model

      Imagine that you have two jars filled with balls.⁵ Each ball has a number on it. The numbers in one jar represent skill, while the numbers in the other represent luck. Higher numbers are better. You draw one ball from the jar that represents skill, one from the jar that represents luck, and then add them together to get a score. Figure 3-2 shows a case where the numbers for skill and luck follow a classic bell curve. But the numbers can follow all sorts of distributions. The idea is to fill each jar with numbers that capture the essence of the activity you are trying to understand.

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, СКАЧАТЬ