The Success Equation. Michael J. Mauboussin
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is zero, the expected value of the new outcome is 3. That is the idea behind reversion to the mean.
We can do the same exercise for the bottom four outcomes (−3 skill, −4 luck; 0 skill, −4 luck; −3 skill, 0 luck; and 3 skill, −4 luck). They add up to −15, and the contribution from skill alone is −3. Here, also, with a new draw the expected value for luck is zero, so the total goes from −15 to an expected value of −3. In both cases, skill remains the same but the large contributions from either good luck or bad luck shrink toward zero.
While most people seem to understand the idea of reversion to the mean, using the jars and the continuum between luck and skill can add an important dimension to this thinking. In the two-jar exercise, you draw only once from the jar representing skill; after that, your level of skill is assumed to remain the same. This is an unrealistic assumption over a long period of time but very reasonable for the short term. You then draw from the jar representing luck, record your value, and return the number to the jar. As you draw again and again, your scores reflect stable skill and variations in luck. In this form of the exercise, your skill ultimately determines whether you wind up a winner or a loser.
The position of the activity on the continuum defines how rapidly your score goes toward an average value, that is, the rate of reversion to the mean. Say, for example, that an activity relies entirely on skill and involves no luck. That means the number you draw for skill will always be added to zero, which represents luck. So each score will simply be your skill. Since the value doesn't change, there is no reversion to the mean. Marion Tinsley, the greatest player of checkers, could win all day long, and luck played no part in it. He was simply better than everyone else.
Now assume that the jar representing skill is filled with zeros, and that your score is determined solely by luck; that is, the outcomes will be dictated solely by luck and the expected value of every incremental draw for skill will be the same: zero. So every subsequent outcome has an expected value that represents complete reversion to the mean. In activities that are all skill, there is no reversion to the mean. In activities that are all luck, there is complete reversion to the mean. So if you can place an activity on the luck-skill continuum, you have a sound starting point for anticipating the rate of reversion to the mean.
In real life, we don't know for sure how skill and luck contribute to the results when we make decisions. We can only observe what happens. But we can be more formal in specifying the rate of reversion to the mean by introducing the James-Stein estimator with a focus on what is called the shrinking factor.18 This construct is easiest to understand by using a concrete example. Say you have a baseball player, Joe, who hits .350 for part of the season, when the average of all players is .265. You don't really believe that Joe will average .350 forever because even if he's an above-average hitter, he's likely been the beneficiary of good luck recently. You'd like to know what his average will look like over a longer period of time. The best way to estimate that is to reduce his average so that it is closer to .265. The James-Stein estimator includes a factor that tells you how much you need to shrink the .350 while Joe's average is high so that his number more closely resembles his true ability in the long run. Let's go straight to the equation to see how it works:
Estimated true average = Grand average + shrinking factor (observed average − grand average)
The estimated true average would represent Joe's true ability. The grand average is the average of all of the players (.265), and the observed average is Joe's average during his period of success (.350). In a classic article on this topic, two statisticians named Bradley Efron and Carl Morris estimated the shrinking factor for batting averages to be approximately .2. (They used data on batting averages from the 1970 season with a relatively small sample, so consider this as illustrative and not definitive.)19 Here is how Joe's average looks using the James-Stein estimator:
Estimated true average = .265 + .2 (.350 − .265)
According to this calculation, Joe is most likely going to be batting .282 for most of the season. The equation can also be used for players who have averages below the grand average. For example, the best estimate of true ability for a player who is hitting only .175 for a particular stretch is .247, or .265 + .2 (.175 − .265).
For activities that are all skill, the shrinking factor is 1.0, which means that the best estimate of the next outcome is the prior outcome. When Marion Tinsley was playing checkers, the best guess about who would win the next game was Marion Tinsley. If you assume that skill is stable in the short term and that luck is not a factor, this is the exact outcome that you would expect.
For activities that are all luck, the shrinking factor is 0, which means that the expected value of the next outcome is the mean of the distribution of luck. In most American casinos, the mean distribution of luck in the game of roulette is 5.26 percent, the house edge, and no amount of skill can change that. You may win a lot for a while or lose a lot for a while, but if you play long enough, you will lose 5.26 percent of your money. If skill and luck play an equal role, then the shrinking factor is 0.5, halfway between the two. So we can assign a shrinking factor to a given activity according to where that activity lies on the continuum. The closer the activity is to all skill, the closer the factor is to 1. The larger the role that luck plays, the closer the factor is to zero. We will see a specific example of how these shrinking factors correlate with skill in chapter 10.
The James-Stein estimator can be useful in predicting the outcome of any activity that combines skill and luck. To use one example, the return on invested capital for companies reverts to the mean over time. In this case, the rate of reversion to the mean reflects a combination of a company's competitive position and its industry. Generally speaking, companies that deal in technology (and companies whose products have short life cycles) tend to revert more rapidly to the mean than established companies with stable demand for their well-known consumer products. So Seagate Technology, a maker of hard drives for computers, will experience more rapid reversion to the mean than Procter & Gamble, the maker of the best-selling detergent, Tide, because Seagate has to constantly innovate, and even its winning products have a short shelf life. Put another way, companies that deal in technology have a shrinking factor that is closer to zero.
Similarly, investing is a very competitive activity, and luck weighs heavily on the outcomes in the short term. So if you are using a money manager's past returns to anticipate her future results, a low shrinkage factor is appropriate. Past performance is no guarantee of future results because there is too much luck involved in investing.
Understanding the rate of reversion to the mean is essential for good forecasting. The continuum of luck and skill, as our experience with the two jars has shown, provides a practical way to think about that rate and ultimately to measure it.
So far, I have assumed that the jars contain numbers that follow a normal distribution, but in fact, distributions are rarely normal. Furthermore, the level of skill changes over time, whether you're talking about an athlete, a company, or an investor. But using jars to create a model is a method that can accommodate those different distributions. Chapters 5 and 6 will examine how skill changes over time and what forms luck can take.
Visualizing luck and skill as a continuum provides a simple concept that can carry a lot of intellectual freight. It allows us to understand when luck can make your level of skill irrelevant, especially in the short term, as we saw with the Playboy Playmates. It allows us to think about extreme performance, as in the cases of Bill Joy and Joe DiMaggio. And makes it possible for us to calibrate the rate of reversion to the mean, as we did with batting averages. Each of these ideas is essential to making intelligent predictions.
Chapter 4 looks at techniques for placing activities on the continuum. It's time to make the ideas from the continuum operational.
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