Randomness Is Over-Rated

In the preceding post (“Fooled by Non-Randomness“), I had much to say about Nassim Nicholas Taleb’s Fooled by Randomness. The short of is this: Taleb over-rates the role of randomness in financial markets. In fact, his understanding of randomness seems murky.

My aim here is to offer a clearer picture of randomness (or the lack of it), especially as it relates to human behavior. Randomness, as explained in the preceding post, has almost nothing to do with human behavior, which is dominated by intention. Taleb’s misapprehension of randomness leads him  to overstate the importance of a thing called survivor(ship) bias, to which I will turn after dealing with randomness.

WHERE IS RANDOMNESS FOUND?

Randomness — true randomness — is to be found mainly in the operation of fair dice, fair roulette wheels, cryptograhic pinwheels, and other devices designed expressly for the generation of random values. But what about randomness in human affairs?

What we often call random events in human affairs really are non-random events whose causes we do not and, in some cases, cannot know. Such events are unpredictable, but they are not random. Such is the case with such things as rolls (throws) of fair dice — which are considered random events. Dice-rolls are “random” only because it is impossible to perceive the precise conditions of each roll in “real time,” even though knowledge of those conditions would enable a sharp-eyed observer to forecast the outcome of each throw with some accuracy, if the observer were armed with — and had instant access to — analyses of the results of myriad throws whose precise conditions had been captured by various recording devices.

An observer who lacks such information, and who considers the throws of fair dice to be random events, will see that the total number of pips showing on both dice converges on the following frequency distribution:

Rolled Freq.
2 0.028
3 0.056
4 0.083
5 0.111
6 0.139
7 0.167
8 0.139
9 0.111
10 0.083
11 0.056
12 0.028

This frequency distribution is really a shorthand way of writing 28 times out of 1,000; 56 times out of 1,000; etc.

Stable frequency distributions, such as the one given above, have useful purposes. In the case of craps, for example, a bettor can minimize his losses to the house (over a long period of time) if he takes the frequency distribution into account in his betting. Even more usefully, perhaps, an observed divergence from the normal frequency distribution (over many rolls of the dice) would indicate bias caused by (a) an unusual and possibly fraudulent condition (e.g., loaded dice) or (b) a player’s special skill in manipulating dice to skew the frequency distribution in a certain direction.

Randomness, then, is found in (a) the results of non-intentional actions, where (b) we lack sufficient knowledge to understand the link between actions and results.

THE REAL WORLD OF HUMAN AFFAIRS: IT IS WHAT IT IS

You will have noticed the beautiful symmetry of the frequency distribution for dice-rolling. Two-thirds of a large number of dice-rolls will have values of 5 through 9. Values 3 and 4 together will comprise about 14 percent of the rolls, as will values 10 and 11 together. Values 2 and 12 each will comprise less than 3 percent of the rolls.

In other words, the frequency distribution for dice-rolls closely resembles a normal distribution (bell curve). The virtue of this regularity is that it makes predictable the outcome of a large number of dice-rolls; and it makes obvious (over many dice-rolls) a rigged game involving dice. A statistically unexpected distribution of dice-rolls would be considered non-random or, more plainly, rigged — that is, intended by the rigging party.

To state the underlying point explicitly: It is unreasonable to reduce intentional human behavior to probabilistic formulas. Humans don’t behave like dice, roulette balls, or similar “random” devices. But that is what Taleb (and others) do when they ascribe unusual success in financial markets to “luck.” For example, here is what Taleb says on page 136:

I do not deny that if someone performed better than the crowd in the past, there is a presumption of his ability to do better in the future. But the presumption might be weak, very weak, to the point of being useless in decision making. Why? Because it all depends on two factors: The randomness-content of his profession and the number of [persons in the profession].

What Taleb means is this:

  • Success in a profession where randomness dominates outcomes is likely to have the same kind of distribution as that of an event that is considered random, like rolling dice.
  • That being the case, a certain percentage of the members of the profession will, by chance, seem to have great success.
  • If a profession has relatively few members, than a successful person in that profession is more of a standout than a successful person in a profession with, say, thousands of members.

Let me count the assumptions embedded in Taleb’s argument:

  1. Randomness actually dominates some professions. (In particular, he is thinking of the profession of trading financial instruments: stocks, bonds, derivatives, etc.)
  2. Success in a randomness-dominated profession therefore has almost nothing to do with the relevant skills of a member of that profession, nor with the member’s perspicacity in applying those skills.
  3. It follows that a very successful member of a randomness-dominated profession is probably very successful because of luck.
  4. The probability of stumbling across a very successful member of a randomness-dominated profession depends on the total number of members of the profession, given that the probability of success in the profession is distributed in a non-random way (as with dice-rolls).

One of the ways in which Taleb illustrates his thesis is to point to the mutual-fund industry, where far fewer than half the industry’s actively managed funds fail to match the performance of benchmark indices (e.g., S&P 500) over periods of 5 years and longer. But broad, long-term movements in financial markets are not random — as I show in the preceding post.

Nor is trading in financial instruments random; traders do not roll dice or flip coins when they make trades. (Well, the vast majority don’t.) That a majority (or even a super-majority) of actively managed funds does less well than an index fund has nothing to do with randomness and everything to do with the distribution of stock-picking skills. The research required to make informed decisions about financial instruments is arduous and expensive — and not every fool can do it well. Moreover, decision-making — even when based on thorough research — is clouded by uncertainty about the future and the variety of events that can affect the prices of financial instruments.

It is therefore unsurprising  that the distribution of skills in the financial industry is skewed; there are relatively few professionals who have what it takes to succeed over the long run, and relatively many professionals (or would-be professionals) who compile mediocre-to-awful records.

I say it again: The most successful professionals are not successful because of luck, they are successful because of skill. There is no statistically predetermined percentage of skillful traders; the actual percentage depends on the skills of entrants and their willingness (if skillful) to make a career of it. A relevant analogy is found in the distribution of incomes:

In 2007, all households in the United States earned roughly $7.896 trillion [25]. One half, 49.98%, of all income in the US was earned by households with an income over $100,000, the top twenty percent. Over one quarter, 28.5%, of all income was earned by the top 8%, those households earning more than $150,000 a year. The top 3.65%, with incomes over $200,000, earned 17.5%. Households with annual incomes from $50,000 to $75,000, 18.2% of households, earned 16.5% of all income. Households with annual incomes from $50,000 to $95,000, 28.1% of households, earned 28.8% of all income. The bottom 10.3% earned 1.06% of all income.

The outcomes of human endeavor are skewed because the distribution of human talents is skewed. It would be surprising to find as many as one-half of traders beating the long-run average performance of the various markets in which they operate.

BACK TO BASEBALL

To drive the point home, I return to the example of baseball, which I treated at length in the preceding post. Baseball, like most games, has many “random” elements, which is to say that baseball players cannot always predict accurately such things as the flight of a thrown or batted ball, the course a ball will take when it bounces off grass or an outfield fence, the distance and direction of a throw from the outfield, and so on. But despite the many unpredictable elements of the game, skill dominates outcomes over the course of seasons and careers. Moreover, skill is not distributed in a neat way, say, along a bell curve. A good case in point is the distribution of home runs:

  • There have been 16,884 players and 253,498 home runs in major-league history (1876 – present), an average of 15 home runs per person who have played in the major leagues since 1876. About 2,700 players have more than 15 home runs; about 14,000 players have fewer than 15 home runs; and about 100 players have exactly 15 home runs. Of the 2,700 players with more than 15 home runs, there are (as of yesterday) 1,006 with 74 or more home runs, and 25 with 500 or more home runs. (I obtained data about the frequency of career home runs with this search tool at Baseball-Reference.com.)
  • The career home-run statistic, in other words, has an extremely long, thin “tail” that, at first, rises gradually from 0 to 15. This tail represents the home-run records of about 89 percent of all the men who have played in the major leagues. The tail continues to broaden until, at the other end, it becomes a very short, very fat hump, which represents the 0.15 percent of players with 500 or more home runs.
  • There may be a standard statistical distribution which seems to describe the incidence of career home runs. But to say that the career home-run statistic matches any kind of distribution is merely to posit an after-the-fact “explanation” of a phenomenon that has one essential explanation: Some hitters are better at hitting home runs than other players; those better home-run hitters are more likely to stay in the major leagues long enough to compile a lot of home runs. (Even 74 home runs is a lot, relative to the mean of 15.)

And so it is with traders and other active “players” in financial markets. They differ in skill, and their skill differences cannot be arrayed neatly along a bell curve or any other mathematically neat frequency distribution. To adapt a current coinage, they are what they are — nothing more, nothing less.

TALEB’S A PRIORI WORLDVIEW, WITH A BIAS

Taleb, of course, views the situation the other way around. He sees an a priori distribution of “winners” and losers,” where “winners” are determined mainly by luck, not skill. Moreover, we — the civilians on the sidelines — labor under the false impression about the relative number of “winners” because

it is natural for those who failed to vanish completely. Accordingly, one sees the survivors, and only the survivors, which imparts such a mistaken perception of the odds [favoring success]. (p. 137)

Here, Taleb is playing a variation on a favorite theme: survivor(ship) bias. What is it? Here are three quotations that may help you understand it:

Survivor bias is a prominent form of ex-post selection bias. It exists in data sets that exclude a disproportionate share of non-surviving firms…. (“Accounting Information Free of Selection Bias: A New UK Database 1953-1999 “)

Survivorship bias causes performance results to be overstated
because accounts that have been terminated, which may have
underperformed, are no longer in the database. This is the most
documented and best understood source of peer group bias. For example, an unsuccessful management product that was
terminated in the past is excluded from current peer groups.
This screening out of losers results in an overstatement of past
performance. A good illustration of how survivor bias can skew
things is the “marathon analogy”, which asks: If only 100 runners out of a 1,000?contestant marathon actually finish, is the 100th the last? Or in the top ten percent? (“Warning! Peer Groups Are Hazardous to Our Wealth“)

It is true that a number of famous successful people have spent 10,000 hours practising. However, it is also true that many people we have never heard of because they weren’t successful also practised for 10,000 hours. And that there are successful people who were very good without practising for 10,000 hours before their breakthrough (the Rolling Stones, say). And Gordon Brown isn’t very good at being Prime Minister despite preparing for 10,000 hours. (“Better Services without Reform? It’s Just a Con“)

First of all, there are no “odds” favoring success — even in financial markets. Financial “players” do what they can do, and most of them — like baseball players — simply don’t have what it takes for great success. Outcomes are skewed, not because of (fictitious) odds but because talent is distributed unevenly.

CONCLUSION

The real lesson for us spectators is not to assume that the “winners” are merely lucky. No, the real lesson is to seek out those “winners” who have proven their skills over a long period of time, through boom and bust and boom and bust.

Those who do well, over the long run, do not do so merely because they have survived. They have survived because they do well.

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