Category: Probability

The Probability Distribution of the Future

The best colloquial definition of risk may be the following:

“Risk means more things can happen than will happen.”

We found it through the inimitable Howard Marks, but it's a quote from Elroy Dimson of the London Business School. Doesn't that capture it pretty well?

Another way to state it is: If there were only one thing that could happen, how much risk would there be, except in an extremely banal sense? You'd know the exact probability distribution of the future. If I told you there was a 100% probability that you'd get hit by a car today if you walked down the street, you simply wouldn't do it. You wouldn't call walking down the street a “risky gamble” right? There's no gamble at all.

But the truth is that in practical reality, there aren't many 100% situations to bank on. Way more things can happen than will happen. That introduces great uncertainty into the future, no matter what type of future you're looking at: An investment, your career, your relationships, anything.

How do we deal with this in a pragmatic way? The investor Howard Marks starts it this way:

Key point number one in this memo is that the future should be viewed not as a fixed outcome that’s destined to happen and capable of being predicted, but as a range of possibilities and, hopefully on the basis of insight into their respective likelihoods, as a probability distribution.

This is the most sensible way to think about the future: A probability distribution where more things can happen than will happen. Knowing that we live in a world of great non-linearity and with the potential for unknowable and barely understandable Black Swan events, we should never become too confident that we know what's in store, but we can also appreciate that some things are a lot more likely than others. Learning to adjust probabilities on the fly as we get new information is called Bayesian updating.


Although the future is certainly a probability distribution, Marks makes another excellent point in the wonderful memo above: In reality, only one thing will happen. So you must make the decision: Are you comfortable if that one thing happens, whatever it might be? Even if it only has a 1% probability of occurring? Echoing the first lesson of biology, Warren Buffett stated that “In order to win, you must first survive.” You have to live long enough to play out your hand.

Which leads to an important second point: Uncertainty about the future does not necessarily equate with risk, because risk has another component: Consequences. The world is a place where “bad outcomes” are only “bad” if you know their (rough) magnitude. So in order to think about the future and about risk, we must learn to quantify.

It's like the old saying (usually before something terrible happens): What's the worst that could happen? Let's say you propose to undertake a six month project that will cost your company $10 million, and you know there's a reasonable probability that it won't work. Is that risky?

It depends on the consequences of losing $10 million, and the probability of that outcome. It's that simple! (Simple, of course, does not mean easy.) A company with $10 billion in the bank might consider that a very low-risk bet even if it only had a 10% chance of succeeding.

In contrast, a company with only $10 million in the bank might consider it a high-risk bet even if it only had a 10% of failing. Maybe five $2 million projects with uncorrelated outcomes would make more sense to the latter company.

In the real world, risk = probability of failure x consequences. That concept, however, can be looked at through many lenses. Risk of what? Losing money? Losing my job? Losing face? Those things need to be thought through. When we observe others being “too risk averse,” we might want to think about which risks they're truly avoiding. Sometimes risk is not only financial. 


Let's cover one more under-appreciated but seemingly obvious aspect of risk, also pointed out by Marks: Knowing the outcome does not teach you about the risk of the decision.

This is an incredibly important concept:

If you make an investment in 2012, you’ll know in 2014 whether you lost money (and how much), but you won’t know whether it was a risky investment – that is, what the probability of loss was at the time you made it.

To continue the analogy, it may rain tomorrow, or it may not, but nothing that happens tomorrow will tell you what the probability of rain was as of today. And the risk of rain is a very good analogue (although I’m sure not perfect) for the risk of loss.

How many times do we see this simple dictum violated? Knowing that something worked out, we argue that it wasn't that risky after all. But what if, in reality, we were simply fortunate? This is the Fooled by Randomness effect.

The way to think about it is the following: The worst thing that can happen to a young gambler is that he wins the first time he goes to the casinoHe might convince himself he can beat the system.

The truth is that most times we don't know the probability distribution at all. Because the world is not a predictable casino game — an error Nassim Taleb calls the Ludic Fallacy — the best we can do is guess.

With intelligent estimations, we can work to get the rough order of magnitude right, understand the consequences if we're wrong, and always be sure to never fool ourselves after the fact.

If you're into this stuff, check out Howard Marks' memos to his clients, or check out his excellent book, The Most Important Thing. Nate Silver also has an interesting similar idea about the difference between risk and uncertainty. And lastly, another guy that understands risk pretty well is Jason Zweig, who we've interviewed on our podcast before.


If you liked this article you'll love:

Nassim Taleb on the Notion of Alternative Histories — “The quality of a decision cannot be solely judged based on its outcome.”

The Four Types of Relationships — As Seneca said, “Time discovers truth.”

Daniel Kahneman in Conversation with Michael Mauboussin on Intuition, Causality, Loss Aversion and More

Ever want to be the fly on the wall for a fascinating conversation. Well, here's your chance. Santa Fe Institute Board of Trustees Chair Michael Mauboussin interviews Nobel Prize winner Daniel Kahneman. The wide-ranging conversation talks about disciplined intuition, causality, base rates, loss aversion and so much more. You don't want to miss this.

Here's an excerpt from Kahneman I think you'll enjoy. You can read the entire transcript here.

The Sources of Power is a very eloquent book on expert intuition with magnificent examples, and so he is really quite hostile to my point of view, basically.

We spent years working on that, on the question of when can intuitions be trusted? What's the boundary between trustworthy and untrustworthy intuitions?

I would summarize the answer as saying there is one thing you should not do. People's confidence in their intuition is not a good guide to their validity. Confidence is something else entirely, and maybe we can talk about confidence separately later, but confidence is not it.

What there is, if you want to know whether you can trust intuition, it really is like deciding on a painting, whether it's genuine or not. You can look at the painting all you want, but asking about the provenance is usually the best guide about whether a painting is genuine or not.

Similarly for expertise and intuition, you have to ask not how happy the individual is with his or her own intuitions, but first of all, you have to ask about the domain. Is the domain one where there is enough regularity to support intuitions? That's true in some medical domains, it certainly is true in chess, it is probably not true in stock picking, and so there are domains in which intuition can develop and others in which it cannot. Then you have to ask whether, if it's a good domain, one in which there are regularities that can be picked up by the limited human learning machine. If there are regularities, did the individual have an opportunity to learn those regularities? That primarily has to do with the quality of the feedback.

Those are the questions that I think should be asked, so there is a wide domain where intuitions can be trusted, and they should be trusted, and in a way, we have no option but to trust them because most of the time, we have to rely on intuition because it takes too long to do anything else.

Then there is a wide domain where people have equal confidence but are not to be trusted, and that may be another essential point about expertise. People typically do not know the limits of their expertise, and that certainly is true in the domain of finances, of financial analysis and financial knowledge. There is no question that people who advise others about finances have expertise about finance that their advisees do not have. They know how to look at balance sheets, they understand what happens in conversations with analysts.

There is a great deal that they know, but they do not really know what is going to happen to a particular stock next year. They don't know that, that is one of the typical things about expert intuition in that we know domains where we have it, there are domains where we don't, but we feel the same confidence and we do not know the limits of our expertise, and that sometimes is quite dangerous.


The Psychology of Risk and Reward

The Psychology of Risk and Reward

An excerpt from The Aspirational Investor: Taming the Markets to Achieve Your Life's Goals that I think you'd enjoy.

Most of us have a healthy understanding of risk in the short term.

When crossing the street, for example, you would no doubt speed up to avoid an oncoming car that suddenly rounds the corner.

Humans are wired to survive: it’s a basic instinct that takes command almost instantly, enabling our brains to resolve ambiguity quickly so that we can take decisive action in the face of a threat.

The impulse to resolve ambiguity manifests itself in many ways and in many contexts, even those less fraught with danger. Glance at the (above) picture for no more than a couple of seconds. What do you see?

Some observers perceive the profile of a young woman with flowing hair, an elegant dress, and a bonnet. Others see the image of a woman stooped in old age with a wart on her large nose. Still others—in the gifted minority—are able to see both of the images simultaneously.

What is interesting about this illusion is that our brains instantly decide what image we are looking at, based on our first glance. If your initial glance was toward the vertical profile on the left-hand side, you were all but destined to see the image of the elegant young woman: it was just a matter of your brain interpreting every line in the picture according to the mental image that you already formed, even though each line can be interpreted in two different ways. Conversely, if your first glance fell on the central dark horizontal line that emphasizes the mouth and chin, your brain quickly formed an image of the older woman.

Regardless of your interpretation, your brain wasn’t confused. It simply decided what the picture was and filled in the missing pieces. Your brain resolved ambiguity and extracted order from conflicting information.

What does this have to do with decision making? Every bit of information can be interpreted differently according to our perspective. Ashvin Chhabra directs us to investing. I suggest you reframe this in the context of decision making in general.

Every trade has a seller and a buyer: your state of mind is paramount. If you are in a risk-averse mental framework, then you are likely to interpret a further fall in stocks as additional confirmation of your sell bias. If instead your framework is positive, you will interpret the same event as a buying opportunity.

The challenge of investing is compounded by the fact that our brains, which excel at resolving ambiguity in the face of a threat, are less well equipped to navigate the long term intelligently. Since none of us can predict the future, successful investing requires planning and discipline.

Unfortunately, when reason is in apparent conflict with our instincts—about markets or a “hot stock,” for example—it is our instincts that typically prevail. Our “reptilian brain” wins out over our “rational brain,” as it so often does in other facets of our lives. And as we have seen, investors trade too frequently, and often at the wrong time.

One way our brains resolve conflicting information is to seek out safety in numbers. In the animal kingdom, this is called “moving with the herd,” and it serves a very important purpose: helping to ensure survival. Just as a buffalo will try to stay with the herd in order to minimize its individual vulnerability to predators, we tend to feel safer and more confident investing alongside equally bullish investors in a rising market, and we tend to sell when everyone around us is doing the same. Even the so-called smart money falls prey to a herd mentality: one study, aptly titled “Thy Neighbor’s Portfolio,” found that professional mutual fund managers were more likely to buy or sell a particular stock if other managers in the same city were also buying or selling.

This comfort is costly. The surge in buying activity and the resulting bullish sentiment is self-reinforcing, propelling markets to react even faster. That leads to overvaluation and the inevitable crash when sentiment reverses. As we shall see, such booms and busts are characteristic of all financial markets, regardless of size, location, or even the era in which they exist.

Even though the role of instinct and human emotions in driving speculative bubbles has been well documented in popular books, newspapers, and magazines for hundreds of years, these factors were virtually ignored in conventional financial and economic models until the 1970s.

This is especially surprising given that, in 1951, a young PhD student from the University of Chicago, Harry Markowitz, published two very important papers. The first, entitled “Portfolio Selection,” published in the Journal of Finance, led to the creation of what we call modern portfolio theory, together with the widespread adoption of its important ideas such as asset allocation and diversification. It earned Harry Markowitz a Nobel Prize in Economics.

The second paper, entitled “The Utility of Wealth” and published in the prestigious Journal of Political Economy, was about the propensity of people to hold insurance (safety) and to buy lottery tickets at the same time. It delved deeper into the psychological aspects of investing but was largely forgotten for decades.

The field of behavioral finance really came into its own through the pioneering work of two academic psychologists, Amos Tversky and Daniel Kahneman, who challenged conventional wisdom about how people make decisions involving risk. Their work garnered Kahneman the Nobel Prize in Economics in 2002. Behavioral finance and neuroeconomics are relatively new fields of study that seek to identify and understand human behavior and decision making with regard to choices involving trade-offs between risk and reward. Of particular interest are the human biases that prevent individuals from making fully rational financial decisions in the face of uncertainty.

As behavioral economists have documented, our propensity for herd behavior is just the tip of the iceberg. Kahneman and Tversky, for example, showed that people who were asked to choose between a certain loss and a gamble, in which they could either lose more money or break even, would tend to choose the double down (that is, gamble to avoid the prospect of losses), a behavior the authors called “loss aversion.” Building on this work, Hersh Shefrin and Meir Statman, professors at the University of Santa Clara Leavey School of Business, have linked the propensity for loss aversion to investors’ tendency to hold losing investments too long and to sell winners too soon. They called this bias the disposition effect.

The lengthy list of behaviorally driven market effects often converge in an investor’s tale of woe. Overconfidence causes investors to hold concentrated portfolios and to trade excessively, behaviors that can destroy wealth. The illusion of control causes investors to overestimate the probability of success and underestimate risk because of familiarity—for example, causing investors to hold too much employer stock in their 401(k) plans, resulting in under-diversification. Cognitive dissonance causes us to ignore evidence that is contrary to our opinions, leading to myopic investing behavior. And the representativeness bias leads investors to assess risk and return based on superficial characteristics—for example, by assuming that shares of companies that make products you like are good investments.

Several other key behavioral biases come into play in the realm of investing. Framing can cause investors to make a decision based on how the question is worded and the choices presented. Anchoring often leads investors to unconsciously create a reference point, say for securities prices, and then adjust decisions or expectations with respect to that anchor. This bias might impede your ability to sell a losing stock, for example, in the false hope that you can earn your money back. Similarly, the endowment bias might lead you to overvalue a stock that you own and thus hold on to the position too long. And regret aversion may lead you to avoid taking a tough action for fear that it will turn out badly. This can lead to decision paralysis in the wake of a market crash, even though, statistically, it is a good buying opportunity.

Behavioral finance has generated plenty of debate. Some observers have hailed the field as revolutionary; others bemoan the discipline’s seeming lack of a transcendent, unifying theory. This much is clear: behavioral finance treats biases as mistakes that, in academic parlance, prevent investors from thinking “rationally” and cause them to hold “suboptimal” portfolios.

But is that really true? In investing, as in life, the answer is more complex than it appears. Effective decision making requires us to balance our “reptilian brain,” which governs instinctive thinking, with our “rational brain,” which is responsible for strategic thinking. Instinct must integrate with experience.

Put another way, behavioral biases are nothing more than a series of complex trade-offs between risk and reward. When the stock market is taking off, for example, a failure to rebalance by selling winners is considered a mistake. The same goes for a failure to add to a position in a plummeting market. That’s because conventional finance theory assumes markets to be inherently stable, or “mean-reverting,” so most deviations from the historical rate of return are viewed as fluctuations that will revert to the mean, or self-correct, over time.

But what if a precipitous market drop is slicing into your peace of mind, affecting your sleep, your relationships, and your professional life? What if that assumption about markets reverting to the mean doesn’t hold true and you cannot afford to hold on for an extended period of time? In both cases, it might just be “rational” to sell and accept your losses precisely when investment theory says you should be buying. A concentrated bet might also make sense, if you possess the skill or knowledge to exploit an opportunity that others might not see, even if it flies in the face of conventional diversification principles.

Of course, the time to create decision rules for extreme market scenarios and concentrated bets is when you are building your investment strategy, not in the middle of a market crisis or at the moment a high-risk, high-reward opportunity from a former business partner lands on your desk and gives you an adrenaline jolt. A disciplined process for managing risk in relation to a clear set of goals will enable you to use the insights offered by behavioral finance to your advantage, rather than fall prey to the common pitfalls. This is one of the central insights of the Wealth Allocation Framework. But before we can put these insights to practical use, we need to understand the true nature of financial markets.

The Lucretius Problem


It's always good to re-read books and to dip back into them periodically. When reading a new book, I often miss out on crucial information (especially books that are hard to categorize with one descriptive sentence). When you come back to a book after reading hundreds of others you can't help but make new connections with the old book and see it anew.

It has been a while since I read Anti-fragile. In the past I've talked about an Antifragile Way of Life, Learning to Love Volatility, the Definition of Antifragility , Antifragile life of economy, and the Noise and the Signal.

But upon re-reading Antifragile I came across the Lucretius Problem and I thought I'd share an excerpt. (Titus Lucretius Carus was a Roman poet and philosopher, best-known for his poem On the Nature of Things). Taleb writes:

Indeed, our bodies discover probabilities in a very sophisticated manner and assess risks much better than our intellects do. To take one example, risk management professionals look in the past for information on the so-called ​worst-case scenario ​and use it to estimate future risks – this method is called “stress testing.” They take the worst historical recession, the worst war, the worst historical move in interest rates, or the worst point in unemployment as an exact estimate for the worst future outcome​. But they never notice the following inconsistency: this so-called worst-case event, when it happened, exceeded the worst [known] case at the time.

I have called this mental defect the Lucretius problem, after the Latin poetic philosopher who wrote that the fool believes that the tallest mountain in the world will be equal to the tallest one he has observed. We consider the biggest object of any kind that we have seen in our lives or hear about as the largest item that can possibly exist. And we have been doing this for millennia.

Taleb brings up an interesting point, which is that our documented history can blind us. All we know is what we have been able to record.

We think because we have sophisticated data collecting techniques that we can capture all the data necessary to make decisions. We think we can use our current statistical techniques to draw historical trends using historical data without acknowledging the fact that past data recorders had fewer tools to capture the dark figure of unreported data. We also overestimate the validity of what has been recorded before and thus the trends we draw might tell a different story if we had the dark figure of unreported data.

Taleb continues:

The same can be seen in the Fukushima nuclear reactor, which experienced a catastrophic failure in 2011 when a tsunami struck. It had been built to withstand the worst past historical earthquake, with the builders not imagining much worse— and not thinking that the worst past event had to be a surprise, as it had no precedent. Likewise, the former chairman of the Federal Reserve, Fragilista Doctor Alan Greenspan, in his apology to Congress offered the classic “It never happened before.” Well, nature, unlike Fragilista Greenspan, prepares for what has not happened before, assuming worse harm is possible.

So what do we do and how do we deal with the blindness?

Taleb provides an answer which is to develop layers of redundancy to act as a buffer against oneself. We overvalue what we have recorded and assume it tells us the worst and best possible outcomes. Redundant layers are a buffer against our tendency to think what has been recorded is a map of the whole terrain. An example of a redundant feature could be a rainy day fund which acts as an insurance policy against something catastrophic such as a job loss that allows you to survive and fight another day.

Antifragile is a great book to read and you might learn something about yourself and the world you live in by reading it or in my case re-reading it.

Fooled By Randomness

fooled by randomness

I don't want you to make the same mistake I did.

I waited too long before reading Fooled by Randomness: The Hidden Role of Chance in Life and in the Markets by Nassim Taleb. He wrote the book before the Black Swan and Antifragile, which propelled him into intellectual celebrity. Interestingly, Fooled by Randomness contains semi-explored gems of the ideas that would later go on to become the best-selling books The Black Swan and Antifragile.

Hindsight Bias

Part of the argument that Fooled by Randomness presents is that when we look back at things that have happened we see them as less random than they actually were.

It is as if there were two planets: the one in which we actually live and the one, considerably more deterministic, on which people are convinced we live. It is as simple as that: Past events will always look less random than they were (it is called the hindsight bias). I would listen to someone’s discussion of his own past realizing that much of what he was saying was just backfit explanations concocted ex post by his deluded mind.

The Courage of Montaigne

Writing on Montaigne as the role model for the modern thinker, Taleb also addresses his courage:

It certainly takes bravery to remain skeptical; it takes inordinate courage to introspect, to confront oneself, to accept one’s limitations— scientists are seeing more and more evidence that we are specifically designed by mother nature to fool ourselves.


Fooled by Randomness is about probability, not in a mathematical way but as skepticism.

In this book probability is principally a branch of applied skepticism, not an engineering discipline. …

Probability is not a mere computation of odds on the dice or more complicated variants; it is the acceptance of the lack of certainty in our knowledge and the development of methods for dealing with our ignorance. Outside of textbooks and casinos, probability almost never presents itself as a mathematical problem or a brain teaser. Mother nature does not tell you how many holes there are on the roulette table , nor does she deliver problems in a textbook way (in the real world one has to guess the problem more than the solution).

Outside of textbooks and casinos, probability almost never presents itself as a mathematical problem” which is fascinating given how we tend to solve problems. In decisions under uncertainty, I discussed how risk and uncertainty are different things, which creates two types of ignorance.

Most decisions are not risk-based, they are uncertainty-based and you either know you are ignorant or you have no idea you are ignorant. There is a big distinction between the two. Trust me, you'd rather know you are ignorant.

Randomness Disguised as Non-Randomness

The core of the book is about luck that we understand as skill or “randomness disguised as non-randomness (that is determinism).”

This problem manifests itself most frequently in the lucky fool, “defined as a person who benefited from a disproportionate share of luck but attributes his success to some other, generally very precise, reason.”

Such confusion crops up in the most unexpected areas, even science, though not in such an accentuated and obvious manner as it does in the world of business. It is endemic in politics, as it can be encountered in the shape of a country’s president discoursing on the jobs that “he” created, “his” recovery, and “his predecessor’s” inflation.

These lucky fools are often fragilistas — they have no idea they are lucky fools. For example:

[W]e often have the mistaken impression that a strategy is an excellent strategy, or an entrepreneur a person endowed with “vision,” or a trader a talented trader, only to realize that 99.9% of their past performance is attributable to chance, and chance alone. Ask a profitable investor to explain the reasons for his success; he will offer some deep and convincing interpretation of the results. Frequently, these delusions are intentional and deserve to bear the name “charlatanism.”

This does not mean that all success is luck or randomness. There is a difference between “it is more random than we think” and “it is all random.”

Let me make it clear here : Of course chance favors the prepared! Hard work, showing up on time, wearing a clean (preferably white) shirt, using deodorant, and some such conventional things contribute to success— they are certainly necessary but may be insufficient as they do not cause success. The same applies to the conventional values of persistence, doggedness and perseverance: necessary, very necessary. One needs to go out and buy a lottery ticket in order to win. Does it mean that the work involved in the trip to the store caused the winning? Of course skills count, but they do count less in highly random environments than they do in dentistry.

No, I am not saying that what your grandmother told you about the value of work ethics is wrong! Furthermore, as most successes are caused by very few “windows of opportunity,” failing to grab one can be deadly for one’s career. Take your luck!

That last paragraph connects to something Charlie Munger once said: “Really good investment opportunities aren't going to come along too often and won't last too long, so you've got to be ready to act. Have a prepared mind.

Taleb thinks of success in terms of degrees, so mild success might be explained by skill and labour but outrageous success “is attributable variance.”

Luck Makes You Fragile

One thing Taleb hits on that really stuck with me is that “that which came with the help of luck could be taken away by luck (and often rapidly and unexpectedly at that). The flipside, which deserves to be considered as well (in fact it is even more of our concern), is that things that come with little help from luck are more resistant to randomness.” How Antifragile.

Taleb argues this is the problem of induction, “it does not matter how frequently something succeeds if failure is too costly to bear.”

Noise and Signal

We are confused between noise and signal.

…the literary mind can be intentionally prone to the confusion between noise and meaning, that is, between a randomly constructed arrangement and a precisely intended message. However, this causes little harm; few claim that art is a tool of investigation of the Truth— rather than an attempt to escape it or make it more palatable. Symbolism is the child of our inability and unwillingness to accept randomness; we give meaning to all manner of shapes; we detect human figures in inkblots.

All my life I have suffered the conflict between my love of literature and poetry and my profound allergy to most teachers of literature and “critics.” The French thinker and poet Paul Valery was surprised to listen to a commentary of his poems that found meanings that had until then escaped him (of course, it was pointed out to him that these were intended by his subconscious).

If we're concerned about situations where randomness is confused with non randomness should we also be concerned with situations where non randomness is mistaken for randomness, which would result in signal being ignored?

First, I am not overly worried about the existence of undetected patterns. We have been reading lengthy and complex messages in just about any manifestation of nature that presents jaggedness (such as the palm of a hand, the residues at the bottom of Turkish coffee cups, etc.). Armed with home supercomputers and chained processors, and helped by complexity and “chaos” theories, the scientists, semiscientists, and pseudoscientists will be able to find portents. Second, we need to take into account the costs of mistakes; in my opinion, mistaking the right column for the left one is not as costly as an error in the opposite direction. Even popular opinion warns that bad information is worse than no information at all.

If you haven't yet, pick up a copy of Fooled by Randomness: The Hidden Role of Chance in Life and in the Markets. Don't make the same mistake I did and wait to read this important book.

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Nassim Taleb on the Notion of Alternative Histories

Russian Roulette
“The quality of a decision cannot be solely judged based on its outcome.”

Writing in Fooled by Randomness: The Hidden Role of Chance in Life and in the Markets Nassim Taleb hits on the notion of alternative histories.

We see what's visible and available. Often this is nothing more than randomness and yet we wrap a narrative around it. The trader who is rich must know what he is doing. A good outcome means we made the right decisions, right? Not so quick. If we were wise we would not judge performance in any field by results (outcome bias).

Taleb argues that we should judge people by the costs of the alternative, that is if history played out in another way. These “substitute courses of events are called alternative histories.”

Clearly, the quality of a decision cannot be solely judged based on its outcome, but such a point seems to be voiced only by people who fail (those who succeed attribute their success to the quality of their decision).

One can illustrate the strange concept of alternative histories as follows. Imagine an eccentric (and bored) tycoon offering you $ 10 million to play Russian roulette, i.e., to put a revolver containing one bullet in the six available chambers to your head and pull the trigger. Each realization would count as one history, for a total of six possible histories of equal probabilities. Five out of these six histories would lead to enrichment; one would lead to a statistic, that is, an obituary with an embarrassing (but certainly original) cause of death. The problem is that only one of the histories is observed in reality; and the winner of $ 10 million would elicit the admiration and praise of some fatuous journalist (the very same ones who unconditionally admire the Forbes 500 billionaires). Like almost every executive I have encountered during an eighteen-year career on Wall Street (the role of such executives in my view being no more than a judge of results delivered in a random manner), the public observes the external signs of wealth without even having a glimpse at the source (we call such source the generator.) Consider the possibility that the Russian roulette winner would be used as a role model by his family, friends, and neighbors.

While the remaining five histories are not observable, the wise and thoughtful person could easily make a guess as to their attributes. It requires some thoughtfulness and personal courage. In addition, in time, if the roulette-betting fool keeps playing the game, the bad histories will tend to catch up with him . Thus, if a twenty-five-year-old played Russian roulette, say, once a year, there would be a very slim possibility of his surviving until his fiftieth birthday— but, if there are enough players, say thousands of twenty-five-year-old players, we can expect to see a handful of (extremely rich) survivors (and a very large cemetery).

The reader can see my unusual notion of alternative accounting: $ 10 million earned through Russian roulette does not have the same value as $ 10 million earned through the diligent and artful practice of dentistry. They are the same, can buy the same goods, except that one’s dependence on randomness is greater than the other.

Reality is different from roulette. First in the simplistic example above, while the result is unknown you know the odds, most of life is dealing with uncertainty. Bullets are infrequent, “like a revolver that would have hundreds, even thousands, of chambers instead of six.” After a while you forget about the bullet. You can't see the chamber and we generally think of risk in terms of what is visible.

Interestingly this is the core of the black swan problem, which is really the induction problem. No amount of evidence can allow the inference that something is true whereas one counter example can refute a conclusion. This idea is also related to the “denigration of history,” where we think things that happen to others would not happen to us.

If you haven't yet, pick up a copy of Fooled by Randomness: The Hidden Role of Chance in Life and in the Markets.

(image source)