Antifragile

Awesome book. It alleviated some of my scariest anxieties.

Future anxiety. Before I read it I thought I would miss something new if I’m not going to read about new shiny things every day. Turns out, new developments are not that important. People cannot solve the same problems for centuries, so it’s better to read classics to anticipate future.

Health anxiety. Health is complex. Doctors don’t know much about it and almost always create problems on the long run. Well known example is antibiotics. Also when you think you’re so tired you gonna die you’re not even close to dying. It boosted my productivity.

Before the book I wasn’t sure why I don’t like government, education system and media. Turns out they’re ruled by fragilistas. That’s why those institutions produce so much bullshit.

The book made me realize that life gets much easier and better when you deliberately remove stuff from it. It doesn’t get better if you patch it constantly. Sure, you’re busy upgrading your lifestyle and you even might acquire comfort. But comfort kills people.

It takes time for the antifragility concept to sink in. The author is very repetitive from time to time. That’s why I put 4 out of 5 stars. But it’s an eye-opening book with unique voice. Definitely worth reading.

We have the illusion that the world functions thanks to programmed design, university research, and bureaucratic funding, but there is compelling—very compelling—evidence to show that this is an illusion, the illusion I call lecturing birds how to fly.
In short, the fragilista (medical, economic, social planning) is one who makes you engage in policies and actions, all artificial, in which the benefits are small and visible, and the side effects potentially severe and invisible.
But simplicity is not so simple to attain. Steve Jobs figured out that “you have to work hard to get your thinking clean to make it simple.” The Arabs have an expression for trenchant prose: no skill to understand it, mastery to write it.
It is worth re-explaining the following: the robust or resilient is neither harmed nor helped by volatility and disorder, while the antifragile benefits from them. But it takes some effort for the concept to sink in. A lot of things people call robust or resilient are just robust or resilient, the other half are antifragile.
Hormesis, a word coined by pharmacologists, is when a small dose of a harmful substance is actually beneficial for the organism, acting as medicine.
Like Mithridates, Agrippina eventually died by more mechanical methods as her son (supposedly) had assassins slay her, thus providing us with the small but meaningful lesson that one cannot be robust against everything. And, two thousand years later, nobody has found a method for us to get “fortified” against swords.
Imagine someone gifted in learning languages but unable to transfer concepts from one tongue to another, so he would need to relearn “chair” or “love” or “apple pie” every time he acquires a new language. He would not recognize “house” (English) or “casa” (Spanish) or “byt” (Semitic). We are all, in a way, similarly handicapped, unable to recognize the same idea when it is presented in a different context.
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 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.
Information is antifragile; it feeds more on attempts to harm it than it does on efforts to promote it. For instance, many wreck their reputations merely by trying to defend it.
So here is a simple rule of thumb (a heuristic): to estimate the quality of research, take the caliber of the highest detractor, or the caliber of the lowest detractor whom the author answers in print—whichever is lower.
As my grandfather, his eldest son, was starting his administrative and hopefully political career, his father summoned him to his deathbed. “My son, I am very disappointed in you,” he said. “I never hear anything wrong said about you. You have proven yourself incapable of generating envy.”
The bold conjecture made here is that everything that has life in it is to some extent antifragile (but not the reverse). It looks like the secret of life is antifragility.
For an idea of how harmful a low-level stressor without recovery can be, consider the so-called Chinese water torture: a drop continuously hitting the same spot on your head, never letting you recover.
Further, my characterization of a loser is someone who, after making a mistake, doesn’t introspect, doesn’t exploit it, feels embarrassed and defensive rather than enriched with a new piece of information, and tries to explain why he made the mistake rather than moving on. These types often consider themselves the “victims” of some large plot, a bad boss, or bad weather. He who has never sinned is less reliable than he who has only sinned once. And someone who has made plenty of errors—though never the same error more than once—is more reliable than someone who has never made any.
We saw that antifragility in biology works thanks to layers. This rivalry between suborganisms contributes to evolution: cells within our bodies compete; within the cells, proteins compete, all the way through.
Now, by disrupting the model, as we will see, with bailouts, governments typically favor a certain class of firms that are large enough to require being saved in order to avoid contagion to other business. This is the opposite of healthy risk-taking; it is transferring fragility from the collective to the unfit.
Nietzsche’s famous expression “what does not kill me makes me stronger” can be easily misinterpreted as meaning Mithridatization or hormesis. It may be one of these two phenomena, very possible, but it could as well mean “what did not kill me did not make me stronger, but spared me because I am stronger than others; but it killed others and the average population is now stronger because the weak are gone.” In other words, I passed an exit exam.
Procrustes was an inn-keeper in Greek mythology who, in order to make the travelers fit in his bed, cut the limbs of those who were too tall and stretched those who were too short. But he had the bed fitting the visitor with total perfection.
Like many rich people, they feel entitled to laugh at their own jokes. These (dull) people are not looking for personal shelter: it is their assets that are seeking refuge.
When randomness gets distributed across a large number of small units, along with small recurrent political disorder, we get the first type, the benign Mediocristan. When randomness concentrates, we get the second type, the sneaky Extremistan.
Systematically preventing forest fires from taking place “to be safe” makes the big one much worse. For similar reasons, stability is not good for the economy: firms become very weak during long periods of steady prosperity devoid of setbacks, and hidden vulnerabilities accumulate silently under the surface—so delaying crises is not a very good idea. Likewise, absence of fluctuations in the market causes hidden risks to accumulate with impunity. The longer one goes without a market trauma, the worse the damage when commotion occurs.
Consider the method of annealing in metallurgy, a technique used to make metal stronger and more homogeneous. It involves the heating and controlled cooling of a material, to increase the size of the crystals and reduce their defects. Just as with Buridan’s donkey, the heat causes the atoms to become unstuck from their initial positions and wander randomly through states of higher energy; the cooling gives them more chances of finding new, better configurations.
The famously mistreated Austro-Hungarian doctor Ignaz Semmelweis had observed that more women died giving birth in hospitals than giving birth on the street. He called the establishment doctors a bunch of criminals—which they were: the doctors who kept killing patients could not accept his facts or act on them since he “had no theory” for his observations. Semmelweis entered a state of depression, helpless to stop what he saw as murders, disgusted at the attitude of the establishment. He ended up in an asylum, where he died, ironically, from the same hospital fever he had been warning against.

Field Example of interventionism Iatrogencs/Costs
Medicine, Health Overtreatment
Steady feeding, thermal stability, etc – denying the human body randomness
Prahmaceutical addition, not substraction
Fragility
Medical Error
Sicker (but longer-living) humans, richer pharma, antibiotic-resistant bacteria
Ecology Micromanaging forest fires Worsening total risks – larger “big ones”
Politics Central planning
U.S. supporting rotten regimes “for the sake of stability”
Informational opacity
Chaos after revolution
Economics “No more boom and Bust”
(Greenspan (US), Labor (UK))
Great Moderation (Bernanke)
State interventionism
Optimization
Illusion of pricing rare events, value-at-risk methodologies, illusion of economies of scale, ignorance of second-order effects
Fragility
Deeper crises when they happen
Support for established, state-friendly corporations; stifling of entrepreneurs
Vulnerability, pseudo-efficiency
Big-time blowups
Business Positive advice (charlatans), focus on return not risk (what to avoid) Richer charlatans, bankrupt businesses
Urbanism City planning Urban blight, inner cities, depressions, crime
Forecasing Forecasting in Black Swan Domain (Fourth Quadrant) in spite of the horrible track record Hidden risks (people take more risks when supplied with a forecast)
Literature Copy editors trying to change your text Blander, more New York Times-style commoditized writing
Parenting Soccer mom removing every random element from children’s lives Touristification of children’s minds
Education The entire concept is grounded in interventionism Ludification – transformation of children’s brains
Technology Neomania Fragility, alienation, nerdification
Media High-frequency sterile information Disruption Of the noise/ signal filtering mechanism
Interventionism
Did you ever wonder why heads of state and very rich people with access to all this medical care die just as easily as regular persons? Well, it looks like this is because of overmedication and excessive medical care.
Now let’s add the psychological to this: we are not made to understand the point, so we overreact emotionally to noise. The best solution is to only look at very large changes in data or conditions, never at small ones.
Just as we are not likely to mistake a bear for a stone (but likely to mistake a stone for a bear), it is almost impossible for someone rational, with a clear, uninfected mind, someone who is not drowning in data, to mistake a vital signal, one that matters for his survival, for noise—unless he is overanxious, oversensitive, and neurotic, hence distracted and confused by other messages. Significant signals have a way to reach you. In the tonsillectomies story, the best filter would have been to only consider the children who were very ill, those with periodically recurring throat inflammation.
“All I want is to live in a world in which predictions such as those by Mr. Kato do not harm you. And such a world has unique attributes: robustness.”
From my experiences of the Lebanese war and a couple of storms with power outages in Westchester County, New York, I suggest stocking up on novels, as we tend to underestimate the boredom of these long hours waiting for the trouble to dissipate. And books, being robust, are immune to power outages.
Fat Tony believed that nerds, administrators, and, mostly, bankers were the ultimate suckers (that was when everyone still thought they were geniuses). And, what’s more, he believed that collectively they were even bigger suckers than they were individually.
We start with the following conflict. We introduced Seneca as the wealthiest person in the Roman Empire. His fortune was three hundred million denarii (for a sense of its equivalence, at about the same period in time, Judas got thirty denarii, the equivalent of a month’s salary, to betray Jesus). Admittedly it is certainly not very convincing to read denigrations of material wealth from a fellow writing the lines on one of his several hundred tables (with ivory legs).
An intelligent life is all about such emotional positioning to eliminate the sting of harm, which as we saw is done by mentally writing off belongings so one does not feel any pain from losses. The volatility of the world no longer affects you negatively.
Seen this way, Stoicism is about the domestication, not necessarily the elimination, of emotions. It is not about turning humans into vegetables. My idea of the modern Stoic sage is someone who transforms fear into prudence, pain into information, mistakes into initiation, and desire into undertaking.
Seneca proposes a complete training program to handle life and use emotions properly—thanks to small but effective tricks. One trick, for instance, that a Roman Stoic would use to separate anger from rightful action and avoid committing harm he would regret later would be to wait at least a day before beating up a servant who committed a violation.
As to growth in GDP (gross domestic product), it can be obtained very easily by loading future generations with debt—and the future economy may collapse upon the need to repay such debt. GDP growth, like cholesterol, seems to be a Procrustean bed reduction that has been used to game systems. So just as, for a plane that has a high risk of crashing, the notion of “speed” is irrelevant, since we know it may not get to its destination, economic growth with fragilities is not to be called growth, something that has not yet been understood by governments. Indeed, growth was very modest, less than 1 percent per head, throughout the golden years surrounding the Industrial Revolution, the period that propelled Europe into domination. But as low as it was, it was robust growth—unlike the current fools’ race of states shooting for growth like teenage drivers infatuated with speed.
Evolutionary theorists claim that females want both economic-social stability and good genes for their children. Both cannot be always obtained from someone in the middle with all these virtues (though good gene providers, those alpha males aren’t likely to be stable, and vice versa). Why not have the pie and eat it too? Stable life and good genes. But an alternative theory may be that they just want to have pleasure—or stable life and good fun.
One finds similar ideas in ancestral lore: it is explained in a Yiddish proverb that says “Provide for the worst; the best can take care of itself.” This may sound like a platitude, but it is not: just observe how people tend to provide for the best and hope that the worst will take care of itself.
Never ask people what they want, or where they want to go, or where they think they should go, or, worse, what they think they will desire tomorrow. The strength of the computer entrepreneur Steve Jobs was precisely in distrusting market research and focus groups—those based on asking people what they want—and following his own imagination. His modus was that people don’t know what they want until you provide them with it.
So consider the asymmetry. You benefit from lower rents, but are not hurt by higher ones. How? Because here again, you have an option, not an obligation. In a way, uncertainty increases the worth of such privilege. Should you face a high degree of uncertainty about future outcomes, with possible huge decreases in real estate value, or huge possible increases in them, your option would become more valuable. The moreuncertainty, the more valuable the option.
Beyond books, consider this simple heuristic: your work and ideas, whether in politics, the arts, or other domains, are antifragile if, instead of having one hundred percent of the people finding your mission acceptable or mildly commendable, you are better off having a high percentage of people disliking you and your message (even intensely), combined with a low percentage of extremely loyal and enthusiastic supporters. Options like dispersion of outcomes and don’t care about the average too much.
Harvard’s former president Larry Summers got in trouble (clumsily) explaining a version of the point and lost his job in the aftermath of the uproar. He was trying to say that males and females have equal intelligence, but the male population has more variations and dispersion (hence volatility), with more highly unintelligent men, and more highly intelligent ones. For Summers, this explained why men were overrepresented in the scientific and intellectual community (and also why men were overrepresented in jails or failures). The number of successful scientists depends on the “tails,” the extremes, rather than the average. Just as an option does not care about the adverse outcomes, or an author does not care about the haters.
No one at present dares to state the obvious: growth in society may not come from raising the average the Asian way, but from increasing the number of people in the “tails,” that small, very small number of risk takers crazy enough to have ideas of their own, those endowed with that very rare ability called imagination, that rarer quality called courage, and who make things happen.
To crystallize, take this description of an option: Option = asymmetry + rationality. The rationality part lies in keeping what is good and ditching the bad, knowing to take the profits. As we saw, nature has a filter to keep the good baby and get rid of the bad. The difference between the antifragile and the fragile lies there. The fragile has no option. But the antifragile needs to select what’s best—the best option.
Mathematics → Ornithological navigation and wing-flapping technologies → (ungrateful) birds fly

So we are blind to the possibility of the alternative process, or the role of such a process, a loop:
Random Tinkering (antifragile) → Heuristics (technology) → Practice and Apprenticeship → Random Tinkering (antifragile) → Heuristics (technology) → Practice and Apprenticeship …

In parallel to the above loop:
Practice → Academic Theories → Academic Theories → Academic Theories → Academic Theories … (with of course some exceptions, some accidental leaks, though these are indeed rare and overhyped and grossly generalized).
The lecturing-birds-how-to-fly effect is an example of epiphenomenal belief: we see a high degree of academic research in countries that are wealthy and developed, leading us to think uncritically that research is the generator of wealth. In an epiphenomenon, you don’t usually observe A without observing B with it, so you are likely to think that A causes B, or that B causes A, depending on the cultural framework or what seems plausible to the local journalist.
One rarely has the illusion that, given that so many boys have short hair, short hair determines gender, or that wearing a tie causes one to become a businessman. But it is easy to fall into other epiphenomena, particularly when one is immersed in a news-driven culture.
So let us call the green lumber fallacy the situation in which one mistakes a source of necessary knowledge—the greenness of lumber—for another, less visible from the outside, less tractable, less narratable.
And note the main Fat Tony statement: “Kuwait and oil are not the same ting [thing].” This will be a platform for our notion of conflation. Tony had greater upside than downside, and for him, that was it.
Expert problems (in which the expert knows a lot but less than he thinks he does) often bring fragilities, and acceptance of ignorance the reverse.3 Expert problems put you on the wrong side of asymmetry. Let us examine the point with respect to risk. When you are fragile you need to know a lot more than when you are antifragile. Conversely, when you think you know more than you do, you are fragile (to error).
  • Look for optionality; in fact, rank things according to optionality;
  • preferably with open-ended, not closed-ended, payoffs;
  • Do not invest in business plans but in people, so look for someone capable of changing six or seven times over his career, or more (an idea that is part of the modus operandi of the venture capitalist Marc Andreessen); one gets immunity from the backfit narratives of the business plan by investing in people. It is simply more robust to do so;
  • Make sure you are barbelled, whatever that means in your business.
  • Much of what other people know isn’t worth knowing.
FAT TONY : “The problem, my poor old Greek, is that you are killing the things we can know but not express. And if I asked someone riding a bicycle just fine to give me the theory behind his bicycle riding, he would fall from it. By bullying and questioning people you confuse them and hurt them.”
What is not intelligible to me is not necessarily unintelligent
The payoff, what happens to you (the benefits or harm from it), is always the most important thing, not the event itself.other. You decide principally based on fragility, not probability. Or to rephrase, You decide principally based on fragility, not so much on True/False.
Examples are pills, terrorists on the plane or nuclear reactors

For the fragile, the cumulative effect of small shocks is smaller than the single effect of an equivalent single large shock.

For the antifragile, shocks bring more benefits (equivalently, less harm) as their intensity increases (up to a point).

If the function is convex (antifragile), then the average of the function of something is going to be higher than the function of the average of something. And the reverse when the function is concave (fragile).

  • Assume that the function under question is the squaring function (multiply a number by itself). This is a convex function. Take a conventional die (six sides) and consider a payoff equal to the number it lands on, that is, you get paid a number equivalent to what the die shows —1 if it lands on 1, 2 if it lands on 2, up to 6 if it lands on 6. The square of the expected (average) payoff is then (1+2+3+4+5+6 divided by 6) 2, equals 3.52 , here 12.25. So the function of the average equals 12.25.
  • But the average of the function is as follows. Take the square of every payoff, 1^2 + 2^2 + 3^2 + 4^2 + 5^2 + 6^2 divided by 6, that is, the average square payoff, and you can see that the average of the function equals 15.17. So, since squaring is a convex function, the average of the square payoff is higher than the square of the average payoff. The difference here between 15.17 and 12.25 is what I call the hidden benefit of antifragility—here, a 24 percent “edge.”
I have used all my life a wonderfully simple heuristic: charlatans are recognizable in that they will give you positive advice, and only positive advice, exploiting our gullibility and sucker-proneness for recipes that hit you in a flash as just obvious, then evaporate later as you forget them. Just look at the “how to” books with, in their title, “Ten Steps for—” (fill in: enrichment, weight loss, making friends, innovation, gettingelected, building muscles, finding a husband, running an orphanage, etc.).
I discovered that I had been intuitively using the less-is-more idea as an aid in decision making (contrary to the method of putting a series of pros and cons side by side on a computer screen). For instance, if you have more than one reason to do something (choose a doctor or veterinarian, hire a gardener or an employee, marry a person, go on a trip), just don’t do it. It does not mean that one reason is better than two, just that by invoking more than one reason you are trying to convince yourself to do something. Obvious decisions (robust to error) require no more than a single reason. Likewise the French army had a heuristic to reject excuses for absenteeism for more than one reason, like death of grandmother, cold virus, and being bitten by a boar. If someone attacks a book or idea using more than one argument, you know it is not real: nobody says “he is a criminal, he killed many people, and he also has bad table manners and bad breath and is a very poor driver.”
But technology can cancel the effect of bad technologies, by self-subtraction. Technology is at its best when it is invisible. I am convinced that technology is of greatest benefit when it displaces the deleterious, unnatural, alienating, and, most of all, inherently fragile preceding technology.
For the perishable, every additional day in its life translates into a shorter additional life expectancy. For the nonperishable, every additional day may imply a longer life expectancy.
I received an interesting letter from Paul Doolan from Zurich, who was wondering how we could teach children skills for the twenty-first century since we do not know which skills will be needed in the twenty-first century—he figured out an elegant application of the large problem that Karl Popper called the error of historicism. Effectively my answer would be to make them read the classics. The future is in the past. Actually there is an Arabic proverb to that effect: he who does not have a past has no future.
As we saw in Chapter 7, what physicists call the phenomenology of the process is the empirical manifestation, without looking at how it glues to existing general theories. Take for instance the following statement, entirely evidence-based: if you build muscle, you can eat more without getting more fat deposits in your belly and can gorge on lamb chops without having to buy a new belt. Now in the past the theory to rationalize it was “Your metabolism is higher because muscles burn calories.” Currently I tend to hear “You become more insulin-sensitive and store less fat.” Insulin, shminsulin; metabolism, shmetabolism: another theory will emerge in the future and some other substance will come about, but the exact same effect will continue to prevail.
The same holds for the statement Lifting weights increases your muscle mass. In the past they used to say that weight lifting caused the “micro-tearing of muscles,” with subsequent healing and increase in size. Today some people discuss hormonal signaling or genetic mechanisms, tomorrow they will discuss something else. But the effect has held forever and will continue to do so.
Curative starvatioin sanatoria
A lesson I learned from this ancient culture is the notion of megalopsychon (a term expressed in Aristotle’s ethics), a sense of grandeur that was superseded by the Christian value of “humility.” There is no word for it in Romance languages; in Arabic it is called Shhm—best translated as nonsmall. If you take risks and face your fate with dignity, there is nothing you can do that makes you small; if you don’t take risks, there is nothing you can do that makes you grand, nothing. And when you take risks, insults by half-men (small men, those who don’t risk anything) are similar to barks by nonhuman animals: you can’t feel insulted by a dog.
The psychologist Gerd Gigerenzer has a simple heuristic. Never ask the doctor what you should do. Ask him what he would do if he were in your place. You would be surprised at the difference.
Suckers try to win arguments, nonsuckers try to win.
Have you noticed that while corporations sell you junk drinks, artisans sell you cheese and wine? And there is a transfer of antifragility from the small in favor of the large— until the large goes bust.
The problem of the commercial world is that it only works by addition (via positiva), not subtraction (via negativa): pharmaceutical companies don’t gain if you avoid sugar; the manufacturer of health club machines doesn’t benefit from your deciding to lift stones and walk on rocks (without a cell phone); your stockbroker doesn’t gain from your decision to limit your investments to what you see with your own eyes, say your cousin’s restaurant or an apartment building in your neighborhood; all these firms have to produce “growth in revenues” to satisfy the metric of some slow thinking or, at best, semi-slow thinking MBA analyst sitting in New York. Of course they will eventually self-destruct, but that’s another conversation.
With the exception of, say, drug dealers, small companies and artisans tend to sell us healthy products, ones that seem naturally and spontaneously needed; larger ones—including pharmaceutical giants—are likely to be in the business of producing wholesale iatrogenics, taking our money, and then, to add insult to injury, hijacking the state thanks to their army of lobbyists. Further, anything that requires marketing appears to carry such side effects. You certainly need an advertising apparatus to convince people that Coke brings them “happiness”—and it works.
We accept that people who boast are boastful and turn people off. How about companies? Why aren’t we turned off by companies that advertise how great they are?

We have three layers of violations:

  • First layer, the mild violation: companies are shamelessly self-promotional, like the man on the British Air flight, and it only harms them.
  • Second layer, the more serious violation: companies trying to represent themselves in the most favorable light possible, hiding the defects of their products—still harmless, as we tend to expect it and rely on the opinion of users.
  • Third layer, the even more serious violation: companies trying to misrepresent the product they sell by playing with our cognitive biases, our unconscious associations, and that’s sneaky. The latter is done by, say, showing a poetic picture of a sunset with a cowboy smoking and forcing an associationbetween great romantic moments and some given product that, logically, has no possible connection to it. You seek a romantic moment and what you get is cancer.
  • A (publicly listed) corporation does not feel shame. We humans are restrained by some physical, natural inhibition.
  • A corporation does not feel pity.
  • A corporation does not have a sense of honor—while, alas, marketing documents mention “pride.”
  • A corporation does not have generosity. Only self-serving actions are acceptable. Just imagine what would happen to a corporation that decided to unilaterally cancel itsreceivables—just to be nice. Yet societies function thanks to random acts of generosity between people, even sometimes strangers. All of these defects are the result of the absence of skin in the game, cultural or biological—an asymmetry that harms others for their benefit.
Now self-ownership for our horizontal friend was vastly more democratic than for his thinking predecessors. It simply meant being the owner of your opinion. And it has nothing to do with wealth, birth, intelligence, looks, shoe size, rather with personal courage.
In other words, for Fat Tony, it was a very, very specific definition of a free person: someone who cannot be squeezed into doing something he would otherwise never do.
Consider this leap in sophistication from Athens to Brooklyn: if for the Greeks, only he who is free with his time is free with his opinion, for our horizontal friend and advisor, only he who has courage is free with his opinion. Sissies are born, not made. They stay sissies no matter how much independence you give them, no matter how rich they get.
I have just presented the mechanism of ethical optionality by which people fit their beliefs to actions rather than fit their actions to their beliefs. Table 8 compares professions with respect to such ethical backfitting.

Comparing professions and activities

Invited to be opportunist (Fits ethics to profession) Protected from playing the pseudoethics game
Gold-digger Prostitute
Networker Social person
Compromises Doesn’t compromise
Someone “here to help” Erudite, dilettante, amateur
Merchant, professional (Classical period) Landowner (Classical period)