How Not to Be Wrong : The Power of Mathematical Thinking (9780698163843)

by Ellenberg, Jordan

  But the simplest explanation doesn’t require much theoretical heavy lifting at all. Simply: buying a lottery ticket, whether you win or not, is, in some small way, fun. Not Caribbean vacation fun, not all-night dance party fun, but one or two dollars’ worth of fun? Quite possibly so. There are reasons to doubt this explanation (for instance, lottery players themselves tend to cite the prospect of winning as their primary reason for playing), but it does an admirable job of explaining the behavior we see.

  Economics isn’t like physics and utility isn’t like energy. It is not conserved, and an interaction between two beings can leave both with more utility than they started with. This is the sunny free-marketeer’s view of the lottery. It’s not a regressive tax, it’s a game, where people pay the state a small fee for a few minutes of entertainment the state can provide very cheaply, and the proceeds keep the libraries open and the streetlights on. Just as when two countries trade with each other, both parties to the transaction come out ahead.

  So yes—play Powerball, if Powerball is fun for you. Math gives you permission!

  There are problems with this view, to be sure. Here’s Pascal again, delivering a typically morose take on the excitement of gambling:

  This man spends his life without weariness in playing every day for a small stake. Give him each morning the money he can win each day, on condition he does not play; you make him miserable. It will perhaps be said that he seeks the amusement of play and not the winnings. Make him then play for nothing; he will not become excited over it, and will feel bored. It is then not the amusement alone that he seeks; a languid and passionless amusement will weary him. He must get excited over it, and deceive himself by the fancy that he will be happy to win what he would not have as a gift on condition of not playing.

  —

  Pascal sees the pleasures of gambling as contemptible. And enjoyed to excess, they can of course be harmful. The reasoning that endorses lotteries also suggests that methamphetamine dealers and their clients enjoy a similar win-win relationship. Say what you want about meth, you can’t deny it is broadly and sincerely enjoyed.*

  But what about another comparison? Instead of strung-out tweakers, think about small-business owners, the pride of America. Opening a store or selling a service isn’t the same thing as buying a lottery ticket; you have some measure of control over your success. But the two enterprises have something in common: for most people, opening a business is a bad bet. It doesn’t matter how delicious you believe your barbecue sauce to be, how disruptively innovative you expect your app to be, how ruthless and borderline felonious you intend your business practices to be—you are much more likely to fail than to succeed. That’s the nature of entrepreneurship: you balance a very, very small probability of making a fortune against a modest probability of eking out a living against a substantially larger probability of losing your pile, and for a large proportion of potential entrepreneurs, when you crunch the numbers, the expected financial value, like that of a lottery ticket, is less than zero. Typical entrepreneurs (like typical lottery customers) overrate their chance of success. Even businesses that survive typically make their proprietors less money than they’d have drawn in salary from an existing company. And yet society benefits from a world in which people, against their wiser judgment, launch businesses. We want restaurants, we want barbers, we want smartphone games. Is entrepreneurship “a tax on the stupid”? You’d be called crazy if you said so. Part of that is because we esteem a business owner more highly than we do a gambler; it’s hard to separate our moral feelings about an activity from the judgments we make about its rationality. But part of it—the biggest part—is that the utility of running a business, like the utility of buying a lottery ticket, is not measured only in expected dollars. The very act of realizing a dream, or even trying to realize it, is part of its own reward.

  That, at any rate, is what James Harvey and Yuran Lu decided. After the downfall of WinFall, they moved west and founded a Silicon Valley startup that sells an online chat system for businesses. (Harvey’s profile page coyly lists “non-traditional investment strategies” among his interests.) As I write, they’re still looking for venture capital funding. Maybe they’ll get it. But if not, I’ll bet you’ll quickly find them starting again, expected value or no expected value, hoping the next ticket they try is a winner.

  Includes: Hereditary genius, the curse of the Home Run Derby, arranging elephants in rows and columns, Bertillonage, the invention of the scatterplot, Galton’s ellipse, rich states vote for Democrats but rich people vote for Republicans, “Is it possible, then, that lung cancer is one of the causes of smoking cigarettes?,” why handsome men are such jerks.

  FOURTEEN

  THE TRIUMPH OF MEDIOCRITY

  The early 1930s, like the present period, was a time of soul-searching for the American business community. Something had gone wrong, that much was plain. But what kind of thing? Was the great crash of 1929, and the subsequent depression, an unpredictable catastrophe? Or was the American economy systemically flawed?

  Horace Secrist was as well placed as anyone could be to answer this question. Secrist was a professor of statistics and director of the Bureau for Business Research at Northwestern, an expert in the application of quantitative methods to business, and the author of a widely used statistics textbook for students and business executives. Since 1920, years before the crash, he had been compiling meticulously detailed statistics on hundreds of business concerns, from hardware stores to railroads to banks. Secrist tabulated expenses, total sales, outlays on wages and rent, and every other piece of data he could get, trying to locate and taxonomize the mysterious variations which made some businesses thrive and others falter.

  So in 1933, when Secrist was ready to reveal the results of his analysis, people in both academia and business were inclined to listen. All the more so when he revealed the striking nature of his results in a 468-page volume, thickly marbled with tables and graphs. Secrist pulled no punches: he called his book The Triumph of Mediocrity in Business.

  “Mediocrity tends to prevail in the conduct of competitive business,” Secrist wrote. “This is the conclusion to which this study of the costs (expenses) and profits of thousands of firms unmistakably points. Such is the price which industrial (trade) freedom brings.”

  How did Secrist arrive at such a doomy conclusion? First, he stratified the businesses in each sector, carefully segregating the winners (high income, low expenses) from the inefficient duds. The 120 clothing stores Secrist studied, for instance, were first ranked by ratio of sales to expenses in 1916, then divided into six groups, or “sextiles,” of twenty shops each. Secrist expected to see the shops in the top sextile consolidate their gains over time, growing ever more superior as they honed their already top-of-market skills. What he found was precisely the opposite. By 1922, the clothing stores in the highest sextile had lost most of their advantage over the typical shop; they were still better-than-average stores, but by and large they were no longer the standouts. What’s more, the lowest sextile—the worst stores—experienced the same effect in the opposite direction, improving their performance toward the average. Whatever genius had propelled the top-sextile stores to excel had mostly exhaused itself in a mere six years. Mediocrity had triumphed.

  Secrist found the same phenomenon in every kind of business. Hardware stores regressed to mediocrity; so did grocery stores. And it didn’t matter what metric you used. Secrist tried measuring his companies by the ratio of wages to sales, the ratio of rent to sales, and whatever other economic stat he could put his hands on. It didn’t matter. With time, the top performers started to look and behave just like the members of the common mass.

  Secrist’s book arrived as a bucket of cold water to the face of an already uncomfortable business elite. Many reviewers saw in Secrist’s graphs and tables a numerical disproof of the mythology that sustained entrepreneurship. Robert Riegel of the University of Buffalo wrote, �
��The results confront the business man and the economist with an insistent and to some degree tragic problem. While there are exceptions to the general rule, the conception of an early struggle, crowned with success for the able and efficient, followed by a long period of harvesting the rewards, is thoroughly dissipated.”

  What force was pushing the outliers toward the middle? It had to have something to do with human behavior, because the phenomenon didn’t seem to show up in the natural world. Secrist, with characteristic thoroughness, had carried out a similar test on the average July temperature for 191 U.S. cities. Here there was no regression. The cities that were hottest in 1922 were just as hot in 1931.

  After decades of recording statistics and studying the operation of American business, Secrist thought he knew the answer. It was built into the nature of competition itself to push down successful businesses and promote their incompetent rivals. Secrist wrote:

  Complete freedom to enter trade and the continuance of competition mean the perpetuation of mediocrity. New firms are recruited from the relatively “unfit”—at least from the inexperienced. If some succeed, they must meet the competitive practices of the class, the market, to which they belong. Superior judgment, merchandising sense, and honesty, however, are always at the mercy of the unscrupulous, the unwise, the misinformed and the injudicious. The results are that retail trade is over-crowded, shops are small and inefficient, volume of business inadequate, expenses relatively high, and profits small. So long as the field of activity is freely entered, and it is; and so long as competition is “free,” and within the limits suggested above, it is; neither superiority nor inferiority will tend to persist. Rather, mediocrity tends to become the rule. The average level of the intelligence of those conducting business holds sway, and the practices common to such trade mentality become the rule.

  Can you imagine a business school professor saying something like this today? It’s unthinkable. In modern discourse, free-market competition is the cleansing blade that cuts down the incompetent and the 10%-less-than-maximally-competent alike. Inferior firms are at the mercy of their betters, not the other way around.

  But Secrist saw the free market, with its firms of different sizes and skill levels jostling against each other, as something like the one-room schoolhouse that was, by 1933, already well on its way to disuse. As Secrist describes it: “Pupils of all ages, of different mentality, and of training, grouped together in a single room, were to be educated. Pandemonium, discouragement, and inefficiency, of course, resulted. Common sense later pointed to the desirability of classification, grading, special treatment—corrections which opened the way for native ability to assert itself, and for superiority to withstand being watered down and diluted by inferiority.”

  That last part sounds a little—well, how should I put it—can you think of anybody else in 1933 who was talking about the importance of superior beings withstanding dilution by inferior ones?

  Given the flavor of Secrist’s views on education, it may come as no surprise that his ideas about regression to mediocrity descend from those of the nineteenth-century British scientist and pioneering eugenicist Francis Galton. Galton was the youngest of seven children and a sort of child prodigy. Galton’s bedridden older sister Adèle took on his education as her chief amusement; he could sign his name at two, and by four was writing her letters like this: “I can cast up any sum in addition and can multiply by 2, 3, 4, 5, 6, 7, 8, 10. I can also say the pence table. I read French a little and I know the Clock.” Galton started medical studies at eighteen, but after his father died, leaving him a substantial fortune, he found himself suddenly less motivated to pursue a traditional career. For a while Galton was an explorer, leading expeditions into the African interior. But the epochal publication of The Origin of Species in 1859 catalyzed a drastic shift in his interests: Galton recalls that he “devoured its contents and assimilated them as fast as they were devoured,” and from then on, the greater share of Galton’s work was devoted to the heredity of human characteristics, both physical and mental. This work led him to a suite of policy preferences that are decidedly unsavory from a modern point of view. The opening of his 1869 book Hereditary Genius gives the flavor:

  I propose to show in this book that a man’s natural abilities are derived by inheritance, under exactly the same limitations as are the form and physical features of the whole organic world. Consequently, as it is easy, notwithstanding those limitations, to obtain by careful selection a permanent breed of dogs or horses gifted with peculiar powers of running, or of doing anything else, so it would be quite practicable to produce a highly-gifted race of men by judicious marriages during several consecutive generations.

  Galton made his case by means of a detailed study of British men of achievement, from clerics to wrestlers, arguing that notable Englishmen* tend to have disproportionately notable relatives. Hereditary Genius met with a great deal of resistance, particularly from the clergy; Galton’s purely naturalistic view of worldly success left little room for a more traditional view of Providence. Especially irksome was Galton’s claim that success in ecclesiastical pursuits was itself subject to hereditary influence: that, as one reviewer complained, “a pious man owes his piety not so much (as we had ever believed) to the direct action of the Holy Ghost on his soul, blowing like the wind where it listeth, but rather to his earthly father’s physical bequest of a constitution adapted to the religious emotions.” Whatever friends Galton had among the religious establishment were surely lost three years later, when he published a short article titled “Statistical Inquiries into the Efficacy of Prayer.” (Executive summary: prayer not so efficacious.)

  By contrast, Galton’s book was received with great excitement, if not uncritical acceptance, by the Victorian scientific community. Charles Darwin wrote Galton in a kind of intellectual frenzy, not even waiting until he’d finished the book:

  DOWN, BECKENHAM, KENT, S.E.

  December 23rd

  MY DEAR GALTON,

  —I have only read about 50 pages of your book (to Judges), but I must exhale myself, else something will go wrong in my inside. I do not think I ever in all my life read anything more interesting and original—and how Well and clearly you put every point! George, who has finished the book, and who expressed himself in just the same terms, tells me that the earlier chapters are nothing in interest to the later ones! It will take me some time to get to these latter chapters, as it is read aloud to me by my wife, who is also much interested. You have made a convert of an opponent in one sense, for I have always maintained that, excepting fools, men did not differ much in intellect, only in zeal and hard work; and I still think this is an eminently important difference. I congratulate you on producing what I am convinced will prove a memorable work. I look forward with intense interest to each reading, but it sets me thinking so much that I find it very hard work; but that is wholly the fault of my brain and not of your beautifully clear style.

  —Yours most sincerely,

  (Signed) CH. DARWIN

  To be fair, Darwin might have been biased, being Galton’s first cousin. What’s more, Darwin truly believed that mathematical methods offered scientists an enriched view of the world, even though his own work was far less quantitative than Galton’s. He wrote in his memoirs, reflecting on his early education,

  I attempted mathematics, and even went during the summer of 1828 with a private tutor (a very dull man) to Barmouth, but I got on very slowly. The work was repugnant to me, chiefly from my not being able to see any meaning in the early steps in algebra. This impatience was very foolish, and in after years I have deeply regretted that I did not proceed far enough at least to understand something of the great leading principles of mathematics, for men thus endowed seem to have an extra sense.

  In Galton, Darwin may have felt he was finally seeing the outset of the extrasensory biology he was mathematically unequipped to launch on his own.

  The critics of Hereditary
Genius contended that, while heredity of intellectual tendencies was real, Galton was overstating its strength relative to other factors affecting achievement. So Galton set out to understand the extent to which our parental inheritance determined our fate. But quantifying the heredity of “genius” wasn’t so easy: how, exactly, was one to measure just how notable his notable Englishmen were? Undeterred, Galton turned to human characteristics that could be more easily placed on a numerical scale, like height. As Galton and everyone else already knew, tall parents tend to have tall children. When a six-foot-two man and a five-foot-ten woman get married, their sons and daughters are likely to be taller than average.

  But now here is Galton’s remarkable discovery: those children are not likely to be as tall as their parents. The same goes for short parents, in the opposite direction; their kids will be tend to be short, but not as short as they themselves are. Galton had discovered the phenomenon now called regression to the mean. His data left no doubt that it was real.

  “However paradoxical it may appear at first sight,” Galton wrote in his 1889 book Natural Inheritance, “it is theoretically a necessary fact,* and one that is clearly confirmed by observation, that the Stature of the adult offspring must on the whole, be more mediocre than the stature of their Parents.”