Size Matters

That men would die was a matter of necessity; which men would die, though, was a matter of circumstance, and Yossarian was willing to be the victim of anything but circumstance.
I do not pretend to understand the moral universe; the arc is a long one, my eye reaches but little ways; I cannot calculate the curve and complete the figure by the experience of sight; I can divine it by conscience. And from what I see I am sure it bends towards justice.
Things refuse to be mismanaged long.
—“Of Justice and the Conscience.


The Casino at Monte Carlo



Once, wrote the baseball statistician Bill James, there was “a time when Americans” were such “an honest, trusting people” that they actually had “an unhealthy faith in the validity of statistical evidence”–but by the time James wrote in 1985, things had gone so far the other way that “the intellectually lazy [had] adopted the position that so long as something was stated as a statistic it was probably false.” Today, in no small part because of James’ work, that is likely no longer as true as it once was, but nevertheless the news has not spread to many portions of academia: as University of Virginia historian Sophia Rosenfeld remarked in 2012, in many departments it’s still fairly common to hear it asserted—for example—that all “universal notions are actually forms of ideology,” and that “there is no such thing as universal common sense.” Usually such assertions are followed by a claim for their political utility—but in reality widespread ignorance of statistical effects is what allowed Donald Trump to be elected, because although the media spent much of the presidential campaign focused on questions like the size of Donald Trump’s … hands, the size that actually mattered in determining the election was a statistical concept called sample size.

First mentioned by the mathematician Jacob Bernoulli made in his 1713 book, Ars Conjectandi, sample size is the idea that “it is not enough to take one or another observation for such a reasoning about an event, but that a large number of them are needed.” Admittedly, it might not appear like much of an observation: as Bernoulli himself acknowledged, even “the most stupid person, all by himself and without any preliminary instruction,” knows that “the more such observations are taken into account, the less is the danger of straying from the goal.” But Bernoulli’s remark is the very basis of science: as an article in the journal Nature put the point in 2013, “a study with low statistical power”—that is, few observations—“has a reduced chance of detecting a true effect.” Sample sizes need to be large enough to be able to eliminate chance as a possible factor.

If that isn’t known it’s possible to go seriously astray: consider an example drawn from the work of Israeli psychologists Amos Tversky (MacArthur “genius” grant winner) and (Nobel Prize-winning) Daniel Kahneman—a study “of two toys infants will prefer.” Let’s say that in the course of research our investigator finds that, of “the first five infants studied, four have shown a preference for the same toy.” To most psychologists, the two say, this would be enough for the researcher to conclude that she’s on to something—but in fact, the two write, a “quick computation” shows that “the probability of a result as extreme as the one obtained” being due simply to chance “is as high as 3/8.” The scientist might be inclined to think, in other words, that she has learned something—but in fact her result has a 37.5 percent chance of being due to nothing at all.

Yet when we turn from science to politics, what we find is that an American presidential election is like a study that draws grand conclusions from five babies. Instead of being one big sample—as a direct popular national election would be—presidential elections are broken up into fifty state-level elections: the Electoral College system. What that means is that American presidential elections maximize the role of chance, not minimize it.

The laws of statistics, in other words, predict that chance will play a large role in presidential elections—and as it happens, Tim Meko, Denise Lu and Lazaro Gamio reported for The Washington Post three days after the election that “Trump won the presidency with razor-thin margins in swing states.” “This election was effectively decided,” the trio went on to say, “by 107,000 people”—in an election in which more than 120 million votes were cast, that means that election was decided by less than a tenth of one percent of the total votes. Trump won Pennsylvania by less than 70,000 votes of nearly 6 million, Wisconsin by less than 30,000 of just less than three million, and finally Michigan by less than 11,000 out of 4.5 million: the first two by just more than one percent of the total vote each—and Michigan by a whopping .2 percent! Just to give you an idea of how insignificant these numbers are by comparison with the total vote cast, according to the Michigan Department of Transportation it’s possible that a thousand people in the five largest counties were involved in car crashes—which isn’t even to mention people who just decided to stay home because they couldn’t find a babysitter.

Trump owes his election, in short, to a system that is vulnerable to chance because it is constructed to turn a large sample (the total number of American voters) into small samples (the fifty states). Science tells us that small sample sizes increase the risk of random chance playing a role, American presidential elections use a smaller sample size than they could, and like several other presidential elections, the 2016 election did not go as predicted. Donald Trump could, in other words, be called “His Accidency” with even greater justice than John Tyler—the first vice-president to be promoted due to the death of his boss in office—was. Yet, why isn’t that point being made more publicly?

According to John Cassidy of The New Yorker, it’s because Americans haven’t “been schooled in how to think in probabilistic terms.” But just why that’s true—and he’s essentially making the same point Bill James did in 1985, though more delicately—is, I think, highly damaging to many of Clinton’s biggest fans: the answer is, because they’ve made it that way. It’s the disciplines where many of Clinton’s most vocal supporters make their home, in other words, that are most directly opposed to the type of probabilistic thinking that’s required to see the flaws in the Electoral College system.

As Stanford literary scholar Franco Moretti once observed, the “United States is the country of close reading”: the disciplines dealing with matters of politics, history, and the law within the American system have, in fact, more or less been explicitly constructed to prevent importing knowledge of the laws of chance into them. Law schools, for example, use what’s called the “case method,” in which a single case is used to stand in for an entire body of law: a point indicated by the first textbook to use this method, Christopher Langdell’s A Selection of Cases on the Law of Contracts. Other disciplines, such as history, are similar: as Emory University’s Mark Bauerlein has written, many such disciplines depend for their very livelihood upon “affirming that an incisive reading of a single text or event is sufficient to illustrate a theoretical or historical generality.” In other words, it’s the very basis of the humanities to reject the concept of sample size.

What’s particularly disturbing about this point is that, as Joe Pinsker documented in The Atlantic last year, the humanities attract a wealthier student pool than other disciplines—which is to say that the humanities tend to be populated by students and faculty with a direct interest in maintaining obscurity around the interaction between the laws of chance and the Electoral College. That doesn’t mean that there’s a connection between the architecture of presidential elections and the fact that—as Geoffrey Harpham, former president and director of the National Humanities Center, has observed—“the modern concept of the humanities” (that is, as a set of disciplines distinct from the sciences) “is truly native only to the United States, where the term acquired a meaning and a peculiar cultural force that it does not have elsewhere.” But it does perhaps explain just why many in the national media have been silent regarding that design in the month after the election.

Still, as many in the humanities like to say, it is possible to think that the current American university and political structure is “socially constructed,” or in other words could be constructed differently. The American division between the sciences and the humanities is not the only way to organize knowledge: as the editors of the massive volumes of The Literary and Cultural Reception of Darwin in Europe pointed out in 2014, “one has to bear in mind that the opposition of natural sciences … and humanities … does not apply to the nineteenth century.” If that opposition that we today find so omnipresent wasn’t then, it might not be necessary now. Hence, if the choice of the American people is between whether they ought to get a real say in the affairs of government (and there’s very good reason to think they don’t), or whether a bunch of rich yahoos spend time in their early twenties getting drunk, reading The Great Gatsby, and talking about their terrible childhoods …well, I know which side I’m on. But perhaps more significantly, although I would not expect that it happens tomorrow, still, given the laws of sample size and the prospect of eternity, I know how I’d bet.

Or, as another sharp operator who’d read his Bernoulli once put the point:

The arc of the moral universe is long, but it bends towards justice.”



All Even

George, I am an old man, and most people hate me.
But I don’t like them either so that makes it all even.

—Mr. Potter. It’s A Wonderful Life (1946).



Because someone I love had never seen it, I rewatched Frank Capra’s 1946 It’s A Wonderful Life the other night. To most people, the film is the story of how one George Bailey comes to perceive the value of helping “a few people get outta [the] slums” of the “scurvy little spider” of the film, the wealthy banker Mr. Potter—but to some viewers, what’s important about the inhabitants of Bedford Falls isn’t that they are poor by comparison to Potter, but instead that some of them are black: the man who plays the piano in the background of one scene, for instance, or Annie, the Bailey family’s maid. To Vincent Nobile, a professor of history at Rancho Cucamonga’s Chaffey College, the casting of these supporting roles not only demonstrates that “Capra showed no indication he could perceive blacks in roles outside the servant class,” but also that Potter is the story’s villain not because he is a slumlord, but because he calls the people Bailey helps “garlic eaters” ( What makes Potter evil, in other words, isn’t his “cold monetary self-interest,” but because he’s “bigoted”: to this historian, Capra’s film isn’t the heartwarming story of how Americans banded together to stop a minority (rich people) from wrecking things, but instead the horrifying tragedy of how Americans banded together to stop a minority (black people) from wrecking things. Unfortunately, there’s two problems with that view—problems that can be summarized by referring to the program for a football game that took place five years before the release of Capra’s classic: the Army-Navy game of 29 November, 1941.

Played at Philadelphia’s Franklin Memorial Stadium (once home of the NFL’s Philadelphia Eagles and still the home of the Penn Relays, one of track and field’s premier events), Navy won the contest 14-6; according to Vintage College Football Programs & Collectibles ( [sic]), the program for that game contains 212 pages. On page 180 of that program there is a remarkable photograph. It is of the USS Arizona, the second and last of the American “Pennsylvania” class of super-dreadnought battleships—a ship meant to be, according to the New York Times of 13 July 1913, “the world’s biggest and most powerful, both offensively and defensively, superdreadnought ever constructed.” The last line of the photograph’s caption reads thusly:

It is significant that despite the claims of air enthusiasts, no battleship has yet been sunk by bombs.”

Slightly more than a week later, of course, on a clear bright Sunday morning just after 8:06 Hawaiian time, the hull of the great ship would rest on the bottom of Pearl Harbor, along with the bodies of nearly 1200 of her crew—struck down by the “air enthusiasts” of the Empire of the Sun. The lesson taught that morning, by aircraft directed by former Harvard student Isoroku Yamamoto, was a simple one: that “a saturation attack by huge numbers of low-value attackers”—as Pando Daily’s “War Nerd” columnist, Gary Brecher, has referred to this type of attack—can bring down nearly any target, no matter how powerful ( (A lesson that the U.S. Navy has received more than once: in 2002, for instance, when during the wargame “Millennium Challenge 2002” Marine Corps Lieutenant General Paul K. Riper (fictionally) sent 16 ships to the bottom of the Persian Gulf with the creative use of, essentially, a bunch of cruise missiles and several dozen speedboats loaded with cans of gasoline driven by gentlemen with, shall we say, a cavalier approach to mortality.) It’s the lesson that the cheap and shoddy can overcome quality—or in other words that, as the song says, the bigger they come, the harder they fall.

It’s a lesson that applies to more than merely the physical plane, as the Irish satirist Jonathan Swift knew: “Falsehood flies, and the Truth comes limping after,” the author of Gulliver’s Travels wrote in 1710. What Swift refers to is how saturation attacks can work on the intellectual as well as physical plane—as Emory University’s Mark Bauerlein (who, unfortunately for the warmth of my argument’s reception, endorsed Donald Trump in this past election) argued, in Partisan Review in 2001, American academia has over the past several generations essentially become flooded with the mental equivalents of Al Qaeda speedboats. “Clear-sighted professors,” Bauerlein wrote then, understanding the conditions of academic research, “avoid empirical methods, aware that it takes too much time to verify propositions about culture, to corroborate facts with multiple sources, to consult primary documents, and to compile evidence adequate to inductive conclusions” ( Discussing It’s A Wonderful Life in terms of, say, the economic differences between banks like the one owned by Potter and the savings-and-loan run by George Bailey—and the political consequences therein—is, in other words, hugely expensive in terms of time and effort invested: it’s much more profitable to discuss the film in terms of its hidden racism. By “profitable,” in other words, I mean not merely because it’s intrinsically easier, but also because such a claim is much more likely to upset people, and thus attract attention to its author: the crass stunt once called épater le bourgeois.

The current reward system of the humanities, in other words, favors those philosopher Isaiah Berlin called “foxes” (who know a great many things) rather than “hedgehogs” (who know one important thing). To the present defenders of the humanities, of course, such is the point: that’s the pro-speedboat argument noted feminist literary scholar Jane Tompkins made so long ago as 1981, in her essay “Sentimental Power: Uncle Tom’s Cabin and the Politics of American Literary History.” There, Tompkins suggested that the “political and economic measures”—i.e., the battleships of American political discourse—“that constitute effective action for us” are, in reality, merely “superficial”: instead, what’s necessary are “not specific alterations in the current political and economic arrangements, but rather a change of heart” ( To those who think like Tompkins—or apparently, Nobile—discussing It’s A Wonderful Life in terms of economics is to have missed the point entirely: what matters, according to them, isn’t the dreadnought clash of, for example, the unit banking system of the antebellum North (speedboats) versus the branch banking system of the antebellum South (battleships) within the sea of the American economy. (A contest that, incidentally, not only did branch banking largely win in 1994, during Bill Clinton’s administration, but a victory that in turn—because it helped to create the enormous “too big to fail” interstate banks of today—arguably played no small role in the crash of 2008). Instead, what’s important is the seemingly-minor attack of a community college teacher upon a Titanic of American culture. Or, to put the point in terms popularized by Silicon Valley: the sheer BS quality of Vincent Nobile’s argument about It’s A Wonderful Life isn’t a bug—it’s a feature.

There is, however, one problem with such tactics—the same problem described by Rear Admiral Chuichi (“King Kong”) Hara of the Imperial Japanese Navy after the Japanese surrender in September 1945: “We won a great tactical victory at Pearl Harbor—and thereby lost the war.” Although, as the late American philosopher Richard Rorty commented before his death in his Achieving Our Country: Leftist Thought in Twentieth Century America, “[l]eftists in the academy” have, in collaboration with “the Right,” succeeded in “making cultural issues central to public debate,” that hasn’t necessarily resulted in a victory for leftists, or even liberals ( Indeed, there’s some reason to suppose that, by discouraging certain forms of thought within left-leaning circles, academic leftists in the humanities have obscured what Elizabeth Drew, in the New York Review of Books, has called “unglamorous structural questions” in a fashion ultimately detrimental not merely to minority communities, but ultimately all Americans (

What Drew was referring to this past August was such matters as how—in the wake of the 2010 Census and the redistricting it entailed in every state in the Union—the Democrats ended up, in the 2012 election cycle, winning the popular vote for Congress “by 1.2 per cent, but still remained in the minority, with two hundred and one seats to the G.O.P.’s two hundred and thirty-four.” In other words, Democratic candidates for the House of Representatives got, as Katie Sanders noted in Politifact in 2013, “50.59 percent of the two-party vote” that November, but “won just 46.21 percent of seats”: only “the second time in 70 years that a party won the majority of the vote but didn’t win a majority of the House seats” ( The Republican advantage didn’t end there: as Rob Richie reported for The Nation in 2014, in that year’s congressional races Republicans won “about 52 percent of votes”—but ended “up with 57 percent of seats” ( And this year, the numbers suggest, the Republicans received less than half the popular vote—but will end up with fifty-five percent (241) of the total seats (435). These losses, Drew suggests, are ultimately due to the fact that “the Democrats simply weren’t as interested in such dry and detailed stuff as state legislatures and redistricting”—or, to put it less delicately, because potentially-Democratic schemers have been put to work constructing re-readings of old movies instead of building arguments that are actually politically useful.

To put this even less delicately, many people on the liberal or left-wing side of the political aisle have, for the past several generations, spent their college educations learning, as Mark Bauerlein wrote back in 2001, how to “scoff[…] at empirical notions, chastising them as ‘näive positivism.’” At the same time, a tiny minority among them—those destined to “relax their scruples and select a critical practice that fosters their own professional survival”—have learned, and are learning, to swim the dark seas of academia, taught by their masters how to live by feeding upon the minds of essentially defenseless undergraduates. The lucky ones, like Vince Nobile, manage—by the right mix of bowing and scraping—to land some kind of job security at some far-flung outpost of academia’s empire, where they make a living entertaining the yokels; the less-successful, of course, write deeply ironic blogs.

Be that as it may, while there isn’t necessarily a connection between the humanistic academy’s flight from what Bauerlein calls “the canons of logic” and the fact that it was so easy—as John Cassidy of The New Yorker observed after this past presidential election—for so many in the American media and elsewhere “to dismiss the other outcome [i.e., Trump’s victory] as a live possibility” before the election, Cassidy at least ascribed the ease with which so many predicted a Clinton victory then to the fact that many “haven’t been schooled in how to think in probabilistic terms” ( That lack of education, which extends from the impact of mathematics upon elections to the philosophical basis for holding elections at all (which extends far beyond the usual seventeenth-century suspects rounded up in even the most erudite of college classes to medieval thinkers like Nicholas of Cusa, who argued in 1434’s Catholic Concordance that the “greater the agreement, the more infallible the judgment”—or in other words that speedboats are more trustworthy than battleships), most assuredly has had political consequences ( While the ever-more abstruse academic turf wars between the sciences and the humanities might be good for the ever-dwindling numbers of tenured college professors, in other words, it’s arguably disastrous, not only for Democrats and the populations they serve, but for the country as a whole. Although Clarence, angel second class, says to George Bailey, “we don’t use money in Heaven”—suggesting the way in which American academics swear off knowledge of the sciences upon entering their secular priesthood—George replies, “it comes in real handy down here, bub.” What It’s A Wonderful Life wants to tell us is that a nation whose leadership balances so precariously upon such a narrow educational foundation is, no matter what the program says, as vulnerable as a battleship on a bright Pacific morning.

Or a skyscraper, on a cloudless September one.

Stormy Weather

They can see no reasons …
—“I Don’t Like Mondays” 
The Boomtown Rats.
The Fine Art of Surfacing. 1979.


“Since Tuesday night,” John Cassidy wrote in The New Yorker this week, “there has been a lot of handwringing about how the media, with all its fancy analytics, failed to foresee Donald Trump’s victory”: as the New York Times headline had it, “How Data Failed Us in Calling an Election.” The failure of Nate Silver and other statistical analysts in the lead-up to Election Day rehearses, once again, a seemingly-ancient argument between what are now known as the sciences and the humanities—an argument sometimes held to be as old as the moment when Herodotus (the “Father of History”) asserted that his object in telling the story of the Greco-Persian Wars of 2500 years ago was “to set forth the reasons why [the Greeks and Persians] wage war on each other.” In other words, Herodotus thought that, to investigate war, it was necessary to understand the motives of the people who fought it—just as Cassidy says the failure of the press to get it right about this election was, Cassidy says, “a failure of analysis, rather than of observation.” The argument both Herodotus and Cassidy are making is the seemingly unanswerable one that it is the interpretation of the evidence, rather than the evidence itself, that is significant—a position that seems inarguable so long as you aren’t in the Prussian Army, dodging Nazi bombs during the last year of the Second World War, or living in Malibu.

The reason why it seems inarguable, some might say, is because the argument both Herodotus and Cassidy are making is inescapable: obviously, given Herodotus’ participation, it is a very ancient one, and yet new versions are produced all the time. Consider for instance a debate conducted by English literature professor Michael Bérubé and philosopher John Searle some years ago, about a distinction between what Searle called “brute fact” and “social fact.” “Brute facts,” Bérubé wrote later, are “phenomena like Neptune, DNA, and the cosmic background radiation,” while the second kind are “items whose existence and meaning are obviously dependent entirely on human interpretation,” such as “touchdowns and twenty-dollar bills.” Like Searle, most people might like to say that “brute fact” is clearly more significant than “social fact,” in the sense that Neptune doesn’t care what we think about it, whereas touchdowns and twenty dollar bills are just as surely entirely dependent on what we think of them.

Not so fast, said Bérubé: “there’s a compelling sense,” the professor of literature argued, in which social facts are “prior to and even constitutive of” brute facts—if social facts are the means by which we obtain our knowledge of the outside world, then social facts could “be philosophically prior to and certainly more immediately available to us humans than the world of brute fact.” The only way we know about Neptune is because a number of human beings thought it was important enough to discover; Neptune doesn’t give a damn one way or the other.

“Is the distinction between social facts and brute facts,” Bérubé therefore asks, “a social fact or a brute fact?” (Boom! Mic drop.) That is, whatever the brute facts are, we can only interpret them in the light of social facts—which would seem to grant priority to those disciplines dealing with social facts, rather than those disciplines that deal with brute fact; Hillary Clinton, Bérubé might say, would have been better off hiring an English professor, rather than a statistician, to forecast the election. Yet, despite the smugness with which Bérubé delivers what he believes is a coup de grâce, it does not seem to occur to him that traffic between the two realms can also go the other way: while it may be possible to see how “social facts” subtly influence our ability to see “brute facts,” it’s also possible to see how “brute facts” subtly influence our ability to see “social facts.” It’s merely necessary to understand how the nineteenth-century Prussian Army treated its horses.

The book that treats that question about German military horsemanship is called The Law of Small Numbers, which was published in 1898 by one Ladislaus Bortkiewicz: a Pole who lived in the Russian Empire and yet conducted a study on data about the incidence of deaths caused by horse kicks in the nineteenth-century Prussian Army. Apparently, this was a cause of some concern to military leaders: they wanted to know whether, say, if an army corp that experienced several horse kick deaths in a year—an exceptional number of deaths from this category—was using bad techniques, or whether they happened to buy particularly ornery horses. Why, in short, did some corps have what looked like an epidemic of horse kick deaths in a given year, while others might go for years without a single death? What Bortkiewicz found answered those questions—though perhaps not in a fashion the army brass might have liked.

Bortkiewicz began by assembling data about the number of fatal horse kicks in fourteen Prussian army corps over twenty years, which he then combined into “corp years”: the number of years together with the number of corps. What he found—as E.J. Gumbel pus it in the International Encyclopedia of the Social Sciences—was that for “over half the corps-year combinations there were no deaths from horse kicks,” while “for the other combinations the number of deaths ranged up to four.” In most years, in other words, no one was killed in any given corps by a horse kick, while in some years someone was—and in terrible years four were. Deaths by horse kick, then, were uncommon, which meant they were hard to study: given that they happened so rarely, it was difficult to determine what caused them—which was why Bortkiewicz had to assemble so much data about them. By doing so, the Russian Pole hoped to be able to isolate a common factor among these deaths.

In the course of studying these deaths, Bortkiewicz ended up independently re-discovering something that a French mathematician, Simeon Denis Poisson, had already, in 1837, used in connection with discussing the verdicts of juries: an arrangement of data now known as the Poisson distribution. And as the mathematics department at the University of Massachusetts is happy to tell us (, the Poisson distribution applies when four conditions are met: first, “the event is something that can be counted in whole numbers”; second, “occurrences are independent, so that one occurrence neither diminishes nor increases the chance of another”; third, “the average frequency of occurrence for the time period in question is known”; and finally “it is possible to count how many events have occurred.” If these things are known, it seems, the Poisson distribution will tell you how often the event in question will happen in the future—a pretty useful feature for, say, predicting the results of an election. But that what wasn’t was intriguing about Bortkiewicz’ study: what made it important enough to outlast the government that commissioned it was that Bortkiewicz found that the Poisson distribution “may be used in reverse”—a discovery ended up telling us about far more than the care of Prussian horses.

What “Bortkiewicz realized,” as Aatish Bhatia of Wired wrote some years ago, was “that he could use Poisson’s formula to work out how many deaths you could expect to see” if the deaths from horse kicks in the Prussian army were random. The key to the Poisson distribution, in other words, is the second component, “occurrences are independent, so that one occurrence neither diminishes nor increases the chance of another”: a Poisson distribution describes processes that are like the flip of a coin. As everyone knows, each flip of a coin is independent of the one that came before; hence, the record of successive flips is the record of a random process—a process that will leave its mark, Bortkiewicz understood.

A Poisson distribution maps a random process; therefore, if the process in question maps a Poisson distribution, then it must be a random process. A distribution that matches the results a Poisson distribution would predict must also be a process in which each occurrence is independent of those that came before. As the UMass mathematicians say, “if the data are lumpy, we look for what might be causing the lump,” while conversely, if  “the data fit the Poisson expectation closely, then there is no strong reason to believe that something other than random occurrence is at work.” Anything that follows a Poisson distribution is likely the result of a random process; hence, what Bortkiewicz had discovered was a tool to find randomness.

Take, for example, the case of German V-2 rocket attacks on London during the last years of World War II—the background, as it happens, to novelist Thomas Pynchon’s Gravity’s Rainbow. As Pynchon’s book relates, the flying missiles were falling in a pattern: some parts of London were hit multiple times, while others were spared. Some Londoners argued that this “clustering” demonstrated that the Germans must have discovered a way to guide these missiles—something that would have been highly, highly advanced for mid-twentieth century technology. (Even today, guided missiles are incredibly advanced: much less than ten percent of all the bombs dropped during the 1991 Gulf War, for instance, had “smart bomb” technology.) So what British scientist R. D. Clarke did was to look at the data for all the targets of V-2s that fell on London. What he found was that the results matched a Poisson distribution—the Germans did not possess super-advanced guidance systems. They were just lucky.

Daniel Kahneman, the Israeli psychologist, has a similar story: “‘During the Yom Kippur War, in 1973,’” Kahneman told New Yorker writer Atul Gawande, he was approached by the Israeli Air Force to investigate why, of two squads that took to the skies during the war, “‘one had lost four planes and the other had lost none.’” Kahneman told them not to waste their time, because a “difference of four lost planes could easily have occurred by chance.” Without knowing about Bortkiewicz, that is, the Israeli Air Force “would inevitably find some measurable differences between the squadrons and feel compelled to act on them”—differences that, in reality, mattered not at all. Presumably, Israel’s opponents were bound to hit some of Israel’s warplanes; it just so happened that they were clustered in one squadron and not the other.

Why though, should any of this matter in terms of the distinction between “brute” and “social” facts? Well, consider what Herodotus wrote more than two millennia ago: what matters, when studying war, is the reasons people had for fighting. After all, wars are some of the best examples of a “social fact” anywhere: wars only exist, Herodotus is claiming, because of what people think about them. But what if it could be shown that, actually, there’s a good case to be made for thinking of war as a “brute fact”—something more like DNA or Neptune than like money or a home run? As it happens, at least one person, following in Bortkiewicz’ footsteps, already has.

In November of 1941, the British meteorologist and statistician Lewis Fry Richardson published, in the journal Nature, a curious article entitled “Frequency of Occurrence of Wars and Other Quarrels.” Richardson, it seems, had had enough of the endless theorizing about war’s causes: whether it be due to, say, simple material greed, or religion, or differences between various cultures or races. (Take for instance the American Civil War: according to some Southerners, the war could be ascribed to the racial differences between Southern “Celtics” versus Northern “Anglo-Saxons”; according to William Seward, Abraham Lincoln’s Secretary of State, the war was due to the differences in economic systems between the two regions—while to Lincoln himself, perhaps characteristically, it was all due to slavery.) Rather than argue with the historians, Richardson decided to instead gather data: he compiled a list of real wars going back centuries, then attempted to analyze the data he had collected.

What Richardson found was, to say the least, highly damaging to Herodotus: as Brian Hayes puts it in a recent article in American Scientist about Richardson’s work, when Richardson compared a group of wars with similar amounts of casualties to a Poisson distribution, he found that the “match is very close.” The British scientist also “performed a similar analysis of the dates on which wars ended—the ‘outbreaks of peace’—with the same result.” Finally, he checked another data set concerning wars, this one compiled by the University of Chicago’s Quincy Wright—“and again found good agreement.” “Thus,” Hayes writes, “the data offer no reason to believe that wars are anything other than randomly distributed accidents.” Although Herodotus argued that the only way to study wars is to study the motivations of those who fought them, there may in reality be no more “reason” for the existence of war than the existence of a forest fire in Southern California.

Herodotus, to be sure, could not have seen that: the mathematics of his time were nowhere near sophisticated enough to run a Poisson distribution. Therefore, the Greek historian was eminently justified in thinking that wars must have “reasons”: he literally did not have the conceptual tools necessary to think that wars may not have reasons at all. That was an unavailable option. But through the work of Bortkiewizc and his successors, that has now become an option: indeed, the innovation of these statisticians has been to show that our default assumption ought to be what statisticians call the “null hypothesis,” which is defined by the Cambridge Dictionary of Statistics to be “the ‘no difference’ or ‘no association’ hypothesis.” Unlike Herodotus, who presumed that explanations must equal causes, we now assume that we ought to be first sure that there is anything to explain before trying to explain it.

In this case, then, it may be that the “brute fact” of the press’ Herodotian commitment to discovering “reasons” that explains why nobody in the public sphere predicted Donald Trump’s victory: because the press is already committed to the supremacy of analysis over observation, it could not perform the observations necessary to think Trump could win. Or, as Cassidy put it, when a reporter saw the statistical election model of choice “registering the chances of the election going a certain way at ninety per cent, or ninety-five per cent, it’s easy to dismiss the other outcome as a live possibility—particularly if you haven’t been schooled in how to think in probabilistic terms, which many people haven’t.” Just how powerful the assumption of the force of analysis over data can be is demonstrated by the fact that—even despite noting the widespread lack of probabilistic thinking—Cassidy still thinks it possible that “F.B.I. Director James Comey’s intervention ten days before the election,” in which Comey announced his staff was still investigating Clinton’s emails, “may have proved decisive.” In other words, despite knowing something about the impact of probability, Cassidy still thinks it possible that a letter from the F.B.I. director was somehow more important to the outcome of this past election than the evidence of their own lives were to million of Americans—or, say, the effect of a system in which the answer to the question where outweighs that of how many?

Probabilistic reasoning, of course, was unavailable to Herodotus, who lived two millennia before the mathematical tools necessary were even invented—which is to say that, while some like to claim that the war between interpretation and data is eternal, it might not be. Yet John Cassidy—and Michael Bérubé—don’t live before those tools were invented, and yet they persist in writing as if they do. While that’s fine, so far as it is their choice as private citizens, it ought to be quite a different thing insofar as it is their jobs as journalist and teacher, respectively—particularly in the case, as say in the 2016 election, when it is of importance to the continued health of the nation as a whole that there be a clear public understanding of events. Some people appear to think that continuing the quarrels of people whose habits of mind, today, would barely qualify them to teach Sunday school is something noble; in reality, it may just be a measure of how far we have yet to travel.


I Think I’m Gonna Be Sad

In all Republics the voice of a majority must prevail.
—Andrew Jackson.

I know no safe depository of the ultimate powers of the society but the people themselves, and if we think them not enlightened enough to exercise that control with a wholesome discretion, the remedy is not to take control from them, but to inform their discretion.
—Thomas Jefferson. “Letter to William Charles Jarvis.” 28 September, 1820



When the Beatles first came to America, in February of 1964—Michael Tomasky noted recently for The Daily Beast—they rode from their gig at Ed Sullivan’s show in New York City to their first American concert in Washington, D.C. by train, arriving two hours and fifteen minutes after leaving Manhattan. It’s a seemingly trivial detail—until it’s pointed out, as Tomasky realized, that anyone trying that trip today would be lucky to do it in three hours. American infrastructure in short is not what it was: as the American Society of Civil Engineers wrote in 2009’s Report Card for American Infrastructure, “years of delayed maintenance and lack of modernization have left Americans with an outdated and failing infrastructure that cannot meet our needs.” But what to do about it? “What’s needed,” wrote John Cassidy, of The New Yorker, recently, “is some way to protect essential infrastructure investments from the vicissitudes of congressional politics and the cyclical ups and downs of the economy.” He suggests, instead, “an independent, nonpartisan board” that could “carry out cost-benefit analyses of future capital-spending proposals.” This board, presumably, would be composed of professionals above the partisan fray, and thus capable of seeing to the long-term needs of the country. It all sounds really jake, and just the thing that the United States ought to do—excepting only for the disappointing fact that the United States already has just such a board, and the existence of that “board” is the very reason why Americans don’t invest in infrastructure.

First though—has national spending on infrastructure declined, and is “politics” the reason for that decline? Many think so: “Despite the pressing infrastructure investment needs of the United States,” businessman Scott Thomasson wrote for the Council on Foreign Relations recently, “federal infrastructure policy is paralyzed by partisan wrangling over massive infrastructure bills that fail to move through Congress.” Those who take that line do have evidence, at least for the first proposition.

Take for instance the Highway Trust Fund, an account that provides federal money for investments in roads and bridges. In 2014, the Fund was in danger of “drying up,” as Rebecca Kaplan reported for CBS News at the time, mostly because the federal gas tax of 18.4 cents per gallon hasn’t been increased since 1993. Gradually, then, both the federal government and the states have, in relative terms, decreased spending on highways and other projects of that sort—so much so that people like former presidential economic advisor and president of Harvard University, Lawrence Summers, say (as Summers did last year) that “the share of public investment [in infrastructure], adjusting for depreciation … is zero.” (That is, spending on infrastructure is effectively less than the rate of inflation—which itself is pretty low.) So, while the testimony of the American Society of Civil Engineers might, to say the least, be biased—asking an engineer whether there ought to be more spending on engineering is like asking an ice cream man whether you need a sundae—there’s a good deal of evidence that the United States could stand more investment in the structures that support American life.

Yet, even if that’s so, is the relative decline in spending really the result of politics—rather than, say, a recognition that the United States simply doesn’t need the same sort of spending on highways and railroads that it once did? Maybe—because “the Internet,” or something—there simply isn’t the need for so much physical building any more. Still, aside from such spectacular examples as the Minneapolis Interstate 35 bridge collapse in 2007 or the failure of the levees in New Orleans during Hurricane Katrina in 2005, there’s evidence that the United States would be spending more money on infrastructure under a different political architecture.

Consider, for example, how the U.S. Senate “shot down … a measure to spend $50 billion on highway, rail, transit and airport improvements” in November of 2011, as The Washington Post’s Rosalind S. Helderman reported at the time. Although the measure was supported by 51 votes in favor to 49 votes against, the measure failed to pass—because, as Helderman wrote, according to the rules of the Senate “the measure needed 60 votes to proceed to a full debate.” Passing bills in the Senate these days requires, it seems, more than majority support—which, near as I can make out, is just what is meant by “congressional gridlock.” What “gridlock” means is the inability of a majority to pass its programs—absent that inability, nearly certainly the United States would be spending more money on infrastructure. At this point, then, the question can be asked: why should the American government be built in a fashion that allows a minority to hold the majority for ransom?

The answer, it seems, might be deflating for John Cassidy’s idea: when the American Constitution was written, it inscribed into its very foundation what has been called (by The Economist, among many, many others) the “dream of bipartisanship”—the notion that, somewhere, there exists a group of very wise men (and perhaps women?) who can, if they were merely handed the power, make all the world right again, and make whole that which is broken. In America, the name of that body is the United States Senate.

As every schoolchild knows, the Senate was originally designed as a body of “notables,” or “wise men”: as the Senate’s own website puts it, the Senate was originally designed to be an “independent body of responsible citizens.” Or, as James Madison wrote to another “Founding Father,” Edmund Randolph, justifying the institution, the Senate’s role was “first to protect the people against their rulers [and] secondly to protect the people against transient impressions into which they themselves might be led.” That last justification may be the source of the famous anecdote regarding the Senate, which involves George Washington saying to Thomas Jefferson that “we pour our legislation into the senatorial saucer to cool it.” While the anecdote itself only appeared nearly a century later, in 1872, still it captures something of what the point of the Senate has always been held to be: a body that would rise above petty politicking and concern itself with the national interest—just the thing that John Cassidy recommends for our current predicament.

This “dream of bipartisanship,” as it happens, is not just one held by the founding generation. It’s a dream that, journalist and gadfly Thomas Frank has said, “is a very typical way of thinking for the professional class” of today. As Frank amplified his remarks, “Washington is a city of professionals with advanced degrees,” and the thought of those professionals is “‘[w]e know what the problems are and we know what the answers are, and politics just get in the way.’” To members of this class, Frank says, “politics is this ugly thing that you don’t really need.” For such people, in other words, John Cassidy’s proposal concerning an “independent, nonpartisan board” that could make decisions regarding infrastructure in the interests of the nation as a whole, rather than from the perspective of this or that group, might seem entirely “natural”—as the only way out of the impasse created by “political gridlock.” Yet in reality—as numerous historians have documented—it’s in fact precisely the “dream of bipartisanship” that created the gridlock in the first place.

An examination of history in other words demonstrates that—far from being the disinterested, neutral body that would look deep into the future to examine the nation’s infrastructure needs—the Senate has actually functioned to discourage infrastructure spending. After John Quincy Adams was elected president in the contested election of 1824, for example, the new leader proposed a sweeping program of investment in roads and canals and bridges, but also a national university, subsidies for scientific research and learning, a national observatory, Western exploration, a naval academy, and a patent law to encourage invention. Yet, as Paul C. Nagel observes in his recent biography of the Massachusetts president, virtually none of Adams’ program was enacted: “All of Adams’ scientific and educational proposals were defeated, as were his efforts to enlarge the road and canal systems.” Which is true, so far as that goes. But Nagel’s somewhat bland remarks do not do justice to the matter of how Adams’ proposals were defeated.

After the election of 1824, which elected the 19th Congress, Adams’ party had a majority in the House of Representatives—one reason why Adams became president at all, because the chaotic election of 1824, split between three major candidates, was decided (as per the Constitution) by the House of Representatives. But while Adams’ faction had a majority in the House, they did not in the Senate, where Andrew Jackson’s pro-Southern faction held sway. Throughout the 19th Congress, the Jacksonian party controlled the votes of 25 Senators (in a Senate of 48 senators, two to a state) while Adams’ faction controlled, at the beginning of the Congress, 20. Given the structure of the U.S. Constitution, which requires agreement between the two houses of Congress as the national legislature before bills can become law, this meant that the Senate could—as it did—effectively veto any of the Adams’ party’s proposals: control of the Senate effectively meant control of the government itself. In short, a recipe for gridlock.

The point of the history lesson regarding the 19th Congress is that, far from being “above” politics as it was advertised to be in the pages of The Federalist Papers and other, more recent, accounts of the U.S. Constitution, the U.S. Senate proved, in the event, hardly to be more neutral than the House of Representatives—or even the average city council. Instead of considering the matter of investment in the future on its own terms, historians have argued, senators thought about Adams’ proposals in terms of how they would affect a matter seemingly remote from the matters of building bridges or canals. Hence, although senators like John Tyler of Virginia, for example—who would later be elected president himself—opposed Adams-proposed “bills that mandated federal spending for improving roads and bridges and other infrastructure” on the grounds that such bills “were federal intrusions on the states” (as Roger Matuz put it in his The Presidents’ Fact Book), many today argue that their motives were not so high-minded. In fact, they were actually as venial as any motive could be.

Many of Adams’ opponents, that is—as William Lee Miller of the University of Virginia wrote in his Arguing About Slavery: John Quincy Adams and the Great Battle in the United States Congress—thought that the “‘National’ program that [Adams] proposed would have enlarged federal powers in a way that might one day threaten slavery.” And, as Miller also remarks, the “‘strict construction’ of the Constitution and states’ rights that [Adams’] opponents insisted upon”— were, “in addition to whatever other foundations in sentiment and philosophy they had, barriers of protection against interference with slavery.” In short—as historian Harold M. Hyman remarked in his magisterial A More Perfect Union: The Impact of the Civil War and Reconstruction on the Constitution—while the “constitutional notion that tight limits existed on what government could do was a runaway favorite” at the time, in reality these seemingly-resounding defenses of limited government were actually motivated by a less-than savory interest: “statesmen of the Old South,” Hyman wrote, found that these doctrines of constitutional limits were “a mighty fortress behind which to shelter slavery.” Senators, in other words, did not consider whether spending money on a national university would be a worthwhile investment for its own sake; instead, they worried about the effect that such an expenditure would have on slavery.

Now, it could still reasonably be objected at this point—and doubtless will be—that the 19th Congress is, in political terms, about as relevant to today’s politics as the Triassic: the debates between a few dozen, usually elderly, white men nearly two centuries ago have been rendered impotent by the passage of time. “This time, it’s different,” such arguments could, and probably will, say. Yet, at a different point in American history, it was well-understood that the creation of such “blue-ribbon” committees or the like—such as the Senate—were in fact simply a means for elite control.

As Alice Sturgis, of Stanford University, wrote in the third edition of her The Standard Code of Parliamentary Procedure (now in its fourth edition, after decades in print, and still the paragon of the field), while some “parliamentary writers have mistakenly assumed that the higher the vote required to take an action, the greater the protection of the members,” in reality “the opposite is true.” “If a two-thirds vote is required to pass a proposal and sixty-five members vote for the proposal and thirty-five members vote against it,” Sturgis went on to write, “the thirty-five members make the decision”—which then makes for “minority, not majority, rule.” In other words, even if many circumstances in American life have changed since 1825, it still remains the case that the American government is (still) largely structured in a fashion that solidifies the ability of a minority—like, say, oligarchical slaveowners—to control the American government. And while slavery was abolished by the Civil War, it still remains the case that a minority can block things like infrastructure spending.

Hence, since infrastructure spending is—nearly by definition—for the improvement of every American, it’s difficult to see how making infrastructure spending less democratic, as Cassidy wishes, would make it easier to spend money on infrastructure. We already have a system that’s not very democratic—arguably, that’s the reason why we aren’t spending money on infrastructure, not because (as pundits like Cassidy might have it), “Washington” has “gotten too political.” The problem with American spending on infrastructure, in sum, is not that it is political. In fact, it is precisely the opposite: it isn’t political enough. That people like John Cassidy—who, by the way, is a transplanted former subject of the Queen of England—think the contrary is itself, I’d wager, reason enough to give him, and people like him, what the boys from Liverpool called a ticket to ride.

Lions For Lambs

And the remnant of Jacob shall be among the Gentiles in the midst of many people as a lion among the beasts of the forest, as a young lion among the flocks of sheep …
Micah 5:8

Micah was the first prophet to predict the downfall of Jerusalem. According to him, the city was doomed because its beautification was financed by dishonest business practices, which impoverished the city’s citizens. He also called to account the prophets of his day, whom he accused of accepting money for their oracles.
“Micah.” Wikipedia.


“Before long I’ll be dead, and you and your brother and your sister and all of her children, all of us dead, all of us rotting underground,” says the villainous patriarch of the aristocratic Lannister clan, Tywin, to his son Jaime in a conversation during the first season of the hit HBO show, Game of Thrones. “It’s the family name that lives on,” Tywin continues—a sentence that not only does much to explain the popularity of the show, but also overturns the usual explanation for that interest: the narrative uncertainty, or the way in which, at least in the first several seasons, it was never obvious which characters were the heroes, and so would survive to the end of the tale. But if Tywin is right, the attraction of the show isn’t that it is so unpredictable. It’s rather that the show’s uncertainty about the various characters’ fates is balanced by a matching certainty that they are in peril: either from the political machinations that end up destroying many of the characters the show had led us to think were protagonists (Ned and his son Robb Stark in particular)—or from the horror that, the opening minutes of the show’s very first episode display, has awakened in the frozen north of Thrones’ fictional world. Hence, the uncertainty about what is going to happen is mirrored by a certainty that something will happen—a certainty signified by the motto of the family to which many fan-favorite characters belong, House Stark: “Winter is Coming.” It’s that motto, I think, that furnishes much of the show’s power—because it is such a direct riposte to much of today’s conventional wisdom, a dogma that unites the supposed “radical left” of the contemporary university with their seeming ideological opposites: the financial elite of Wall Street.

To put it plainly, the relevant division in America today is not between Republicans and Democrats, but instead between those who (still) think the notion encapsulated by the phrase “Winter Is Coming” matters—and those who don’t. For the idea contained within the phrase “Winter Is Coming,” after all, is much older than George Martin’s series of fantasy novels. It is, for example, much the same as an idea expressed by the English writer George Orwell, author of 1984 and Animal Farm, in 1946:

… we are all capable of believing things which we know to be untrue, and then, when we are finally proved wrong, impudently twisting the facts so as to show that we were right. Intellectually, it is possible to carry on this process for an indefinite time: the only check on it is that sooner or later a false belief bumps up against solid reality, usually on a battlefield.

What Orwell expresses here, I’d say, is the Stark idea—the idea that, sooner or later, one’s beliefs run up against reality, whether that reality comes in the form of the weather or war or something else. It’s the notion that, sooner or later, things converge towards reality: a notion that many contemporary intellectuals have abandoned. To them, the view expressed by Orwell and the Starks is what’s known as “foundationalism”: something that all recent students in the humanities have been trained, over the past several generations, to boo and hiss.

“Foundationalism,” according to Pennsylvania State University literature professor Michael Bérubé, for example—a person I often refer to because, unlike the work of a lot others, he at least expresses what he’s saying clearly, and also because he represents a university well-known for its commitment to openness and transparency and occasionally less-than-enthusiastic opposition to child abuse—is the notion that there is a “principle that is independent of all human minds.” That is opposed, for people who think about this sort of thing, to “antifoundationalism”: the idea that a lot of stuff (maybe everything) is simply a matter of “human deliberation and consensus.” Also known as “social constructionism,” it’s an idea that Orwell, or the Starks, would have looked at slant-eyed: winter, for instance, doesn’t particularly care what people think about it, and while war is like both a seminar and a hurricane, the things that happen in war—like, say, having the technology to turn an entire city into a fireball—are not appreciably different from the impact of a tsunami.

Within the humanities however the “anti-foundationalist” or “social constructionist” idea has largely taken the field. “Notwithstanding,” as literature professor Mark Bauerlein of Emory University has remarked, “the diversity trumpeted by humanities departments these days, when it comes to conceptions of knowledge, one standpoint reigns supreme: social constructionism.” To those who hold it, it is a belief that straightforwardly powers what Bauerlein calls “a moral obligation to social justice”: in this view, either you are on the side of antifoundationalism, or you are a yahoo who thinks that the problem with the world is that there isn’t enough Donald Trump in it. Yet antifoundationalism, or the idea that everything is a matter of human discussion, is not necessarily so obviously on the side of good and not evil as the professors of the nation’s universities appear to believe.

In fact, while Bauerlein says that this dogma is “a party line, a tribal glue distinguishing humanities professors from their colleagues in the business school, the laboratory, the chapel, and the computing center, most of whom believe that at least some knowledge is independent of social conditions,” there’s actually good reason to think that a disbelief in an underlying reality isn’t all that unfamiliar to the business school. Arguably, there’s no portion of the university that pays more homage to the dogma of “social construction” than the business school.

Take, for instance, the idea Eugene Fama has built his career upon: the “random walk” theory of the stock market, also known as the “efficient market hypothesis.” Today, Fama is a Nobel Prize-laureate (well, winner of the Swedish National Bank’s Prize in Economic Sciences in Memory of Alfred Nobel, a prize not established by Alfred Nobel in his 1895 will), a professor at the University of Chicago’s Booth School of Business, and the so-called “Father of Finance, ” but in 1965 he was an obscure graduate student—at least, until he wrote the paper that established him within his profession that year, “The Behavior of Stock-Market Prices.” In that paper, Fama argued that “the future path of the price level of a security is no more predictable than the path of a series of cumulated random numbers,” which had the consequence that “the series of price changes has no memory.” (Which is what stock prospectuses mean when they say that “past performance cannot predict future performance.”) What Fama meant was that, no matter how many times he went back over the data, he could find no means by which to predict the future path of a particular stock. Hence he concluded that, when it comes to the market, “the past cannot be used to predict the future in any meaningful way”—an idea with some notably anti-foundationalist consequences.

Those consequences can be be viewed in such papers as Fama’s 2010 study with colleague Kenneth French: “Luck versus Skill in the Cross-Section of Mutual Fund Returns”—a study that set out to examine whether it was true that the managers of mutual funds can actually do what they claim they can do, and outperform the stock market. In “Luck versus Skill,” Fama and French say that the evidence shows those managers can’t: “For fund investors the … results are disheartening,” because “few active funds produce … returns that cover their costs.” Maybe there are really intelligent people out there who are smarter than the market, Fama is suggesting—but if there are, he can’t find them.

Now, so far Fama’s idea might sound pretty unexceptional: to readers of this blog, it might even sound like common sense. It’s a fairly close idea to the one explored, for instance, by psychologist Amos Tversky and his co-authors in the paper, “The Hot Hand in Basketball,” which was about how what appeared to be a “hot,” or “clutch,” basketball shooter was simply an effect of randomness: if your skill level is such that you expect to make a certain percentage of your shots, then—simply through the laws of probability—it is likely that you will make a certain number of baskets in a row. Similarly, if there are enough mutual funds in the market, some number of them will have gaudy track records to report: “Given the multitude of funds,” as Fama writes, “many have extreme returns by chance.” If there’s enough participants in any competition, some will be winners—or to put it another way, if a monkey throws enough shit at a wall, some of it will stick.

That, Fama might say, doesn’t mean that the monkey has somehow gotten in touch with Reality: if no one person can outperform the market, then there is nothing anyone can know that would help them to become a better stock-picker. What that must mean in turn is (as the Wikipedia article on the subject notes) that “market prices reflect all available information,” or that “stocks always trade at their fair value”—which is right about where that the work of seemingly-conservative professors in economics departments and business schools, and their seeming-liberal opponents in departments of the humanities begins to converge.

Fama, after all, denies the existence of what are known as “bubbles”: “speculative bubbles, market bubbles, price bubbles, financial bubbles, speculative manias or balloons” as Wikipedia terms them. “Bubbles” describe situations in which a given asset—like, I don’t know, a house—is traded “at a price or price range that strongly deviates from the corresponding asset’s intrinsic value.” The classic example is the Dutch tulip craze of the seventeenth century, during which a single tulip bulb might have sold for ten times the yearly wage of a workman. (Other instances might be closer to the reader’s mind than that.) But according to Fama there can be no such thing as a “bubble”: when John Cassidy of The New Yorker said to Fama in an interview that the chief problem during the financial crisis of 2008 was that “there was a credit bubble that inflated and ultimately burst,” Fama replied by saying, “I don’t know what a credit bubble means. I don’t even know what a bubble means. These words have become popular. I don’t think they have any meaning.” Although a careful reader might note that what Fama is saying here is something like that there is a bubble in the concept of bubbles, what he intends is to deny that there are bubbles, and thus that there is any “intrinsic value” to a given asset.

It’s at this point, I think, that the connection between Eugene Fama’s contention about the “efficient market hypothesis” and the doctrine in the humanities known as “antifoundationalism” becomes clear: both are denials of the Starks’ “Winter Is Coming” motto. After all, a bubble only makes sense if there is some kind of “intrinsic,” or “foundational,” value to something; similarly, a “foundationalist” thinks that there is some nonhuman reality. But why does this obscure and esoteric doctrinal dispute among a few intellectuals matter, aside from being the latest turn of the wheel of fashion within the walls of the academy?

Well, it matters because what they are really discussing—the real meaning of “intrinsic value”—is whether to allow ordinary people to have any say about the future of their lives.

Many liberals, for instance, have warned about the Republican assault on the right to vote in such matters as the Supreme Court’s 2013 ruling in Shelby County vs. Holder, which essentially gutted the Voting Rights Act of 1965, or the passage of “voter ID laws” in many states—sold as “protections” but in reality a means of preventing voting. What’s far less-often discussed, however, is that intellectuals of the supposed academic left have begun—quietly, to be sure—to question the very idea of voting.

Oxford don Mary Beard, for example—a scholar of the ancient world and avowed feminist—recently wrote a column for the London Review of Books concerning the “Brexit” referendum, in which the people of Great Britain decided whether to stay in the European Union or not. Beard’s sort—educated, with “progressive” opinions—thought that Britain ought to remain in the Union; when the results came in, however, the nation had decided to leave, or “Brexit.” “Handing us a referendum,” Beard wrote in response, “is not a way to reach a responsible decision”—“for God’s sake,” one can almost hear Beard lecturing, “how can you let an important decision be up to the [insert condescending adjective here] voters?” But while that might sound like a one-time response to a very particular situation, in fact many smart people who share Beard’s general views also share her distrust of elections.

What is an election, anyway, but an event analogous to a battle, or a hurricane? To people inclined to dismiss the significance of real events, it’s easy enough to dismiss the notion of elections. “Importantly”— wrote Princeton University’s Lawrance S. Rockefeller Professor of Politics, Stephen Macedo, recently—“majority rule is not a fundamental principle of either democracy or fairness, nor is it required by any basic principle of democracy or fairness.” According to Macedo, “the basic principle of democracy” isn’t elections, but instead “political equality,” or a “respect [for] minority rights and … fair and inclusive deliberation.” In other words, so long as “minority rights” are respected and there is “fair and inclusive deliberation,” it doesn’t matter if anyone votes or not—which is to say that to very many smart, and supposedly “liberal” or “leftist” people, the very notion that voting has any kind of “intrinsic value” to it at all has become irrelevant.

That, more or less, is what the characters on Game of Thrones think too. After all, as Tywin says to Jaime at one point during the conversation I began this essay with, a “lion doesn’t concern himself with the opinion of a sheep.” Which, one supposes, is not a very surprising sentiment on a show that, while it sometimes depicts depicts dragons and magic, mostly concerns the doings of a handful of aristocrats in a feudal age. What might be pretty surprising, however—depending on your level of distrust—is that, today, a great many of the people entrusted to be society’s shepherds appear to agree with them.

Mr. Tatum’s Razor

Arise, awake, and learn by approaching the exalted ones, for that path is sharp as a razor’s edge, impassable, and hard to go by, say the wise.
Katha Upanishad 1-III-14

Plurality is never to be posited without necessity.
—William of Ockham. Questions on the Sentences of Peter Lombard. (1318).

“The United States had lost. And won.” So recently wrote the former European and present naturalized American John Cassidy when Team USA advanced out of the “group stage” in the World Cup soccer tournament despite losing its last game of that stage. (To Germany, 1-0.) So even though they got beat, it’s the first time the U.S. has advanced out of the group stage in back-to-back Cups. But while the moment represented a breakthrough by the team, Cassidy warns it hasn’t been accompanied by a breakthrough in the fandom: “don’t ask [Americans] to explain how goal difference works,” he advises. He’s right that most are unfamiliar with the rule that allowed the Americans to play on, but he’s wrong if he’s implying that Americans aren’t capable of understanding it: the “sabermetric revolution”—the statistical study of the National Pastime—begins by recognizing the same principle that also backs goal difference. Yet while thus there’s precedent to think that Americans could understand goal difference—and, maybe, accept soccer as a big-time sport—there’s one reason to think America can’t: the American political system. And, though that might sound wacky enough for any one piece of writing, golf—a sport equally at home in America and Europe—is ideally suited to explain why.

Goal difference is a procedure that applies at the opening stage of the World Cup, which is organized differently than other large sporting tournaments. The NCAA college basketball tournament, for instance, is an “elimination” type tournament: sorts each of its 64 teams into four different brackets, then seeds each bracket from a #1 ranked team to a #16 ranked team. Each team then plays the team on the opposite side of the bracket, so that the the best team plays the lowest ranked team, and so on. Winning allows a team to continue; losing sends that team home, which is what makes it an “elimination” type of tournament.

The World Cup also breaks its entrants into smaller groups, and for the same reason—so that the best teams don’t play each other too early—but that’s where the similarities end. The beginning, “group” stage of the tournament is conducted in a round-robin format: each team in a group plays every other team in a group. Two teams from each group then continue to the next part of the competition.

Because the group stage is played under a round-robin, rather than elimination, structure losing a game doesn’t result necessarily in exiting the tournament—which is not only how the United States was not eliminated from competition by losing to Germany, but also is what makes the World Cup un-American in Cassidy’s estimation. “Isn’t cheering a team of losers,” Cassidy writes, “an un-American activity?” But there’s at least two questionable ideas packed into this sentence: one is that a team that has lost—a “loser”—is devoid of athletic ability, or what we might call value, and secondly that “losers” are un-American, or anyway that cheering for them is.

The round-robin format of the group stage after all just means that the tournament does not think a loss of a game necessarily reveals anything definitive about the value of a team: only a team’s record against all the other teams in its group does that. If the tournament is still unsure about the value of a team—that is, if two or more teams are tied for best, or second-best (two teams advance) record—then the tournament also looks at other ways to determine value. That’s what “goal difference,” or differential, is: as Ken Boehlke put it on CBSports website (“Understanding FIFA World Cup Procedures”), goal difference is “found by simply subtracting a team’s goals against from its goals scored.” What that means is that by the way the World Cup reckons things, it’s not only important whether a team lost a close game, but it’s also important if that team wins a blow-out.

Goal difference was, as Cassidy says, the reason why the American team was able to be one of the two teams of each group to advance. It’s true that the Americans were tied by win-loss record with another team in their group, Portugal. But the Americans only lost to Germany by one goal, while earlier in the stage the Portuguese lost 4-0. That, combined with some other results, meant that the United States advanced and Portugal did not. What the World Cup understands, is that just winning games isn’t necessarily a marker of a team’s quality, or value: what also matters is how many goals a team allows, and scores.

Now, John Cassidy appears to think that this concept is entirely foreign to Americans, and maybe he’s right—except for any of the Americans who happen to have seen the movie Moneyball, which not only grossed over $75 million dollars in the United States and was nominated for six Oscars but also starred Brad Pitt. “What are you really worth?” was the film’s tagline, and in the speech that is the centerpiece of the movie, the character Peter Brand (played by Jonah Hill, another fairly well-known actor) says to his boss—general manager of the Oakland A’s Billy Beane (played by Pitt)—that “Your goal … should be to buy wins. And in order to buy wins, you need to buy runs.” And while Moneyball, the film, was released just a few years ago, the ideas that fuel it have been around since the 1970s.

To be sure, it’s hardly news that scoring points results in winning games—the key insight is that, as Graham MacAree put it on the website FanGraphs, it is “relatively easy to predict a team’s win-loss record using a simple formula,” a formula that was invented a man named Bill James in the 1970s. The formula resembled the classic Pythagorean Theorem that James called it the Pythagorean Expectation: what it expressed was that the ratio of a team’s past runs scored to runs allowed is a better predictor of future success (i.e., future wins and losses) than that team’s past ratio of wins to losses. What it meant was that, to quote MacAree again, “pure pythagorean expectancy is probably a better way of gauging a team than actual wins and losses.” Or to put it another way, knowing how many runs a team scored versus how many that team’s opponents scored is more valuable than knowing how many games it won.

What the Pythagorean Expectation model and the goal difference model do, then, concentrate focus on what is the foundational act of their respective sports: scoring goals and scoring runs. Conversely, both weaken attention on winning and losing. That might appear odd: isn’t the point of playing a game to win, not (just) to score? But what both these methods realize is that a focus on winning and losing, instead of scoring, is vulnerable to a particular statistical illusion called a Simpson’s Paradox.

As it happens, an episode of the television series Numb3rs used a comparison of the batting averages of Derek Jeter and David Justice in the middle 1990s to introduce the idea of what a Simpon’s Paradox is, which seems tailor-made for the purpose. Here is a table—a more accurate one than the television show used—that shows those averages during the 1995, 1996, and 1997 seasons:





Derek Jeter









David Justice









Compare the year-by-year averages: Jeter, you will find, has a worse average than Justice in every year. Then compare the two players’ totals: Jeter actually has a slightly better average than Justice. A Simpson’s Paradox results, as the Stanford Encyclopedia of Philosophy puts it, a when the “structures that underlie” a set of facts “invalidate … arguments that many people, at least initially, take to be intuitively valid.” Or as the definition on Wikipedia describes it, a bit more elegantly, the paradox occurs when “appears that two sets of data separately support a certain hypothesis, but, when considered together, they support the opposite hypothesis.” In this case, if we consider the data year-by-year, it seems like Justice is a better hitter than Jeter—but when we consolidate all of the data, it supports the notion that Jeter is better than Justice.

There’s at least two ways we can think that the latter hypothesis is the more likely: one is the simple fact that 1995 was Derek Jeter’s first appearance in the major leagues, because he was born in 1974, whereas Justice was already a veteran player who was born eight years earlier. Jeter is younger. Quite obviously then from the perspective of a general manager looking at these numbers after the 1997 season, buying Jeter is a better move because more of Jeter’s career is available to be bought: since Jeter is only retiring this year (2014), that means that in 1997 there was 17 seasons of Derek Jeter available, whereas since David Justice retired in 2002, there were only 5 more seasons of David Justice available. Of course, none of that information would have been available in 1997—and injuries are always possible—but given the age difference it would have been safe to say that, assuming you valued each player relatively equally on the field, Jeter was still more valuable. In one sense though that exercise isn’t very helpful, because it doesn’t address just what Simpson’s Paradox has to do with thinking about Derek Jeter.

In another though it has everything to do with it. The only question that matters about a baseball player, says Bill James, is “If you were trying to win a pennant, how badly would you want this guy?” Or in other words, don’t be hypnotized by statistics. It sounds like a simple enough lesson, which in a way it is—but it’s terribly difficult to put into practice. In this case, it is terribly easy to become mystified by the two players’ batting averages, but what James might advise is to look at the events that these numbers represent: instead of looking at the averages, look at the components of those averages.

 What looking at the raw numbers reveals is that Jeter had more hits than Justice over the three seasons: 385 to 312. That difference matters because—unlike the difference in batting average over the same period, which is only a couple of points—78 more hits is a lot more hits, and as James wrote in The New Bill James Historical Baseball Abstract, the “essential measure of a hitter’s success is how many runs he has created.” Further, without getting too far into the math of it, smart people who’ve studied baseball have found that a single hit is worth nearly half a run. (Joe Posnanski, former Senior Writer at Sports Illustrated and one of those people, has a nice post summarizing the point called “Trading Walks For Hits” at What that would mean is that Jeter may have created more runs than Justice did over the same period: depending on the particular method used, perhaps more than twenty more runs. And since runs create wins (that conversion being calculated as about ten runs to the win) that implies that Jeter likely helped his team to two more wins than Justice did over the same period.

To really know which player contributed more to winning would require a lot more investigation than that, but the point is that following James’ method leads towards the primary events that generate outcomes, and away from the illusions that a focus on outcomes foster. Wins are generated by runs, so focus on runs; runs are created by hits, so focus on hits. So too does goal difference mean that while the World Cup recognizes wins, it also recognizes the events—goals—that produce wins. Put that way, it sounds quite commonsensical—but in fact James was lucky in a sense to stumble upon it: because there are two ways to organize sport, and only one of those types is amenable to this kind of analysis. It was fortunate, both to James and to baseball, that he was a fan of a game that could easily be analyzed this way.

In sports like baseball, there’s a fairly predictable relationship between scoring and winning. In other sports though there isn’t, and that’s why golf is very important. It is a sport that under one way to play it the sport is very amenable to means of analysis like the World Cup’s goal difference or Bill James’ Pythagorean Expectation. Golf however also has another way to play, and that way does not have a predictable relationship between scores and wins. What the evidence will show is that having two different forms to the sport isn’t due to a mistake on the part of the designers’: it’s that each form of the game was designed for a different purpose. And what that will show, I will argue, is that whether a game has one sort of scoring system or the other predicts what the purpose of the design is—and vice versa.

On the PGA Tour, the standard tournament consists of four rounds, or 72 holes, at the end of which the players who have made it that far add up their scores—their number of strokes—and the lowest one wins. In the Rules of Golf, this format is known as “stroke play.” That’s what makes it like the group stage of the World Cup or Bill James’ conception of baseball: play begins, the players attempt some action that produces a “score” (however that is determined), and at the end of play each of those scoring events is added together and compared. The player or team that produces the right amount of these “scoring events” is then declared the winner. In short, under the rules of stroke play—just as to the World Cup’s group stage, or to Bill James’ notion of baseball—there is a direct relationship between the elemental act of the game, scoring, and winning.

But the format most often used by golf’s professionals is not the only method available: many amateur tournaments, such as the United States Amateur, use the rules format known as “match play.” Under this format, the winner of the contest is not necessarily the player who shoots the lowest overall score, as in stroke play. Instead, as John Van der Borght has put the matter on the website of the United States Golf Association, the official rule-making body of the sport, in match play the “winner is the player who wins the most holes.” It’s a seemingly minor difference—but in fact it creates such a difference that match play is virtually a different sport than stroke play.

Consider, for instance, this year’s Accenture Match Play tournament, held at the Dove Mountain course near Tucson, Arizona. (The only tournament on the PGA Tour to be held under match play rules.)  “Factoring in conceded putts,” wrote Doug Ferguson of the Associated Press earlier this season, “Pablo Larrazabal shot a 68 and was on his way back to Spain,” while “Ernie Els shot 75 and has a tee time at Dove Mountain on Thursday.” In other words, Larrazabal lost his match and Els won his, even though Larrazabal played better than Els. Intuitively, Larrazabal was the better player at this tournament, which would lead to thinking Larrazabal continued to play and Els exited—but the actual results conclude the reverse. It’s a Simpson’s Paradox, and unlike stroke play—which cannot generate Simpson’s Paradoxes—match play produces them all the time. That’s why match play golf does not resemble baseball or soccer, as golf does in stroke play, but instead a sport whose most prestigious tournament—Wimbledon—just concluded. And tennis is the High Church of Simpson’s Paradox.

Simpson’s Paradox, for example, is why many people don’t think Roger Federer is not the greatest tennis player who ever lived. That’s because the Swiss has won 17 major championships, a record, among other career accomplishments. “But,” as Michael Steinberger wrote in the New York Times not long ago, “he has a losing record against [Rafael] Nadal, and a lopsided one at that.” (Nadal leads 23-10.) “How can you be considered the greatest player ever if you were arguably not even the best player of your own era?” Steinberger asks. Heroically, Steinberger attempts to answer that question in favor of Federer—the piece is a marvel of argumentation, where the writer sets up a seemingly-insurmountable rhetorical burden, the aforementioned question, then seeks to overcome it. What’s interesting, though—and in several searches through the Internet I discovered many other pieces tackling more or less the same subject—neither Steinberger nor anyone else attempted what an anonymous blogger did in 2009.

He added up the points.

The blog is called SW19, which is the United Kingdom’s postal code for the district Wimbledon is in. The writer, “Rahul,” is obviously young—he (or she) stopped posting in December of 2009, because of the pressures of college—but yet Rahul did something I have not seen any other tennis journalist attempt: in a post called “Nadal vs. Federer: A Pythagorean Perspective,” Rahul broke down “the Federer/Nadal rivalry on a point-by-point basis, just to see if it really is as lopsided as one would expect.” That is, given Nadal’s dominant win-loss record, the expectation would be that Nadal must win an equally-impressive number of points from Federer.

By July of 2009—the time of publication—Nadal led Federer by 13-7 in terms of their head-to-head record, a 65 percent winning percentage. The two champions had played 4,394 total points across those 20 matches—one of them the 2008 French Open, won by Nadal in straight sets, 6-1, 6-3, 6-0. (Nadal has, as of 2014, now won 9 French Opens, a majors record, while Federer has only won the French once—the very next year after Nadal blew him off the court: 2009.) Now, if there was a straightforward relation between points and wins, Nadal’s percentage of those points ought to be at least somewhat similar to his winning percentage of those matches.

But what Rahul found was this: of the total points, Nadal had won 2,221 and Federer 2,173. Nadal had only beaten Federer on 48 points, total, over their careers to that point, including the smackdown at Roland Garros in 2008. It’s less than one percent of all the points. And if you took that match out of the total, Nadal had won a grand total of eight more points than Federer, out of over 4,000 points and 19 other matches. It is not 65 percent. It is not even 55 percent.

Still, it’s the final nugget that Rahul uncovered that is likely of the most relevance. In three of the twenty matches won by Nadal to that moment in their careers, Federer had actually won more points: two matches in 2006, in Dubai and Rome, and once at the Australian Open in 2009. As Rahul points out, “if Federer had won those three matches, the record would sit at 10-10”—and at least in 2009, nobody would have been talking about Federer’s Achilles heel. I don’t know what the current Pythagorean record stands between the two players at the moment, but it’s interesting that nobody has taken up this detail when discussing Federer’s greatness—though nub of it has been taken up as a serious topic concerning tennis as a whole.

In January in The Atlantic, Professor Ryan Rodenberg of the Florida State University noted that not only did Federer have the 17 Grand Slam titles and the 302 weeks ranked No. 1 in the world, but he also held another distinction: “the worst record among players active since 1990 in so-called ‘Simpson’s Paradox’ matches—those where the loser of the match wins more points than the winner.” Federer’s overall record in these matches is like that of his record against Nadal: not good. The Swiss is only 4-24.

To tennis aficionados, it’s a point that must appear irrelevant—at least, no one until Professor Rodenberg appears to have mentioned it online. To be sure, it does seem questionably relevant: Federer has played nearly 1200 matches professionally; 28 is a pittance. But Rodenberg, along with his co-authors, found that matches like the Isner-Mahut match, where the loser out-scored the winner, constituted “about 4.5 percent” of “61,000 men’s ATP and Grand Slam matches dating back to 1990.” That’s over 3,000 matches—and given that, in exactly zero soccer matches or baseball games over that time frame or any other time, did the losing side net more goals or plate more runs than the other, it at the least raises some questions.

How, after all, is it possible for one side of the net to win—despite losing more of the points? The answer, as Rodenberg puts it, is  “tennis’ decidedly unique scoring system.” In sports like baseball, sports psychologist Allen Fox wrote recently on for the website for the magazine Tennis, “score is cumulative throughout the contest … and whoever has the most points at the end wins.” Sports like tennis or match play golf are different however: in tennis, as Fox says, “[i]f you reach game point and win it, you get the entire game while your opponent gets nothing—all the points he or she won in the game are eliminated.” In the same fashion, once a hole is over in match play golf it doesn’t matter what either competitor scored on that hole: each total is struck out, and the match in effect begins again. What that in turn means is that certain points, certain scoring events, have more value than others: in golf, what matters is the stroke that takes a hole, just as in tennis what matters is the point that takes a game, or a set, or a match. Those points are more valuable than other points—a fact of tremendous importance.

It’s this scoring mechanism that is what allows tennis and match play golf to produce Simpson’s Paradox games: a system whereby the competition as a whole is divided into smaller competitions that function independently of the others. In order to get Simpson’s Paradox results, having a system like this is essential. The $64,000 question however is: just who would design a system like that, a system that can in effect punish a player who does the thing that defines the sport better than the other player more often than the player who doesn’t? It isn’t enough just to say that results like that are uncommon, because why allow that to happen at all? In virtually every other sport, after all, no result like these would ever come up. The only serious answer must be that tennis and match play golf were specifically designed to produce Simpson’s Paradoxes—but why? The only way to seek that answer, I’d say, is to search back through history.

The game we today call tennis in reality is correctly termed “lawn tennis,” which is why the formal name of the organization that sponsors the Wimbledon tournament is the “All England Lawn Tennis and Croquet Club.” The sport is properly called that in order to distinguish it from the older game known as “real tennis” or, in French, Jeu de Paume. Whereas our game of tennis, or lawn tennis, is generally played outdoors and on a single plane, Jeu de Paume is played indoors, in unique, non-standardized courts where strange bounces and funny angles are the norm. And while lawn tennis only came into existence in 1874, Jeu de Paume goes well back into the Middle Ages. “World titles in the sport were first competed in 1740,” as Rolf Potts noted in a piece about the game in the online magazine, The Smart Set, “and have continued to the present day, making Jeu de Paume men’s singles the oldest continuous championship event in sport.” Jeu de Paume, thus, is arguably the oldest sport in the world.

Aside from its antiquity, the game is also, and not unrelatedly, noted for its roots in the ancien regime: “Nearly all French royalty were familiar with the sport from the 13th century on,” as Rolf Potts notes. And not just French royalty: Henry VIII of England is regularly described as a great player by historians. These are not irrelevant facts, because the status of the players of Jeu de Paume in fact may be directly relevant to how tennis is scored today.

“When modern tennis,” writes Potts, “was simplified into its popular form in 1874, it appropriated the scoring system of the ancient French game.” So our game of tennis did not invent its own method of scoring; it merely lifted another game’s method. And that game’s method may be connected to the fact that it was played by aristocrats in the fact that so much about Jeu de Paume is connected to gambling.

“In October of 1532,” Potts reports, Henry VIII lost 50 pounds on tennis matches: “about a thousand times the sum most Englishmen earned in a week.” Anne Boleyn, Henry’s second wife, by some accounts “was betting on a tennis game when Henry’s men arrested her in May of 1536,” while others say that her husband received the news of her execution while he himself was playing a match. Two centuries earlier, in 1355, King John II of France had been recorded paying off a bet with “two lengths of Belgian cloth.” And in Rob Lake’s academic paper, “Real Tennis and the Civilising Process,” published in the academic journal Sport in History, Lake claims that “the game provided opportunities for nobles to engage in conspicuous consumption … through gambling displays.”

So much so, in fact, that Potts also reports that “some have speculated that tennis scoring was based on the gros denier coin, which was said to be worth 15 deniers.” Be that as it may, two facts stand out: the first is that the game’s “gradual slide into obscurity began when fixed games and gambling scandals sullied its reputation in the late 17th century,” and the second that “games are still regulated by a complicated handicapping system … so that each player begins the game with an equal expectation of winning.” So elaborate is that handicap system, in fact, that when Rolf Potts plays the first match of his life, against a club professional who is instructing him, he “was able to play a close game.” Gambling, in seems, was—as Potts says—“intrinsic to Jeu de Paume.” And since the sport still has a handicap system, which is essential to gambling, so it still is.

We can think about why that is by comparing Jeu de Paume to match play golf, which also has an early connection both to feudalism and gambling. As Michael Bohn records in Money Golf: 600 Years Of Bettin’ On Birdies, the “earliest record of a golf bet in Scotland was in 1503,” when on February 3 King James IV paid out 42 shillings to the Earl of Bothwell in “play at the golf.” And as John Paul Newport of the Wall Street Journal writes, “historically all the early recorded competitions—King James IV in 1503, for example, or the Duke of York, later King James II [of England], in 1681—were match play.” That is likely not a coincidence, because the link between the aristocracy, gambling, and match play is not difficult to explain.

In the first place, the link between the nobility and gambling is not difficult to understand since aristocrats were virtually the only people with both money and the time for sport—the opportunity, as a prosecutor would say. “With idle people amusement is the business of life,” as  the London magazine The Spectator noted in 1837; and King James’ bet with the Earl of Bothwell—42 shillings, or a little over £2—would have bought roughly six month’s work from a laborer during the sixteenth century. Not merely that: the aristocracy were practically the only people who, legally speaking, could gamble in during the Renaissance: as Nicholas Tosney notes in a paper for the University of Nevada, Las Vegas in 2010—“Gaming in Britain and America: Some Historical Comparisons”—gambling in England was outlawed in 1541 for anyone not at least a gentleman.

Yet just having the ability does not carry a case. It’s also required to be able to posit a reason—which of course isn’t all that hard to find when it comes to gambling. Aside from the obvious financial inducement, though, aristocratic men had something extra pushing them toward gaming. As the same 1837 Spectator article noted, gambling was widely thought to be “a necessary accomplishment of a young man in fashionable circles.” After all, what better way to demonstrate belonging to the upper classes by that form of conspicuous consumption that buys—nothing? The literature on the subject is so extensive as to not need bothering with trolling out in its entirety: nobles had both the means and the motive to gamble, so it therefore seems reasonable to suppose that a game adopted by gamblers would be ideal for gambling.

And examined closely, match play does have such features. Gambling after all would best explain why match play consists of what John Van der Borght calls “18 one-hole contests.” According to John Paul Newport, that’s so “an awful hole here or there doesn’t spoil the day”—but a better explanation is likely because doing things that way allows the previous hole’s loser to bet again. Multiplying contests obviously increases the opportunity to bet—and thus for a sucker to lose more. And that’s why it is significant that the match play format should have a link to the nobility and gambling: because it helps to demonstrate that the two formats of golf are not just different versions of the same game, but in fact have two different purposes—purposes that are so different they are virtually different sports.

That difference in purpose is likely why, as Newport observes, it isn’t “until the mid-18th century are there records of stroke-play competitions.” One reason for the invention of the stroke play format was, Newport tells us, “to make tournaments involving larger numbers of golfers feasible.” The writer for the Wall Street Journal—make of that connection what you will—presents the new format as simply demanded by the increasing number of players (a sign, though Newport does not mention it, that the game was spreading beyond the boundaries of the nobility). But in reality stroke play was invented to serve a different purpose than match play, a purpose even now recognized by the United States Golf Association.

About the best definition of the purpose of stroke play—and thus, it’s difference from match play—can be found in the reply Sandy Tatum, then the executive director of the United States Golf Association, gave to a reporter at the 1974 U.S. Open at Winged Foot. That tournament would become known as “the Massacre at Winged Foot,” because even the winner, Hale Irwin, finished over par (+7). So when the extent of how tough the golf course was playing became obvious, one reporter asked Tatum if the USGA was trying to embarrass the best players in the world. What Tatum said in reply to the reporter is about as succinct an explanation of the purpose of the U.S. Open, and stroke play, as is possible.

“Our objective is not to humiliate the best golfers in the world,” Tatum said in response to the question: “It’s to identify them.”And identifying the greatest golfers is still the objective of the USGA: That’s why, when Newport went to interview the current executive director of the USGA, Mike Davis, about the difference between stroke play and match play for his article, Davis said “If all you are trying to do is determine who is playing the best over a relatively short period of time, [then] 72 holes of stroke play is more equitable [than match play].” The position of the USGA is clear: if the purpose of the competition is to “identify,” as Tatum said, or “determine,” as Davis said, the best player, then the best format for that purpose is stroke play, and not match play.

One reason why the USGA can know this is that it is obviously not in the interest of gamblers to identify themselves as great players. Consider, for instance, a photo printed along with Golf magazine’s excerpt of Kevin Cook’s book, Titanic Thompson: The Man Who Bet On Everything. The photo depicts one Alvin “Titanic Thompson” Thomas, swinging a club late in life. Born in 1892, Cook says that “Titanic was the last great player to ignore tournament golf”—or stroke play golf, anyway. Not because he couldn’t: Cook says that Byron Nelson, who among other exploits won 11 tournaments on the PGA Tour in a row in the summer of 1945, and thus seems an excellent judge, said “there was ‘no question’ that Titanic could have excelled on Tour, ‘but he didn’t have to.’”—because Titanic “‘was at a higher level, playing for $25,000 a nine while we [Tour players] played for $150.’” Thomas, or Thompson was the greatest of golf gamblers; hence the caption of the photo: “Few golf photos exist of Thompson,” it reads, “for obvious reasons.” Being easily identifiable as a great golfer, after all, is not of much use to a gambler—so a format designed for gambling would have little incentive to “out” better players.

To put it simply then the game of tennis today has the structure that it does today because it descends from a different game—a game whose intent was not to identify the best player, but rather to enable the best player to maximize his profits. Where the example of tennis, or match play golf, should then lead specifically, is to the hypothesis that any point-driven competition that has non-continuous scoring—which is to say divided into sub-competitions whose results are independent of all the others—and where some parts of the competition have a higher value than other parts, ought to raise doubt, at the least, as to the validity of the value of the competition’s results.

The nature of such structures make it elementary to conceal precisely that which the structure is ostensibly designed to reveal: the ultimate value that underlies the whole operation, whether that is the athletic ability of an individual or a team—or something else entirely. Where goal difference and Pythagorean Expectation and stroke play all consolidate scores in order to get at the true value those scoring events represent, tennis’ method and match play divide scores to obscure value.

That’s why match play is so appealing to golf gamblers—it allows the skilled player to hide his talent, and thus maximize income. Conversely, that’s why the U.S. Open uses stroke play: because the USGA wants to reveal the best player. Some formats of play lend themselves to one purpose or the other—and what that leads to is a kind of thought experiment. If the notion advanced here is correct, then there are two kinds of ways a given sport may score itself, and concurrently two different purposes those different means of scoring may serve. If a sport is more like golf’s match play than it is like golf’s stroke play, in short, it can be predicted that it’s likely to be vulnerable to gamblers.

As it happens, it’s widely believed that professional tennis has a gambling problem. “Everyone knows,” said last year’s Wimbledon winner, Andy Murray, “that match-fixing takes place in professional tennis”—all the way back in October of 2007. A story in the Guardian that year summed up the scandal that broke over the sport that August, which began when the world’s largest online betting exchange, Betfair, reported “irregular gambling patterns” on a match between Nikolay Davydenko—once ranked as high as #3 in the world—and Martin Arguello—at the time ranked #87—at the Polish Open. At the end of September 2007, Novak Djokovic—this year’s Wimbledon champion—said “he was offered £10,000 to lose in a tournament in St. Petersburg” the previous year. In late October of 2007—after Murray’s comment to the press—“French undercover police” were “invited into the Paris Masters amid suspicions of match-fixing in tennis.” But what Simpson’s Paradox would tell the police—or tennis’ governing bodies—is that looking for fixed matches is exactly what the cunning gambler would want the authorities to do.

“The appeal of tennis to gamblers,” wrote Louisa Thomas for Grantland earlier this year, “makes total sense” for a number of reasons. One is that “tennis is played everywhere, all the time”: there’s likely a tournament, somewhere in the world, any time anyone feels the urge to bet, unlike a lot of other sports. That ubiquity makes tennis vulnerable to crooked gamblers: as Thomas observes, there are “tens of thousands of professional matches, hundreds of thousands of games, millions of points”—a spread of numbers so wide that the volume alone discourages detection by any authority.

Another reason why tennis should be appealing to gamblers is that “bettors can make wagers during play itself”: you can get online while watching a match and lay down some action. As The Australian reported this year—when a young man was arrested at the Australian Open with an electronic device designed to transmit scores quicker than the official tournament did—there are “websites that allow bets to be laid on individual events such as whether a player faults on serve.” Now, essentially the scam that the man at the Australian Open was arrested for is the same con as depicted in the film The Sting, which itself tells something of a tale about the sport.

But the real scandal of tennis, though perhaps Thomas does not emphasize this enough, is that it is vulnerable to manipulation simply because  “broken into discrete points, games, sets, matches, and tournaments.” It’s a point, however, that one of Professor Rodenberg’s students understands.

What Benjamin Wright—a graduate student in Rodenberg’s department at the Florida State University—knows is that because of tennis’ scoring system, the sport doesn’t need to have crooked players throwing matches to be corrupt. “Governing bodies must be aware,” says Wright—in his master’s thesis, “Best of N Contests: Implications of Simpson’s Paradox in Tennis”—“that since tennis does not use a running score like other sports intentionally losing points, games, and sets is plausible since such acts may not have long-term implications.” In other words, “a player would not need to lose an entire match intentionally.” All that’s necessary—especially since it’s possible to bet on tennis in real time—is for a player to lose “points during specific periods of a match.” All a gambler needs to know, that is, is that a player will throw the second point of the fourth game of the second set—knowledge that is nearly undetectable because under the rules of the game it is entirely possible for a player to shave points without risking a loss.

“Who’s to say,” says Thomas about the undetectability of corruption, a player is “not just having a really rotten day?” But what Thomas doesn’t appear to grasp fully is that the actual disgrace is the question of how a player could be accused of corruption if she has won her match? That’s the real scandal: how even apparently well-trained journalists can miss the point. “Although tennis is perceived as a genteel sport,” wrote Joe Drape of the New York Times about the Davydenko scandal in 2007, “it has always confronted the same problem as other contests based on individual competition like boxing.” That problem, Drape said, is that a “fixer needs to sway only one person, and taking a dive is hard to detect.” Drape is, to be sure, right about what he says—so far as that goes. But Drape does not point out—I think likely because he does not understand—why “taking a dive” is so difficult to unmask in tennis: because it’s possible to throw a point—or a game, or a set—without affecting the outcome of the match.

Now, this is so obviously crooked that the gall of it is simply breathtaking. Yet the reality is simply that, aside from a few very naive people who could probably stand to have a few dollars taken from them by shrewd, and likely Russian, mobsters, no one really loses much by this arrangement. There are far worse scams in the world, and people who bet on tennis are probably not very sympathetic victims. But what knowing what we now know about tennis, and match play golf, allows us to now do is to evaluate all competitions: any contest which has the characteristics we have isolated (non-cumulative scoring, unequal points) will necessarily produce Simpson’s Paradox results. Further, any contest that produces Simpson’s Paradox results does so by design: there’s no reason to add an extra layer of complexity to a competition unless it’s in somebody’s interests. Lastly, since the only reason to add that layer of complexity, and thus produce Simpson’s Paradoxes, is to conceal value, it’s more likely than not that those interests are not entirely legitimate.

Now, it so happens that there is a competition that has those two characteristics and has demonstrably produced at least one paradoxical result: one where the “winner” lost and the “loser” won.

That competition is called an American presidential election.

The Mark of Z

“One way to characterize professional golf,” wrote John Cassidy earlier this summer in The New Yorker, “is to say that it has reached parity—there are so many good players, and they all have a roughly equal chance of winning.” Cassidy called it the “random golfer theory,” and has trotted it out after Webb Simpson’s win at Olympic and Ernie Els’ win at Lytham. The idea is that anybody within the top 100 has a shot of winning any major: an idea that is, more or less, borne out by the fact that of the past 17 majors, there has been 17 winners. Until now, which is to say that Rory’s win at the PGA has blown that idea up just as surely as the events of the past five years has blown up both the Black-Scholes formula and the hype of this year’s Ryder Cup at Medinah to what will, especially in the Fleet Street press, be absurd levels.

The cry will be, as it’s been since McIlroy won the U.S. Open at Congressional a year ago, for a Tiger vs. Mac showdown during Sunday’s singles matches, only with an even heightened pitch now that Rory’s won his first two majors at a more rapid clip than Tiger won his first two. And as it happens, Tiger’s second major was also a PGA, and, also, it was at Medinah. Which, as it further happens, was also the first time Tiger faced a competitor who seemed to have all the tools he did, but was from Europe—and younger to boot. And after that PGA, in 1999, Sergio Garcia, like Rory’s fans today, demanded to play Tiger in that year’s Ryder Cup.

Obviously, European fans are hoping for a different outcome this time around: that Ryder Cup was at the Country Club in Brookline, and the Euros got smoked in singles; that was the year that the American captain, Ben Crenshaw, said the night before the finale, “I got a good feeling about this.” It was also the year of the “excessive celebration” after Justin Leonard made his putt on the 17th hole of regulation—which came before Jose Olazabal had a chance to make his putt, which would have at least continued the match, a point that, if you believe the London papers, all of Europe has been brooding about for the past nearly-decade-and-a-half. Not that Europeans are well-known to carry around centuries-long grudges or anything.

In any case, this year’s Ryder Cup is shaping up, at least from the wrong end of the Atlantic, to be a kind of revanchist’s dream, only without soaking the fields of Flanders in blood. In place of Sergio, they have Rory, who actually wins tournaments, and even majors, without regripping his club twenty-five times or casually insulting entire states. And most alarmingly, at least from this side of the Atlantic, our main guy not only has never made a big deal out of these kinds of team events—Tiger is on record as saying he doesn’t regard the Ryder Cup as being the same as one of the four majors—but he hasn’t won a major in four years. Or, in other words, since their kid starting winning them. Which is where the Black-Scholes point comes in.

“If Capital One was trading at $30 a share,” says Michael Lewis in The Big Short: Inside the Doomsday Machine, the Black-Scholes model for pricing options—those obscure financial instruments that have had so much say in our lives recently— “assumed that, over the next two years, the stock was more likely to get to $35 a share than to $40, and more likely to get to $40 than to $45, and so on.” This makes sense to us, intuitively: we like to think that “slow and steady wins the race,” for instance. But the real world does not always behave in that slow and incremental way: everyone would have bet that dinosaurs would be the dominant species on the planet for eons, until a meteorite crashed in southern Mexico. Sometimes things can change quite suddenly—and not reach any intermediate stops. Once, there were billions of dinosaurs. Then, there weren’t.

Once, there was a Tiger, and now there’s a Rory. In between there’s been a collection of Keegan Bradleys and Webb Simpsons, a collection that has largely made the golf press uneasy at best and, at worst, spooked. Golf is, after all, one of the few sports—the other that I can think of at the moment being horse racing—where nobody likes an underdog, at least until the point where it seems like the underdog can actually win; or, in short, become the overdog. Rory, with his eight-shot win at the PGA, might just have reached that point: a point that, as it happens, the wonks over at Grantland have quantified using a measure they call “Z-Score,” which is apparently a standard part of the average mathematician’s toolbag.

“Z-Score” is calculated by taking the winner’s score and subtracting the average score of all the players who finished the tournament, then dividing that against “the variance between the scores and the average performance,” as Grantland’s resident golf stat-head, Bill Barnwell, says. In other words, a tournament where the winner shot “20-under-par and the last-place finisher shot 20-over-par” would have a higher value than a tournament “in which the winner shot 3-under-par and the duffer in last shot 4-over.” Of the top ten scores ever figured, Tiger Woods and Jack Nicklaus have three apiece, with Tiger Woods’ performance at the 2000 U.S. Open, where he blew away the field by fifteen shots, achieving the highest “Z-Score” ever recorded at -4.12 (meaning that he was more than four standard deviations better than the average performance in the tournament.

It’s a good methodology in that it factors out things like weather (everyone plays in similar conditions, within reason) and so on, and to a degree allows us to compare performances across the generations. For instance, it’s now arguable that Jack Nicklaus’ performance at the 1965 Masters might be better than Woods’ win in 1997, even though Woods broke Nicklaus’ scoring record (271, or -11 to par, versus 270, or -12 to par), because while Woods’ “Z-Score” in 1997 was -3.24 Nicklaus’ “Z-Score” was -3.48. Or in other words, Woods was only a bit more than three times better than his competitors in 1997, while Nicklaus was nearly three-and-a-half times better. Obviously, this doesn’t really matter much (though Davis Love’s win at the 1997 PGA, which he took by five shots and produced a Z-Score of 3.54, looks a lot better after running it through this formula), but it’s fun to compare scores across eras.

Like, for instance, the scores Tiger Woods produced in his prime versus the scores Rory McIlroy has produced in his two major wins: last year’s U.S. Open at Congressional and this year’s PGA. McIlroy won both tournaments by eight shots, which is the kind of performance necessary to place on the Z-Score leaderboard, but Z-Score isn’t factored by how much the second-place guy shot, but rather by how much the field as a whole shot. Rory’s Z-Score for the tournaments places him comfortably within the top twenty Z-Scores ever recorded, but his -3.07 score for Congressional, together with his -3.15 score for Kiawah, aren’t enough to place him very close to Tiger’s epic win in 2000. The Congressional score, in fact, doesn’t even place Rory close to Jack Nicklaus’ -3.22 at Turnberry in 1977—you know, the “Duel In The Sun” Jack lost to Tom Watson.

Rory’s wins, that is, have been big—but they haven’t been that big, at least by comparison to Jack and Tiger. The win at Congressional, at least as measured by Z-Score, isn’t even as good as Padraig Harrington’s British Open win in 2008, which the Irishman won at 3-OVER par, only four shots better than his nearest competitor—Harrington rang up a -3.09 Z-Score during what was a famously-windblown tournament. Still, Rory’s fans might cite Barnwell’s observation that through “his first nine majors, McIlroy has put up an average Z-Score 0.97 standard deviations below the mean,” an average only exceeded by Seve Ballesteros (-1.04) and Ernie Els (-1.25) in anyone’s first nine majors. Rory is, obviously, still very young; it’s quite possible we still haven’t seen his best stuff.

Still, what the Z-Score tale tells us is that while Rory is a very, very good golfer, he doesn’t go to the same dimension-bending, dinosaur-slaying, places Tiger Woods could go in his prime. But if we haven’t yet seen Rory’s best, there are few places Rory could demonstrate that to better effect than Medinah, the course Tiger has tamed twice for two of his fourteen major titles and a membership in the club itself. It’s no honorary membership, either: Tiger has the same rights as any other full member, an honor the club presented him with after his second win in 2006, which is to say that, in a sense perhaps more real than any other course, Medinah really is Tiger’s home turf. For Rory to beat Tiger there would be, one suspects, a grievous blow to the competitive Tiger—all the implacable laws of sport, which are even more inflexible than any mathematical model, thus demand that there is only one possible final match for the Ryder Cup’s finale at the end of September: Woods v. McIlroy, for all the stakes that there are. May the best Z-Score win—and to hell with the “random golfer theory.”