At Play In The Fields Of The Lord

Logo for 2015 US Amateur at Olympia Fields Country Club
Logo for 2015 US Amateur at Olympia Fields Country Club

 

Behold, I send you forth as sheep in the midst of wolves:
be ye therefore wise as serpents, and harmless as doves.
—Matthew 10:16

Now that the professional, Open tournaments are out of the way, the U.S. Amateur approaches. A tournament that has always been a symbol of wealth and discrimination—the Amateur was a tournament invented specifically to keep out the riff-raff of professional golfers—the site of this year’s edition might be considered particularly unfortunate considering that this year the tournament will fall just more than a year after the Michael Brown shooting in Ferguson, Missouri: Olympia Fields, in Chicago’s south suburbs, is a relatively wealthy enclave among a swath of exceedingly poor villages and towns very like the terrain of the St. Louis suburbs just a few hundred miles away. Yet there’s a deeper irony at work here that might be missed even by those who’d like to point out that similarity of setting: the format of the tournament, match-play, highlights precisely what the real message of the Brown shooting was. That real message, the one that is actually dangerous to power, wasn’t the one shouted by protestors—that American police departments are “racist.” The really dangerous message is the one echoed by the Amateur: a message that, read properly, tells us that our government’s structure is broken.

The later rounds of U. S. Amateur are played under golf’s match play, rather than stroke play, rules—a difference that will seem arcane to those unfamiliar with the sport, but is a very significant difference nevertheless. In stroke play, competitors play whatever number of holes are required—in professional tournaments, usually 72 holes—and count up however many strokes each took: the player with the fewest strokes wins. Match play however is not the same: in the first place, because in stroke play each golfer is effectively playing against every other player in the field, because all the strokes of every player count. But this is not so in match play.

In the first place, match play consists of, as the name suggests, matches: that is, once the field is cut to the 64 players with the lowest score after an initial two-day stroke play tournament, each of those 64 contestants plays an 18-hole match against one other contestant. The winner of each of these matches then proceeds to move on, until there is a champion—a single-elimination tournament that is exactly like the NCAA basketball tournament held every year in March. The winner of each match in turn, as John Van der Borght says on the website of the United States Golf Association, “is the player who wins the most holes.” That is, what matters on every hole is just whether the golfer has shot a lower score than the opponent for that hole, not overall. Each hole starts the competition again, in other words—like flipping coins, what happened in the past is irrelevant. It’s a format that might sound hopeful, because on each hole whatever screw-ups a player commits are consigned to the dustbin of history. In fact, however, it’s just this element that makes match-play the least egalitarian of formats—and ties it to Ferguson.

Tournaments conducted under match play rules are always subject to a kind of mathematical oddity called a Simpson’s Paradox: such a paradox occurs when, as the definition on Wikipedia says, it “appears that two sets of data separately support a certain hypothesis, but, when considered together, they support the opposite hypothesis.” For example, as I have mentioned in this blog before, in the first round of the PGA Tour’s 2014 Accenture Match Play tournament in Tucson, an unknown named Pedro Larrazabal shot a 68 to Hall-of-Famer Ernie Els’ 75—but because they played different opponents, Larrazabal was out of the tournament and Els was in. Admittedly, even with such an illustration the idea might still sound opaque, but the meaning can be seen by considering, for example, the tennis player Roger Federer’s record versus his rival Rafael Nadal.

Roger Federer has won 17 major championships in men’s tennis, a record—and yet many people argue that he is not the Greatest Of All Time (G.O.A.T.). The reason those people can argue that is because, as Michael Steinberger pointed in the New York Times not long ago, Federer “has a losing record against Nadal, and a lopsided one at that.” Steinberger then proceeded to argue why that record should be discarded and Federer should be called the “GOAT” anyway. But weirdly, Steinberger didn’t attempt—and neither, so far as I can tell, has anyone else—what an anonymous blogger did in 2009: a feat that demonstrates just what a Simpson’s Paradox is, and how it might apply both to the U.S. Amateur and Ferguson, Missouri.

What that blogger did, on a blog entitled SW19—a reference to the United Kingdom’s postal code for Wimbledon, the great tennis arena—was he counted up the points.

Let me repeat: he counted up the points.

That might sound trivial, of course, but as the writer of the SW19 blog realized, tennis is a game that abounds in Simpson’s Paradoxes: that is, it is a game in which it is possible to score fewer points than your opponent, but still win the match. Many people don’t realize this: it might be expected, for example, that because Nadal has an overwhelmingly-dominant win-loss record versus Federer, he must also have won an equally-dominant number of points from the Swiss champion. But an examination of the points scored in each of the matches between Federer and Nadal demonstrates that in fact the difference between them was miniscule.

The SW19 blogger wrote his post in 2009; at that time Nadal led Federer by 13 matches to 7 matches, a 65 percent winning edge for the Spaniard, Nadal. Of those 20 matches, Nadal won the 2008 French Open—played on Nadal’s best surface, clay—in straight sets, 6-1, 6-3, 6-0. In those 20 matches, the two men played 4,394 total points: that is, where one player served and the two volleyed back and forth until one player failed to deliver the ball to the other court according to the rules. If tennis had a straightforward relationship between points and wins—like golf’s stroke play format, in which every “point” (stroke) is simply added to the total and the winner has the fewest points—then it might be expected that Nadal has won about 65 percent of those 4,394 points played, which would be about 2,856 points. In other words, to get a 65 percent edge in total matches, Nadal should have about a 65 percent edge in total points: the point total, as opposed to the match record, between the two ought to be about 2,856 to 1,538.

Yet this, as the SW19 blogger realized, is not the case: the real margin between the two players was Nadal, 2,221, and Federer, 2,173. In other words, even including the epic beating at Roland Garros in 2008, Nadal had only beaten Federer by a total of 48 points over the course of their careers–a total of less than one percent of all the points scored. Not merely that, but if that single match at the 2008 French Open is excluded, then the margin becomes eight points.  The mathematical difference between Nadal and Federer, thus, is the difference between a couple of motes of dust on the edge of a coin while it’s being flipped—if what is measured is the act that is the basis of the sport, the act of scoring points. In terms of points scored, Nadal’s edge is about a half of percentage point—and most of that percentage was generated by a single match. But Nadal had a 65 percent edge in their matches.

How did that happen? The answer is that the structure of tennis scoring is similar to that of match play in golf: the relation between wins and points isn’t direct. In fact, as the SW19 blogger shows, of the twenty matches Nadal and Federer had played to that moment in 2009, Federer had actually scored more points than Nadal in three of them—and still lost the match. If there were a direct relation between points and wins in tennis, that is, the record between Federer and Nadal would actually stand even, at 10-10, instead of what it was in reality, 13-7—a record that would have accurately captured the real point differential between them. But because what matters in tennis isn’t—exactly—the total number of points you score, but instead the numbers of games and sets you win, it is entirely possible to score more points than your opponent in a tennis match—and still lose. (Or, the converse.)

The reason why that is possible, as Florida State University professor Ryan Rodenberg put it in The Atlantic not long ago, is due to “tennis’ decidedly unique scoring system.” (Actually, not unique, because as might be obvious by now match play golf is scored similarly.) In sports like soccer, baseball, or stroke play golf, as sports psychologist Allen Fox once wrote in Tennis magazine, “score is cumulative throughout the contest … and whoever has the most [or, in the case of stroke play golf, least] points at the end wins.” But in tennis things are different: “[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.” Just in the same way that what matters in tennis is the game, not the point, in match play golf all that matters is the hole, and not the stroke.

Such scoring systems breed Simpson’s Paradoxes: that is, results that don’t reflect the underlying value a scoring system is meant to reflect—we want our games to be won by the better player, not the lucky one—but instead are merely artifacts of the system used to measure. The point (ha!) can be shown by way of an example taken from a blog written by one David Smith, head of marketing for a company called Revolution Analytics, about U.S. median wages. In that 2013 post, Smith reported that the “median US wage has risen about 1%, adjusted for inflation,” since 2000. But was that statistic important—that is, did it measure real value?

Well, what Smith found was that wages for high school dropouts, high school graduates, high school graduates with some college, college graduates, and people with advanced degrees all fell over the same period. Or, as Smith says, “within every educational subgroup, the median wage is now lower than it was in 2000.” But how can it be that “overall wages have risen, but wages within every subgroup have fallen?” The answer is similar to the reason why Rafael had a 65 percent winning margin against Federer: although there are more college graduates now than in 2000, the wages of college graduates haven’t fallen (1.2%) as far as, say, high school dropouts (7.9%). So despite the fact that everyone is poorer—everyone is receiving lower wages, adjusted for inflation—than in 2000, mendacious people can say wages are actually up. Wages are up—if you “compartmentalize” the numbers in just the way that reflects the story you’d like to tell.

Now, while the story about American wages might suggest a connection to Ferguson—and it does—that isn’t the connection between the U.S. Amateur and Ferguson, Missouri, I’d like to discuss. That connection is this one: if the trouble about the U.S. Amateur is that it is conducted under match play—a format that permits Simpson’s Paradox results—and Simpson’s Paradoxes are, at heart, boundary disputes—arguments about whether to divide up the raw data into smaller piles or present them as one big pile—then that suggests the real link to Ferguson because the real issue behind Darren Wilson’s shooting of Michael Brown then isn’t racism—or at least, the way to solve it isn’t to talk about racism. Instead, it’s to talk borders.

After Ferguson police officer Darren Wilson shot Michael Brown last August, the Department of Justice issued a report that was meant, as Zoë Carpenter of The Nation wrote this past March, to “address the roots of the police force’s discriminatory practices.” That report held that those practices were not “simply the result of racist cops,” but instead stemmed “from the way the city preys on residents financially, relying on the fines that accompany even minor offenses to balance its budget.” The report found an email from Ferguson’s finance director to the town’s police chief that, Carpenter reported, said “unless ticket writing ramps up significantly before the end of the year, it will be hard to significantly raise collections next year.” The finance director’s concerns were justified: only slightly less than a quarter of Ferguson’s total budget was generated by traffic tickets and other citations. The continuing operation of the town depends on revenue raised by the police—a need, in turn, that drives the kind of police zealotry that the Department of Justice said contributed to Brown’s death.

All of which might seem quite far from the concerns of the golf fans watching the results of the matches at the U.S. Amateur. Yet consider a town not far from Ferguson: Beverly Hills, Missouri. Like Ferguson, Beverly Hills is located to the northwest of downtown St. Louis, and like Ferguson it is a majority black town. But where Ferguson has over 20,000 residents, Beverly Hills has only around 600 residents—and that size difference is enough to make the connection to the U.S. Amateur’s format of play, match play, crystalline.

Ferguson after all is not alone in depending so highly on police actions for its revenues: Calverton Park, for instance, is another Missouri “municipality that last fiscal year raised a quarter of its revenue from traffic fines,” according to the St. Louis Post-Dispatch. Yet while Ferguson, like Calverton Park, also raised about a quarter of its budget from police actions, Beverly Hills raised something like half of its municipal budget on traffic and other kinds of citations, as a story in the Washington Post. All these little towns, all dependent on traffic tickets to meet their budgets; “Most of the roughly ninety municipalities in St. Louis County,” Carpenter reports in The Nation, “have their own courts, which … function much like Ferguson’s: for the purpose of balancing budgets.” Without even getting into the issue of the fairness of property taxes or sales taxes as a basis for municipal budgeting, it seems obvious that depending on traffic tickets as a major source of revenue is poor planning at best. Yet without the revenue provided by cops writing tickets—and, as a result of Ferguson, the state of Missouri is considering limiting the percentage of a town’s budget that can be raised by such tickets, as the St. Louis Dispatch article says—many of these towns will simply fail. And that is the connection to the U.S. Amateur.

What these towns are having to consider in other words is, according to the St. Louis Post-Dispatch, an option mentioned by St. Louis County Executive Steve Stenger last December: during an interview, the official said that “the consolidation of North County municipalities is what we should be talking about” in response to the threat of cutting back reliance on tickets. Small towns like Beverly Hills may simply be too small: they create too little revenue to support themselves without a huge effort on the part of the police force to find—and thus, in a sense, create—what are essentially taxable crimes. The way to solve the problem of a “racist” police department, in other words, might not be to conduct workshops or seminars in order to “retrain” the officers on the frontline, but instead to redrawn the political boundaries of the greater St. Louis metropolitan area.

That, at least, is a solution that our great-grandparents considered, as an article by writer Kim-Mai Cutler for Tech Crunch this past April remarked. Examining the historical roots of the housing crisis in San Francisco, Cutler discovered that in “1912, a Greater San Francisco movement emerged and the city tried to annex Oakland,” a move Oakland resisted. Yet as a consequence of not creating a Bay-wide government, Cutler says, “the Bay Area’s housing, transit infrastructure and tax system has been haunted by the region’s fragmented governance” ever since: the BART (Bay Area Regional Transit) system, for example, as originally designed “would have run around the entire Bay Area,” Cutler says, “but San Mateo County dropped out in 1961 and then Marin did too.” Many of the problems of that part of Northern California could be solved, Cutler thusly suggests via this and other instances—contra the received wisdom of our day—by bigger, not smaller, government.

“Bigger,” that is, in the sense of “more consolidated”: by the metric of sheer numbers, a government built to a larger scale might not employ as many people as do the scattered suburban governments of America today. But what such a government would do is capture all of the efficiencies of economies of scale available to a larger entity—thus, it might be in a sense smaller than the units it replaced, but definitely would be more powerful. What Missourians and Californians—and possibly others—may be realizing then is that the divisions between their towns are like the divisions tennis makes around its points, or match play golf makes around its strokes: dividing a finite resource, whether points or strokes or tax dollars (or votes), into smaller pools creates what might be called “unnatural,” or “artificial,” results—i.e., results that inadequately reflect the real value of the underlying resource. Just like match play can make Ernie Els’ 75 look better than Pedro Larrazabal’s 68, or tennis’ scoring system can make Rafael Nadal look much better than Federer—when in reality the difference between them is (or was) no more than a sliver of a gnat’s eyelash—dozens of little towns dissipate the real value, economic and otherwise, of the people that inhabit a region.

That’s why when Eric Holder, Attorney General for the United States, said that “the underlying culture” of the police department and court system of Ferguson needs to be reformed, he got it exactly wrong. The problems in St. Louis and San Francisco, the evidence suggests, are created not because government is getting in the way, but because government isn’t structured correctly to channel the real value of the people: scoring systems that leave participants subject to the vagaries of Simpson’s Paradox results might be perfectly fine for games like tennis or golf—where the downsides are minimal—but they shouldn’t be how real life gets scored, and especially not in government. Contra Holder, the problem is not that the members of the Ferguson police department are racists. The problem is that the government structure requires them, like occupying soldiers or cowboys, to view their fellow citizens as a kind of herd. Or, to put the manner in a pithier way: A system that depends on the harvesting of sheep will turn its agents into wolves. Instead of drowning the effects of racism—as a big enough government would through its very size—multiplying struggling towns only encourages racism: instead of diffusing racism, a system broken into little towns focuses it. The real problem of Ferguson then—the real problem of America—is not that Americans are systematically discriminatory: it’s that the systems used by Americans aren’t keeping the score right.

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This Pitiless Storm

Poor naked wretches, whereso’er you are,
That bide the pelting of this pitiless storm,
How shall your houseless heads and unfed sides,
Your loop’d and window’d raggedness, defend you,
From seasons such as these?
The Tragedy of King Lear Act III, Scene 4

“Whenever people talk to me about the weather,” the Irish writer Oscar Wilde once remarked, “I always feel quite certain that they mean something else.” As it happens, the weather at this year’s British Open has been delayed by high winds and will not be finished with the regulation 72 holes until Monday at the earliest. Which raises a question: why does the Open need to finish all 72 holes? The answer concerns something called a “Simpson’s Paradox”—an answer that also demonstrates just how talk about the weather at the British Open is in fact talk about something else. Namely, the 2016 American presidential election.

To see how, it’s first necessary to see the difference between the British Open and other professional golf tournaments, which are perfectly fine with shortening themselves. Take for instance the 2005 Northern Trust Open in Los Angeles: Adam Scott won in a playoff against Chad Campbell after the tournament was shortened to 36 holes due to weather. In 2013, the Tournament of Champions at Kapalua in Hawaii was “first cut to 54 holes because of unplayable conditions over the first two days,” according to Reuters, and was under threat of “being further trimmed to 36 holes.” The same story also quoted tour officials as saying “the eventual champion would wind up with an ‘unofficial win’” were the tournament to be shortened to 36 holes. (As things shook out they did end up completing 54 holes, and so Dustin Johnson’s win officially counted.) In a standard PGA tournament then, the “magic number” for an “official” tournament is 54 holes. But if so, then why does the Open need 72?

To answer that, let’s take a closer look at the standard professional golf tournament. Most such tournaments are conducted according to what the Rules of Golf calls “stroke play”: four rounds of golf, or 72 holes, at the end of which the players who have made it that far add up their scores—their number of strokes. The player with the lowest score, it may seem like it goes without saying, wins. But it does need to be said—because that isn’t the only option.

Many amateur tournaments after all, 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, 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, the Accenture Match Play tournament—the only tournament on the PGA Tour to be held under match play rules. The 2014 edition (held at the Dove Mountain course near Tucson, Arizona), had some results that demonstrate just how different match play is than stroke play, as Doug Ferguson of the Associated Press observed. “Pablo Larrazabal shot a 68 and was on his way back to Spain,” Ferguson noted about the first day’s results, 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 was arguably the better player at this tournament—at least, if you consider the “better player” to be the one who puts his ball in the hole most efficiently.

Such a result might seem unfair—but why? It could be argued that while shooting a lower number might be what stroke play golf is, that isn’t what match play golf is. In other words, Larrazabal obviously wasn’t better at whatever it was that this tournament measured: if Larrazabal couldn’t beat his opponent, while Els could, then clearly Els deserved to continue to play while Larrazabal did not. While you might feel that, somehow or other, Larrazabal got jobbed, that’s merely a sentimental reaction to what ought to be a hardhearted calculation: maybe it’s true that under stroke play rules Larrazabal would have won, but that wasn’t the rules of the contest at Dove Mountain. In other words, you could say that golfing ability was, in a sense, socially constructed: what matters isn’t some “ahistorical” ability to golf, but instead how it is measured.

Here’s the $64,000 question a guy named Bill James might ask in response to such an argument, however (couched in terms of baseball players): “If you were trying to win a pennant, how badly would you want this guy?” In other words, based on the evidence presented, what would you conclude about the respective golf ability of Els and Larrazabal? Wouldn’t you conclude that Larrazabal is better at the task of putting his ball in the hole, and that the various rule systems that could be constructed around that task are merely different ways of measuring that ability—an ability that pre-existed those systems of measurement?

“We’re not trying to embarrass the best players in the game,” said Sandy Tatum at the 1974 U.S. Open, the so-called Massacre at Winged Foot: “We’re trying to identify them.” Scoring systems in short should be aimed at revealing, not concealing, ability. I choose Bill James to make the point not just because the question he asks is so pithy, but because he invented an equation that is designed to discover underlying ability: an equation called the Pythagorean Expectation. That equation, in turn, demonstrates just why it is so that match play and stroke play are not just different—yet equally valid—measures of playing ability. In so doing, James also demonstrates just why it is that the Open Championship requires that all 72 holes be played.

So named because it resembles so closely that formula, fundamental to mathematics, called the Pythagorean Theorem, what the Pythagorean Expectation says is that the ratio of a team’s (or player’s) points scored to that team’s (or player’s) points allowed is a better predictor of future success than the team’s (or player’s) ratio of wins to losses. (James used “runs” because he was dealing with baseball.) More or less it works: as Graham MacAree puts it on the website FanGraphs, using James’ formula makes it “relatively easy to predict a team’s win-loss record”—even in sports other than baseball. Yet why is this so—how can a single formula predict future success at any sport? It might be thought, after all, that different sports exercise different muscles, or use different strategies: how can one formula describe underlying value in many different venues—and thus, incidentally, demonstrate that ability can be differentiated from the tools we use to measure it?

The answer to these questions is that adding up the total points scored, rather than the total games won, gives us a better notion of the relative value of a player or a team because it avoids something called the “Simpson’s Paradox”—which is what happens when, according to Wikipedia, it “appears that two sets of data separately support a certain hypothesis, but, when considered together, they support the opposite hypothesis.” Consider what happens for example when we match Ernie Els’ 75 to Pablo Larrazabal’s 68: if we match them according to who won each hole, Els comes out the winner—but if we just compared raw scores, then Larrazabal would. Simpson’s Paradoxes appear, in short, when we draw the boundaries around the raw data differently: the same score looks different depending on what lens is used to view it—an answer that might seem to validate those who think that underlying ability doesn’t exist, but only the means used to measure it. But what Simpson’s Paradox shows isn’t that all boundaries around the data are equal—in fact, it shows just the opposite.

What Simpson’s Paradox shows, in other words, is that drawing boundaries around the data can produce illusions of value if that drawing isn’t done carefully—and most specifically, if the boundaries don’t capture all of the data. That’s why the response golf fans might have to the assertion that Pablo Larrazabal is better than Ernie Els proves, rather than invalidates, the argument so far: people highly familiar with golf might respond, “well, you haven’t considered the total picture—Els, for instance, has won two U.S. Opens, widely considered to be the hardest tournament in the world, and Larrazabal hasn’t won any.” But then consider that what you have done just demonstrates the point made by Simpson’s Paradox: in order to say that Els is better, you have opened up the data set; you have redrawn the boundaries of the data in order to include more information. So what you would have conceded, were you to object to the characterization of Larrazabal as a better golfer than Els on the grounds that Els has a better overall record than Larrazabal, is that the way to determine the better golfer is to cast the net as wide as possible. You have demanded that the sample size be increased.

That then is why a tournament contested over only 36 holes isn’t considered an “official” PGA tournament, while 54 holes isn’t enough to crown the winner of a major tournament like the Open Championship (which is what the British Open is called when it’s at home). It’s all right if a run-of-the-mill tournament be cut to 54 holes, or even 36 (though in that case we don’t want the win to be official). But in the case of a major championship, we want there to be no misunderstandings, no “fluky” situations like the one in which Els wins and Larrazabal doesn’t. The way to do that, we understand, is to maximize chances, to make the data set as wide as possible: in sum, to make a large sample size. We all, I think, understand this intuitively: it’s why baseball has a World Series rather than a World Championship Game. So that is why, in a major championship, it doesn’t matter how long it takes—all the players qualified are going to play all 72 holes.

Here I will, as they say in both golf and baseball, turn for home. What all of this about Simpson’s Paradoxes means, at the end of the day, is that a tournament like the Open Championship is important—as opposed to, say, an American presidential election. In a presidential election as everyone knows, what matters isn’t the total numbers of votes a candidate wins, but how many states. In that sense, American presidential elections are conducted according to what, in golf, would be considered match play instead of stroke play. Now, as Bill James might acknowledge, that begs the question: does that process result in better candidates being elected?

As James might ask in response: would you like to bet?

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:

1995

1996

1997

Totals

Derek Jeter

12/48

.250

183/582

.314

190/654

.291

385/1284

.300

David Justice

104/411

.253

45/140

.321

163/495

.329

312/1046

.298

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 joeposnanski.com.) 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.