Soccernomics (19 page)

“not even blinking,” in the Spanish football phrase. Then, exactly as Ignacio had recommended, Cech dived to his right and duly saved Ronaldo’s shot. Ignacio recalled later, “After that, I started to believe 126

that they were following the advice quite closely.” As for his wife, “I think she was a bit shocked.”

What’s astonishing—though it seems to have passed unnoticed at the time—is what happened after that. Chelsea’s next four penalty takers, Frank Lampard, Ashley Cole, John Terry, and Salomon Kalou,
all
hit the ball to Van der Sar’s left, just as Ballack and Belletti had done. In other words, the first six Chelsea kicks went to the same corner.

Ashley Cole was the only one of the six who partly disregarded Ignacio’s advice. Cole was left-footed, so when he hit the ball to Van der Sar’s left, he was shooting to his own “natural side”—the side that Ignacio had said Van der Sar tended to choose. Indeed, the Dutchman chose correctly on Cole’s kick, and very nearly saved the shot, but it was well struck, low (as Ignacio had recommended), and just wriggled out of the keeper’s grip. But all Chelsea’s right-footed penalty takers had obeyed Ignacio to the letter and kicked the ball to their “unnatural side,” Van der Sar’s left.

So far, Ignacio’s advice had worked very well. Much as the economist had predicted, Van der Sar had dived to his natural side four times out of six. He hadn’t saved a single penalty. Five of Chelsea’s six kicks had gone in, while Terry’s, as the whole world knows, flew out off the post with Van der Sar in the wrong corner.

But after six kicks, Van der Sar, or someone else at Manchester United, figured out that Chelsea was pursuing a strategy. Admittedly, the keeper didn’t quite get its strategy right. Wrongly but understand-ably, he seems to have decided that Chelsea’s strategy was to put all the kicks to his left. After all, that’s where every kick he had faced up to that point had gone.

As Anelka prepared to take Chelsea’s seventh penalty, the gangling keeper, standing on the goal line, extended his arms to either side of him.

Then, in what must have been a chilling moment for Anelka, the Dutchman pointed with his left hand to the left corner. “That’s where you’re all putting it, isn’t it?” he seemed to be saying. (This is where books fall short as a medium. We urge you to watch the shoot-out on YouTube.) T H E E C O N O M I S T ’ S F E A R O F T H E P E N A LT Y K I C K

127

Now Anelka had a terrible dilemma. This was game theory in its rawest form. United had come pretty close to divining Chelsea’s strategy: Ignacio had indeed advised right-footed kickers like Anelka to put the ball to Van der Sar’s left side.

So Anelka knew that Van der Sar knew that Anelka knew that Van der Sar tended to dive right against right-footers. What was Anelka to do? He decided to avoid the left corner, where he had presumably planned to put the ball. Instead, he kicked to Van der Sar’s right. That might have been fine, except that he hit the ball at midheight—exactly the level that Ignacio had warned against. Watching the kick on TV, Ignacio was “very upset.” Perhaps Anelka was at sea because Van der Sar had pressured him to change his plans at the last moment. Van der Sar saved the shot. Alex Ferguson said afterward, “That wasn’t an accident, his penalty save. We knew exactly where certain players were putting the ball.” Anelka’s decision to ignore Ignacio’s advice probably cost Chelsea the Champions League.

RANDOMIZATION: FRANCK RIBÉRY CRACKS GAME THEORY

Crib sheets like Lehmann’s might just work on penalty shoot-outs.

Many of the players who take kicks in a shoot-out aren’t regular penalty takers. (After Gareth Southgate missed England’s crucial kick at Euro ’96, his mother said that the last time he’d taken a penalty was three years before, and he’d missed that one, too.) These inferior penalty takers are not skilled or steady-headed enough to be able to vary their strategy. Quite likely, they will just aim for their favorite corner, hoping that their lack of a track record means the other side won’t know their preference.

But that is not how a good penalty taker—his team’s regular man—

thinks.

Suppose the good kicker always chose the same corner for his penalty (game theorists call this a “pure strategy”). It would be easy to oppose: if the kicker always kicks left, then the goalkeeper knows what to do.

Pure strategies don’t work for penalty taking. As Levitt and Company 128

found, “There are no kickers in our sample with at least four kicks who always kick in one direction.” Take that, Jens Lehmann.

Even a more complicated pure strategy does not work. For example, suppose the kicker always shoots in the opposite corner to the one he chose last time. Then a future opponent studying this player would discover the sequence—left, right, left, right, left, right—and with a bit of thought guess what comes next. The essence of good penalty taking is unpredictability: a good penalty taker will be one whose next penalty cannot be predicted with certainty from his history of penalty taking.

This is a particular kind of unpredictability. It does not mean that the kicker should go left half of the time and right half of the time.

After all, most kickers have a natural side, and favoring that side gives them a higher chance of scoring. But even if you naturally shoot to the keeper’s right, as most right-footed kickers do, sometimes you have to shoot to his left, just to keep him honest. In fact, if a kicker knows his chances of scoring for either corner of the net (depending also on which way the goalkeeper dives), he can choose the proportion of kicks to his natural side that maximizes the probability of scoring. A right-footed kicker won’t put 100 percent of his kicks to his natural “right”

side, because that would give the goalkeeper certainty. Even a small change, like kicking right only 99 percent of the time, would raise the chances of scoring considerably by creating uncertainty in the goalkeeper’s mind.

Kicking to the left half the time would leave the keeper very uncertain. However, it would also entail the kicker hitting many poor shots to his unnatural side. So the kicker does best by hitting somewhere over half his kicks to his natural right side.

Likewise, we can calculate the proportion of times a goalkeeper should dive left or right. (Note that we are assuming the goalkeeper cannot know which way the ball is going before he decides which way to dive.) Kickers and keepers who mix it up like this are pursuing what game theorists call “mixed strategies.”

T H E E C O N O M I S T ’ S F E A R O F T H E P E N A LT Y K I C K

129

Mixed strategies are peculiar because they require the actor to incorporate randomness into decision making. Should I go to the pub or the cinema? A mixed strategy requires me to toss a coin, which sounds odd, since one might expect that I prefer one to the other. With a mixed strategy, you let the coin make the decision for you.

Game theorists have wondered for years whether people in the real world follow mixed strategies. They have found in tests that people tend not to use mixed strategies even when it is profitable for them to do so.

In fact, our behavior seems to fall short of mixed play in a very specific way: in most cases, our sequence of choices is predictable, because people tend to do the opposite of what they have done in the past. For instance, they choose first left, then right, then left, then right, left, right, left, right, confusing
change
with
randomness
. These guinea pigs would not make good penalty takers.

Eventually, game theorists began to test mixed strategies in the natural laboratory of penalty taking. Years before Ignacio Palacios-Huerta advised Chelsea, he collected a database of 1,417 penalties taken between 1995 and 2000. First, he calculated the proportion of successful kicks based on whether the kicker went to his natural side (left or right). The success rate was 95 percent if the kicker went to his natural side and the goalkeeper went to the opposite side (the remaining 5 percent of kicks missed the goal). The success rate was 92

percent if the kicker went to his “unnatural” side and the goalkeeper went to his natural side. Obviously, the kicker’s success rates were lower if the keeper chose correctly: a scoring rate of 70 percent if both keeper and kicker went to the kicker’s natural side, and 58 percent if both went to the other side.

Using these figures, Ignacio calculated the optimal mixed-strategy choices for each player. To maximize his chances of scoring, an imaginary penalty taker would have to hit 61.5 percent of his kicks to his natural side, and 38.5 percent to the other side. In reality, the penalty takers Ignacio observed got pretty close to this: they hit 60 percent to their natural side, and 40 percent the other way.

130

A keeper’s best strategy (if he insists on diving rather than standing still) is to dive to the kicker’s natural side 58 percent of the time and to the other side 42 percent of the time. The actual figures, Ignacio found, were scarily close: 57.7 percent and 42.3 percent. Levitt’s team, using a different database of penalties, found that keepers went to the right 57

percent of the time. So it looks as if keepers as well as penalty takers really do follow mixed strategies.

But what we most want to know are the choices of individual kickers and goalkeepers, not the overall averages. Ignacio studied twenty-two kickers and twenty goalkeepers, each of whom was involved in more than thirty penalties in his database. Again, Ignacio calculated the success rates depending on the side the kicker and goalkeeper chose, and calculated the frequencies in each direction that would maximize the chances of success for kickers and keepers.

In real life, the actual frequencies the players observed were indistinguishable from the best mixed-strategy choices in more than 95 percent of cases. We can say with a high degree of confidence that penalty takers and goalkeepers really do use mixed strategies. Levitt’s paper found the same thing: except for the bizarre keeper who always dived left, almost all the other kickers and keepers played mixed strategies.

Finally, Ignacio tested the most important question of all: are soccer players capable of constructing a truly random sequence in their penalty-taking decisions, as the mixed-strategy theory requires? Careful statistical testing showed that indeed they are. In other words, it is impossible to predict which way a regular penalty taker will kick based on his history of kicks. Each time he chooses his corner without any reference to what he did the last time.

Randomization of penalties is a completely logical theory that against all odds turns out to be true in practice. As long as the penalty taker is a pro, rather than some terrified Southgateian innocent roped in for a job that he doesn’t understand, lists like Lehmann’s are useless.

All this shows the extraordinary amount of subconscious thought that goes into playing top-level soccer. Previous studies in game theory T H E E C O N O M I S T ’ S F E A R O F T H E P E N A LT Y K I C K

131

had shown that people could construct random sequences if the problem was first explained to them in some detail. Nobody is suggesting that soccer players have sat at home reading up on mixed-strategy equi-libria. Rather, the best players intuitively grasp the truth of the theory and are able to execute it. That is what makes them good players.

Franck Ribéry is the designated penalty taker for Bayern Munich and France. Needless to say, the scar-faced little playmaker places his kicks according to a randomized mixed strategy. But more than that, one of his former managers explains, even once Ribéry has embarked on his jagged hither-and-thither run-up,
he himself
does not know which corner he will choose. When the born economist Arsène Wenger was told this, he gushed with admiration.

As good a player as Ribéry is, he might do even better as a game theorist.

THE SUBURBAN NEWSAGENTS

City Sizes and Soccer Prizes

The scene: the VIP room at the Athens Olympic Stadium, a couple of hours before the Champions League final of 2007 between Milan and Liverpool kicks off. Michel Platini and Franz Beckenbauer are being but-tonholed every couple of yards by other middle-aged men in expensive suits. There is a crush at the buffet, and another across the room, where a familiar silver cup with “big ears” stands on a dais. You line up, assume a conquering pose beside the Champions League trophy, and grin. Nice young ladies from UEFA slip the picture into a frame for you.

An Englishman watching the scene, a soccer official, confides that he first got this close to the cup thirty years ago. Where? In Bramcote, a suburb of Nottingham. One of Brian Clough’s brothers ran the local post office-cum-newsagents, and Clough himself would sometimes pop in and serve customers, or just stand behind the counter reading the papers. One Sunday morning when the future official went in with his grandfather, there was the European Cup freshly won by Forest, plunked on top of a pile of
Nottingham Evening Posts
. Behind it stood Brian Clough, holding an open newspaper in front of his face. He neither moved nor spoke, but he knew the boy would remember the scene 133

134

forever. The official remembers: “I was too young and shy to speak to the man, which I regret to this day.”

It’s odd to think of the game’s biggest club trophy ending up in a place like Bramcote (population 7,318). Yet it’s not that exceptional.

Provincial towns like Nottingham, Glasgow, Dortmund, Birmingham, or Rotterdam have all won European Cups, while the seven biggest metropolitan areas in Europe—Istanbul, Paris, Moscow, London, St. Petersburg, Berlin, and Athens—never have. This points to an odd connection between city size, capital cities, and soccer success. Here’s why Arsenal and Chelsea haven’t won the Champions League (but may soon).

GENERAL FRANCO’S TRANSISTOR RADIO:

THE ERA OF TOTALITARIAN SOCCER

The best measure of success in club soccer is a simple list: the names of the clubs that have won the European Cup since the competition began in 1956. Study this list, and you’ll see that the history of the European Cup breaks down into three periods.