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The FIFA World Cup, and specifically the penalty-kick-heavy match I watched a few weeks ago between Colombia and England, reminded me of the first chapter of a book I read last year called How to Think Like a Freak, by Steven Levitt and Stephen Dubner, the hosts of the Freakonomics podcast. The first chapter was about where to aim a penalty kick to maximize the chance of a goal. Because of players’ proximity to the goal when taking a penalty kick, goalies have little option aside from guessing where the player will kick it, and then diving in that direction. Logically, then, both players and goalies should analyze each others’ tendencies to better perform in a penalty kicking scenario.
Subsequent analysis found that when goalies leap, they typically go to the kicker’s strong side (left side for a right-footed kicker and right side for a left-footed kicker) — 57% of the time, in fact. The other 41% of the time, they dive to the kicker’s weak side. It would seem obvious, then, where players should kick the ball: their own weak side. But that strategy ignores that 57% + 41% is only 98%. What happens the other 2% of the time, when goalies dive neither left nor right? The answer is obvious: 2% of the time when trying to block a penalty kick, goalies stay right where they are. So if you’re a soccer player taking a penalty kick, where does it make the most sense to kick the ball? Easy answer, given the data: right at the goalie. Ninety-eight times out of 100, you’ll score.
Now, in a static world, it would be that simple. Goalies’ and players’ strategies would be fixed, with neither adjusting for the behavior of the other. But that’s not how it works. Especially in professional sports, where teams’ and players’ tendencies are both studied and observed prior to competition, players learn from one another. Thus, their behaviors — and specifically those displayed in a one-versus-one scenario, such as a penalty kick — evolve in response to one another. If players taking penalty kicks started routinely kicking the ball at the goalie, goalies everywhere would adjust, and strategies on both sides would find a new equilibrium — but not before those taking penalty kicks would’ve scored just a few more times than usual.
Despite the math, however, players haven’t taken advantage of what would on the surface appear to be an extremely compelling opportunity. The question, then, is why?
At the end of the first chapter of How to Think Like a Freak, the authors explain: there is a higher level of risk — personal and professional — associated with aiming a kick directly at the goalie.
The 57% strong side, 41% weak side, 2% center equilibrium is what is known in economics as a Nash Equilibrium. Essentially, in a two player game, like a penalty kick, each player will play the strategy that maximizes their payoff based on what they think the other player is most likely to do. In soccer, however, this equilibrium is complicated by the presence of highly irrational — and sometimes violent — spectators. There would be no way to explain to a fan whose team had just lost the World Cup, after all, that their star player’s decision to aim their penalty kick directly at the goalie was actually hyper-rational, and not utterly condemnable.
If that same player had aimed top-right and the goalie had blocked it, well, the goalie got lucky (it is rarer for goalies to save penalty kicks than allow them), and at least the kicker was trying to make it. That, while tragic, would be forgiven. If the kicker aimed at the goalie, on the other hand, and the goalie remained there and blocked the kick, all bets would be off. The player who took the kick would be subject to enormous amounts of media controversy, if not hate mail and potential death threats. They kicked it right at the goalie! He didn’t even have to do anything to block it! Left unacknowledged, of course, would be the fact that, mathematically, the kicker’s decision was at worst defensible, and at best irrefutably logical.¹
The numbers make sense. The payoff of kicking the ball in a certain direction lies not just in the upside of scoring, but in the downsides of missing, and more important, the downsides of missing in a certain way. So it is that the Nash equilibrium — 57% strong side, 41% weak side, 2% center — has not changed. This is another characteristic of Nash Equilibria: once in one, no player has any incentive to change their strategy. Barring a major change in soccer rules, then — say, a reduction in the size of the goal, or a lengthening of the distance between the kicker and goalie in a penalty kick scenario — none of them will.
Modern society seems to overlook this truth — that strategies are constantly reaching new equilibriums as they adjust in accordance with reactions to those very strategies.
Let’s take a real world example: Chipotle. At this point, everyone has probably had the experience of asking for double meat in their burrito or burrito bowl. What many have likely noticed is that when you ask the person behind the counter to give you double meat prior to having been given a scoop, you end up getting less than if you ask for double meat after being given the first scoop. The advice that dots Twitter and uncurated Facebook feeds, then — assuming you want to maximize your meat consumption at Chipotle — is to do the latter: ask for a type of meat, watch the person behind the counter give you one scoop, and then ask for a second after the first is already in the bowl. Doing so forces the person behind the counter to “double” what’s already in the bowl: a scoop that is most certainly larger than what you would’ve gotten if you had asked for double meat right off the bat.
Markets react, and we live in one.
I remember reading this advice the first time and thinking: wow, how brilliant, I’ll do this forever and they’ll never know. The reality is this: like kicking the ball directly at a goalie when taking a penalty kick, this advice only works for a limited time, like any strategy in a dynamic marketplace. I know this because every time I go to a Chipotle now (and perhaps this is because I’m a tall, could-crush-a-lot-of-food-quickly looking guy), I’m asked before the person starts scooping the meat whether I’m getting single or double.
In other words, they’re catching on.
Now, this is not to say that I couldn’t just say single when asked how much meat I wanted, get a scoop, then pretend to have changed my mind and get double. I easily could. But Chipotle’s deliberate creation of that friction — my having to ask — indicates two things. First, they have clearly picked up on the trick — likely first through employee reports, and then through balance sheets indicating both shortages of and increased demand for meat — and informed employees accordingly. Second, like those taking penalty kicks have settled on the Nash equilibrium of aiming penalty kicks at certain parts of the net certain percentages of the time (57% strong side, 41% weak side, 2% center), so too does Chipotle’s reaction to consumer savviness mean that now, the Nash equilibrium quantity of meat given at Chipotle has dropped. This is a direct result of Chipotle servers’ understanding that they have what economists call a dominant strategy. They’re able to play that strategy regardless of what they think I, as a consumer, will do. Knowing that I might ask for a second scoop of meat after seeing them put in the first, their dominant strategy is to ask, “Single or double?” prior to giving the first scoop of meat.
Put another way, markets react, and we live in one.
None of this should come as a surprise, but it does offer insight into the world we live in, especially since the world we live in now is more reliant on signals of reality than on reality itself. As I wrote in “The Instagram Generation”:
That we were gifted Instagram after years of being conditioned to emphasize the cosmetic was almost poetic; if the universal obsession with the cosmetic was a stick of dynamite, then Instagram was the spark at the end of the fuse. More than that, Instagram is a reflection of the same truth our generation has been getting stuffed down its throat for years: that is, there is more riding on the image you portray than on your actual character.
There are countless examples of cosmetic alterations that make us more compelling candidates for literally anything — high school, college, employment, marriage, etc. We can take a position on student council in middle school, understanding that when we apply to high school, no one will care about that position aside from its utility as a filter, thinking: okay, he displays leadership/competence, or at worst has parents that know those things matter, which probably signals his own competence — and also solidifies them/him as donors down the road — them with the money, him with the trust fund. It is only fitting that the school will turn around and reinvest that money into more signals: a new gym, swimming pool, or building — signals meant only to attract more like those responsible for them.
It gets especially exciting as an individual. We can visit foreign countries in high school to signal worldliness and a willingness to explore and give back. Best of all, though, we can bring a picture back with us, ideally one in which the people in it with us have a skin color different than our own. We can wear suits to an interview, make eye contact, and prepare literally one question about the company and poof, we’re better candidates than 99.9% of people out there. I won’t pretend to know what those trying to get married would signal; that isn’t me, yet. But I can imagine it would be something that would signal interest in marriage. And if none of this works, there are always connections. What is being well-connected, after all, if not having well-established signals of your own competence in high places?
All of these signals, however, will only work for a time.
Market forces are constantly pushing and pulling on one another, creating new equilibriums moment by moment.
On a soccer pitch, the predictability of the rules creates similarly predictable outcomes, like the Nash equilibrium outlined above. Our world, on the other hand, has an impossibly broad spectrum of competition and thus evolves very quickly everywhere. Unlike the static equilibrium of penalty kick placement, market forces are constantly pushing and pulling on one another, creating new equilibriums moment by moment. This, too, is how once explanatory signals are perverted so quickly in the real world; once something is well-understood to be a signal — like a student council position — it no longer is.
Just as the goalie who fails to recognize a change in penalty kickers’ strategies will not last long on any soccer team, let alone in the World Cup, individuals and businesses who do not respond to the oscillations, emergences, and outright disappearances of equilibriums will not last long in a dynamic marketplace.
I wonder about the student council presidents and high school trip-takers of yesteryear. It is possible that they assumed those positions and experiences because they genuinely wanted to. It is also possible that they didn’t. It makes me wonder if today, we are stuck in a sea of both, unable to tell which is which, unable to tell even when we are either one, searching desperately for a signal of anything — just one — yet to be distorted.
¹ In early 2015, economist Justin Wolfers penned an op-ed in the New York times titled “Game Theory Says Pete Carroll’s Call at Goal Line is Defensible.” The article was about Pete Carroll’s decision to pass the ball three times on the one-yard line in the 2015 Super Bowl against the New England Patriots when his team was down by four with 26 seconds left, even as they possessed one of the most lethal running backs in the history of the game: Marshawn Lynch. It is a perfect example of a decision that made perfect sense “in theory,” but appeared absurd in practice, especially since Russell Wilson’s third down pass — also on the one-yard line — was intercepted. The Seahawks would go on to lose that Super Bowl, 28–24.