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.