Weekend Game Theory Musings: Just for (Penalty) Kicks

When I arrived at my home this past weekend, I was greeted with the familiar sound coming from my family room - lightning fast, British accented, and increasing pitch - Premier League commentators. My family has always been a soccer oriented family, with both of my younger siblings playing at a high level (not to brag, but), leaving me the odd non-soccer-player. I joined my brother on the couch to watch the climax of the game: penalty kicks.

To anybody who has watched soccer, penalty kicks are arguably the most nerve-wracking and dreaded outcome of a 90 minute match. Not only are the players exhausted, but the immense pressure of the one on one face-off is enough to make even the greatest players fail to deliver. This face-off can in fact be modeled as a game in an economic sense.

Both the kicker and the goalie must make decisions simultaneously - it takes on average 0.5 seconds for the ball to reach the goal, and 0.6 seconds for the goalie to move from the center to either side of the goal. The kicker must make their decision without giving any signal to the goalie, because of the power needed for the ball to hit the net such that it wouldn't be considered "an easy save." Consequently, for the goalie, their move will mean completely abandoning any hope of saving the ball should they have guessed incorrectly. When making the decision, both must take into account their own expectations of how the other will move.

What is interesting is that the preferred shooting foot of the kicker is not necessarily the optimal outcome for them. With sports player statistics being a public obsession and information readily available, it is not far-fetched to imagine that goalies at the professional level have an understanding of what side the kicker prefers/is more accurate with, and vice versa for the kicker knowing the goalie's stats. With this symmetric information, suppose we take a right-footed kicker: they are more likely to score on the right than the left, but the goalie may anticipate this preference and be more likely to save the ball by moving to the right. This would mean that on average, anywhere the kicker shot to would have an equally likely chance of resulting in a goal. What it comes down to, then, is how strategic the kicker can be in randomizing their shot location such that the goalie is not signaled and the kicker can be victorious.
This can have interesting economic implications when we consider the considerable monetary value that the players have which depend on the outcome of their matches, and the industry as a whole having incredible financial power. Logically, if penalty kicks will result in a random outcome, more money should be spent on gathering players who can score and defend effectively without resulting in PKs.

I'm hoping I can impart some of this wisdom on my teenage siblings over fall break, but I'm sure this will be another moment where the ignorant older sister tries to be relevant... I'm not ignorant, my opportunity costs of becoming fantastic at the sport are just much too high!