Penalties, Serves, and Being Random in Sports
This summer, I, along with billions of others, watched an event that is possibly one of the purest representations of game theory in real life. It’s the soccer Euro 2020 (in 2021) finals, and after 120+ minutes of play, the game has ended in a tie, meaning that the winner will be settled by penalties. As I waited in anticipation for the shootout to begin, I wondered what was the best strategy to win the shootout (spoiler alert: it wasn’t England’s). Fast forward two months, and I’m sitting in an auditorium learning more about this exact situation.
In lecture, we modeled soccer penalties using a simple non-cooperative zero-sum game. The players in our game are the penalty taker and the goalie, and their choices are the direction in which to aim/dive. In our model, there is no Nash equilibrium in pure strategies. Of course, this is just as true in real high-level penalty shootouts. If penalty-takers knew that the goalie was going to dive left for every single penalty, they would score every time. Thus, the optimal approach to penalties is to use a mixed strategy based on past data. I initially scoffed at the applicability of this idea in real life, because while great in theory, mixed strategies require that the players incorporate randomness into their decision-making, and humans usually aren’t good entropy sources. For me at least, when I play sports, I know that I alternate my choices a lot more than I should if I’m trying to be random. If I go left on one occasion, I’ll naturally want to go right for the next. You can test whether you yourself are a good source of entropy here. Upon re-watching this penalty shootout after learning more about game-theory in class, I wondered if professional athletes are any better than the regular human at being ‘random’. ‘
This question is answered in Ignacio Palacios-Huerta’s paper Professionals Play Minimax. Palacios-Huerta uses real data to determine whether professional soccer players are to be serially independent enough to effectively incorporate randomness into their decision-making. Against all odds, he finds that “professional soccer players are indeed able to generate random sequences; they neither switch strategies too often or too little” (Palacios-Huerta 410).
However, I still wasn’t fully convinced of professional athletes’ capability to be random. Since soccer players don’t generally take two penalties in the same match, they have more time to reset and perhaps even to incorporate randomness artificially by using a computer to truly randomly decide where to shoot. Additionally, there is less of a chance that serial independence comes into play since penalties are often one-offs for each taker. I wanted to examine a situation in which so many zero-sum games were repeated that it was impossible to be random using anything besides the human mind. Luckily for me, this situation exists in tennis.
In professional tennis, servers usually serve out wide or down the middle, and receivers can choose to lean one way or the other. There are around 150 serves per match, so it is impossible for players to memorize serve direction sequences that would make them random. This serves as the perfect scenario to test whether professional athletes can truly be serially independent and optimally follow mixed strategies. In their paper Nash at Wimbledon, Gauriot, Page, and Wooders use precise data from nearly 500,000 professional tennis serves to determine whether tennis players can be random in their serves. They find that tennis players, particularly the men, can be random and serially independent, thus following the optimal mixed strategy. They explain that men’s serves are generally faster than women’s serves, meaning that female tennis receivers who may not be truly random in the direction they lean are punished less often because they have more time to react. The authors also find that higher-ranked tennis players generally follow the Nash Equilibrium more closely.
To me, these studies simply display the amount of mental intuition it takes to be a top athlete. While these players probably aren’t studying the intricacies of game theory and mixed strategies in their free time, they have intuitively grasped these concepts through enough experience and are able to perfectly execute them.
Source 1: http://www.palacios-huerta.com/docs/professionals.pdf
Source 2: http://www.johnwooders.com/papers/NashAtWimbledon.pdf