Game Theory | A Tribute to Roger Federer
Tennis fans mourn as Roger Federer announced his official retirement from the game of tennis. His style of play oozed elegance. His serve was as effortless as it was effective. With a casual slick of his hair, Federer delivered serves that nailed the court with precision and left his opponents bewildered.
Federer’s tactic of serving without predictability is a widely-used technique in tennis. One can consider this strategy through the lens of Game Theory. Nash Equilibrium posits that both players in a game of tennis will seek to choose the optimal strategy against their opponent in order to maximize payoff. In line with this theory, tennis players are advised to keep their serve unpredictable for each point. If the receiver perceives a pattern in their opponent’s serve game, they can then predict this pattern and exploit it.
John Wooder’s paper, Nash at Wimbledon, examines data from over 3000 professional tennis matches through Hawk-Eye, a computerized ball tracking system. It models each point as a 2×2 matrix, where the server chooses to play to the receivers right or left, and the receiver simultaneously chooses between overplaying to the right or left. πsr denotes the probability the server wins (serving in direction s, receiver overplaying in direction r). Since only one player can win any given point, 1- πsr denotes the probability of the receiver winning the point. The study takes into account two extra factors that affect payoff: whether it’s first or second serve, and whether it’s from the ad or deuce court.
Receiver
| Server (below) | L | R |
| L | π (LL) | π (LR) |
| R | π (RL) | π (RR) |
The study finds that the behavior of higher-ranked players conform closely to Nash’s hypothesis. In order to maximize success, higher-ranked players serve with less serial correlation, indicating they base their serving decision independent from previous points. This notion is reinforced by Dr. Leonidas Spilioulous, an expert in tennis data analytics. The following graph displays the predictability of the first serve of tennis players in relation to the previous point. If the percentage is near 0, this means that the player’s serve is independent of the previous event.
Tennis is a game that is equally as mental as it is physical. Federer’s calm, composed nature is evident in his serve game. His predictability percentage from both court sides is near 0. Regardless of the outcome of the previous point, he approaches each serve with indifference and level-headedness, contributing to his immense success.
Perhaps, Federer’s true strength lies not just in the physical game, but in the power of his mind. The Swiss legend leaves behind an indelible legacy that will be remembered forever.
Sources
- http://www.johnwooders.com/papers/NashAtWimbledon.pdf
- https://tennishead.net/exclusive-roger-federer-novak-djokovic-weakness-revealed-by-maths-professor/