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Game, Set, Match: Game Theory in Tennis

The game of tennis is one of the oldest sports in the world and considered one of the most mental games out there, considering that in a singles match, two players are pitted against each other through at least two grueling sets. Each player has the chance to serve, and therefore each player also has the chance to receive. The question becomes whether we can use game theory to predict the behavior of each player on the court. More specifically, the simplest form of these predictive experiments has been to look at the initial serve of the player and the initial decision of the returning player, and the outcome of the point (who wins the point). Research has shown that although each player may have a stronger side (forehand or backhand) or able to serve better to one side, it is very rare to find a Pure-Strategy Nash Equilibrium between any two top players, and therefore each player must constantly be adapting his mixed-strategy Nash Equilibrium to fit prior information.

First we need to set up the 2 x 2 matrix as well as state a couple of assumptions. We assume that the only decision for the server and returner is to go left or right. This assumption is only partially valid, since the server can also decide to serve at the body of the returner, and can often do this effectively (i.e. Federer is known for his highly successful serve). Furthermore, the type of serve is not coded (i.e. whether it is a flat fast serve, or has some special topspin or slice spin). Again, this assumption takes away from some top players who have highly specialized serves and are capable of using them to their advantage. Furthermore, the matrix does not take into account what happens after the returning player has returned the ball. In other words, the point can possibly be over right after the serve (i.e. the server aces the returner, the return is unable to return, or possibly the server faults). The precise point about to be played is also not taken into consideration, although if it is a deuce point or an advantage point (game-point) definitely has influence on the decision of both players. Nonetheless, the matrix should look something as follows (borrowed from the article, the numbers inside the matrix stand for the probability that the Server wins the point):

Matrix

Interestingly, the analysis of ten championship matches between top players shows two interesting conclusions: that the winner of points can be predicted based on the decisions made both players and that players nonetheless switch their “mixed-strategy” more often than is predicted or beneficial for the player. However, one conclusion that can be made is that the more expert a player is, the better his decision making is, especially if his knowledge of the other player is high. The more knowledgeable a player is about the other player, the better his decision making. Nonetheless, although each player’s previous decision as well as previous payoff is important in his successive decisions, unlike a simple game, it is much harder to find a mixed-strategy Nash Equilibrium because of the mental portion of the game. For example, if a player like Federer is returning serve against Nadal, much more is involved in his decision to go left or right. It is possible that his conscience is not completely clear and therefore he may be down on himself for losing previous points, matches or tournaments, and these thought processes will definitely affect his decision-making, most likely negatively. Nonetheless, the most important conclusion from this research is that knowledge of the other player’s capabilities on the court is highly valuable. Furthermore, it is important for each player to be able to base each of their decisions on previous decisions made, and to make sure that they are not stuck with one pure strategy Nash Equilibrium. This mental game between each player is what makes each tennis match exciting to watch, especially between players who have a long history, like the three men who have dominated male tennis in the last decade, Roger Federer, Novak Djokovic and Rafael Nadal. Many people have mixed opinions about tennis, but for me it is always purely enjoyable.

http://www.u.arizona.edu/~jwooders/WimbledonAER.pdf

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