Game Theory and Unpredictability in Tennis
https://tennishead.net/exclusive-roger-federer-novak-djokovic-weakness-revealed-by-maths-professor/
Every professional tennis player must be in peak physical condition to compete at the highest level. However, what sets the good from the great is in the mental side of the game. Some of the greatest players in the world, such as Novak Djokovic and Roger Federer, have the ability to play unpredictably. Dr. Leonidas Spiliopoulos, an expert in tennis data analytics, models the unpredictability of the serves of select tennis players using game theory. According to game theory, the probability that a player serves it down the middle (“T”) and wide (“W”) should be independent of their last serve to optimize unpredictability (in other words, players must be indifferent of their two strategies to maximize their chance of success).
The graphs below, taken from the article, model each players’ unpredictability as the difference in % of serves directed toward W or T for both the ad side of court and the deuce side; in other words, how unpredictable each player tends to be based on a previous event (and this statistic is calculated separately for both sides of the tennis court) . In the first graph, this event is the direction of their last serve, and in the second graph, the event is player winning or losing the previous point. If a player’s serves on one side are perfectly unpredictable and independent of the specified previous event, that % value will be zero.
An observable trend is that some of the highest-ranked and most experienced players tend to serve most unpredictably. This highlights the true mental side of the game: the best players, at least according to the statistics, are the ones that can completely ignore events of the previous play. For example, Roger Federer, who is regarded by some to be the greatest player ever, has a predictability percentage of around zero when serving on both sides of the court. This model demonstrates the power of game theory, and how important indifference to your strategies is to one’s success in these types of games. It also points out which sides each player tends to be easy to read; for example, Rafael Nadal’s deuce-sided serves tend to be more predictable than his ad ones.
This study relates to the concept in class about making a player indifferent in a mixed strategy equilibrium. There is no dominant strategy in tennis; if a player decided to serve the ball in the same place every time, he would be found out easily and not do well. Therefore, to get the best payoff (in this case, points/wins), a player must be indifferent to their strategies. This example is quite similar to the penalty kick game we studied in class. In both cases, the player who is serving wants to put the ball opposite of where the other player is guessing, and vice-versa. The expressions we set up for the expected payoff of each players’ moves were based on the other player rather than themself, which is an important concept. This article suggests that being the best in tennis extends beyond athleticism and talent to the mental capacity to read your opponent and to stay focused on the current play.