## How to improve your return game in tennis – Predicting serve direction using game theory

We can see with this article that tennis matches can attempt to be predicted using complex game theory and Markov Chain model from statistics of players’ serves from previous matches such as the match between Andy Roddick and Younes Al Yanaoui during the 2003 Australian Open. However, we can use even simpler game theory to attempt to improve one’s chance of winning return games as I will discuss below.

As a Cornell University Men’s Varsity tennis player, I have had the opportunity and pleasure of actually conducting a game theory study on my own game. Serving and returning are two of the most crucial aspects of tennis and being able to accurately predict the direction of one’s opponent’s serve can improve one’s chances of neutralizing points during return games and thus winning more points significantly.

Over 2 days, I played tennis matches with Player X and my goal was to be able to improve my chances of winning points while returning. In order to do that I would have to improve the number of serves I return successfully which would need for me to improve my ability to predict my opponent’s serve direction. Returning successfully means returning my opponent’s serve well enough to neutralize or be on the offense after the return. Before I returned each serve, I had two options. The first option is to expect the serve to come to my forehand. The second option is to expect it to come to my backhand. I was able to collect data after playing two full matches (this represents 213 returns hit) and determine the payoffs for each player as a percentage of success (for my opponent, it would be winning the point off his serve or being on the offense after the serve), which can be illustrated by this 2×2 matrix:

 Me\Player X Serve to forehand Serve to backhand Expect forehand 86%, 14% 36%, 64% Expect backhand 71%, 29% 64%, 36%

Now let’s say that player X serves to my forehand with a probability of q and thus serves to my backhand with a probability of 1-q.

My payoff for expecting to hit forehand returns the whole match would be: 0.86*q + 0.36*(1-q) = 0.5*q + 0.36

My payoff for expecting to hit backhand returns the whole match would be: 0.71*q + 0.64*(1-q) = 0.07*q + 0.64

Therefore, my payoff would be the same for expecting forehands or backhands during the match when:

0.5*q + 0.36 = 0.07*q + 0.64 ó 0.43*q = 0.28 ó q = 0.65

This study can help my return game tremendously because all I have to do is analyze whether my opponent serves mostly to my backhand or to my forehand. If my opponent serves roughly over 65% of serves to my forehand then I should expect to hit forehand returns during the whole match. If my opponent hits less than 65% of serves to my forehand then I should expect to hit backhand returns during the whole match.