Tennis and Game Theory

Background

Tennis is one of the oldest games in the world and is argued to be one of the most mental games there is. This plays into the game theory aspect where the decision to hit the ball and whether to challenge a call is trying to be optimized.

First things first, gaining a point and winning the game:

A player gains a point when they hit the ball in bounds on the other side of the net and their opponent doesn’t hit it before the second bounce or when their opponent hits the ball into the net or out of bounds. In order to win the game you must win two sets and in order to win a set you have to win 6 games. Each game is played to 4 points where the winner must win by 2.

Where to hit?

Every time a player serves they can either serve left or right of their opposing player within the area they are supposed to be serving to. The opposing player then has the option to hit the ball back to the left or the right.

This gives a payoff matrix:

The numbers inside this matrix represent the probability the server is going to win the point. Since this probability has a lot of factors depending especially on who the server and receiver are they can’t be known in a general form.

There are a couple different conclusions that were made from the research done. The research was taken from 10 championship matches and shows that the players with more experience will make better choices especially if they have a lot of information on their opponent. This makes a lot of sense and I think could be said about a lot of other sports as well. I know in volleyball we go over film watching the other teams before we play against them to base some of our decisions off what we see from the film. Since tennis is such a mental game as well it would seem it would be really beneficial going in there knowing everything you can about who you’re trying to beat.

Challenges:

One of the things I found really interesting in my research on tennis was the recent introduction to challenges that enable the player to overturn the call made by the umpire regarding whether the ball landed in or out of bounds. This is something that’s more recently been implemented in sporting events and looking at the outcomes from it is something that seems like it could impact the way the game will be played in the future.

First things first, each player is only given a limited number of challenges and once you run out, you won’t get any more. The thing that makes it interesting is that if the call is overturned when you challenge you get to keep that challenge, so you could potentially have all your challenges left after the game is finished even if you challenged at all in the game.

The research I looked at was using the data it had on 35 tennis tournaments around the world that were using the challenge rule. There were 2,784 challenges made within these 35 matches by 179 players in 741 matches where at least one challenge was made… That’s a lot of challenges right there! And seeing that the cameras and programs used called Hawkeye to rule the challenges took about 30 seconds that’s a lot of time spent on challenges. This is one of the reasons why there is a limit on the number of challenges a player can make and the hawkeye system doesn’t just make all the calls. It takes a longer amount of time to run then just having an umpire call it in or out and keeping the game running. This would impede the players game I’m sure as, I’ve already mentioned and seems to be coming up a lot, tennis is a very mental sport.

This graph shows the lost and won challenge outcomes in relation to how far away from the line the ball turned out to be with 0 being right on the line.

As you might’ve guessed there aren’t very many challenges that were lost or won once the ball got a good distance away from the line. At that point you would be able to clearly see if the ball landed in or out and we can just assume that the umpire is a good line judge and would clearly be able to see that as well.

Compiling the results of the challenges it was seen that the players had been using a strategy as to when they would challenge. There are points in the game that can be deemed more or less important. The points in a close game are more important than those in a game that isn’t as close. The results that were found are given below.

As can be seen there are times where the challenges are won more than other points and this shows that the players have used their challenges conservatively. If the game is early in the set one wouldn’t want to “waste” a challenge on something they weren’t sure of since they don’t know if they’ll need that challenge later, whereas later in the game they are more likely to use their challenges for points they aren’t as sure they’ll win. This turns out to be a very good strategy since the points later in the game are deemed more important to winning.

The equation below was formed to show the expected success rate for challenging every time the q > y. F(y) is the distribution of probability of succeeding in a challenge from which players draw when they lose a point.

This equation was used to find the optimal challenge behavior. It was found that once the optimal challenge behavior was found and compared to the behavior the tennis players were using there were a lot of payoffs being generated for the players even though their behavior wasn't exactly optimal. The players were found to not be challenging as much as they optimally should be, but are still reaping the benefits from the challenges since there is a 1.6 percentage point increase in winning with the challenges.

Conclusion:

The strategy the players have with regards to challenges is close to optimal and the direction in which they hit the ball should be based on who they are playing and what happened previously in the game.

Game, Set, Match: Game Theory in Tennis : Networks Course blog for INFO 2040/CS 2850/Econ 2040/SOC 2090

ON THE OPTIMALITY OF LINE CALL CHALLENGES IN PROFESSIONAL TENNIS : https://people.stanford.edu/ranabr/sites/default/files/tennis.pdf