Game Theory and Tennis

Pop quiz: Who is your favorite tennis player?

Introduction

The game of tennis is one of the oldest sports in the world and considered one of the most mental games.  In a singles match, each player has the chance to serve, and therefore each player also has the chance to receive. The server can win points on the serve, and can also lose the point on the serve through a double-fault, or the serve may be returned successfully. The scoring system of a tennis game is the following: each game within a match is won when a player has four or more points and a lead of at least two points. The first player to six games with a lead of at least two games wins a set ( a tiebreaker is played if the score reaches six games apiece), and the winner of three out of five sets wins the match.

On the aspect of game theory, the question becomes whether we can us game theory to predict the behavior of each player on the court. 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/her mixed-strategy Nash Equilibrium to fit prior information. No Pure-Strategy Nash Equilibrium indicates that players should be "unpredictable" by his/her opponents. That is, players should randomize between strategies so that they do not get exploited.

Example

Consider the following example in a Tennis match,


the potential pure strategies for the server can be analyzed as follows:

(1). If the server always aims forehands (F) then the receiver (anticipating the forehand serve) will always move forehands and the payoffs will be (90,10) to receiver and server respectively
(2). If the server always aims Backhands then the receiver (anticipating the backhand serve) will always move backhands and the payoffs will be (60,40).

However, the server will choose neither (1) nor (2) if he/she wants to win the serve. Thus, the server can increase his/her performance by mixing forehands and backhands.

For example, suppose the server aims forehand with 50% chance and backhands with 50% chance. Then the receiver's payoff is
(1). 0.5*90 + 0.5*20 = 55 if the receiver moves forehands and
(2). 0.5*30 + 0.5*60 = 45 if the receiver moves backhands.
Since it is better to move forehands, the receiver will do that and the payoff will be 55. Hence, if the server mixes 50-50 the payoff will be 45, which is already an improvement for the server's payoff.

The next step is to see if we can find a best mix-strategy for the server. Suppose the server aims forehands with q probability and backhands with 1-q probability. Then calculating the receiver's payoff we get:
(1). q*90 + (1-q)*20 = 20 + 70q if the receiver moves forehands and
(2). q*30 + (1-q)*60 = 60 - 30q if the receiver moves backhands.

The receiver will move towards the side that maximize his/her payoff. Therefore, the receiver will move:
(1). forehands if 20 + 70q > 60 - 30q,
(2). backhands if 20 + 70q < 60 - 30q, and
(3). either one if 20 + 70q = 60 - 30q.
So the receiver's payoff = max{20 + 70q, 60 - 30q}.

In order to maximizing the server's payoff, he/she should minimize the receiver's payoff. The server can do this by setting 20 + 70q and 60 - 30q equal:
20 +70q = 60 - 30q so that 100q = 40 and q = 0.4.

The server should aim forehands 40% of the time and backhands 60% of the time. In this case, the receiver's payoff will be 20 + 70*0.4 = 60 - 30*0.4 = 48. Hence, the server's payoff will be 100-42=52.

We can use the similar method to analyze the mix-strategy for the receiver. 

Suppose the receiver moves forehands with probability p. Then the receiver's payoff is:

(1). p*90 + (1-p)*30 = 30 + 60p if the server aims forehands and

(2). p*20 + (1-p)*60 = 60 - 40p if the server aims backhands.

In this situation, the server will aim towards the side that minimizes the receiver's payoff. Hence, the server will aim at:

(1). forehands if 30 + 60p < 60 - 40p

(2). backhands if 30 + 60p > 60 - 40p

(3). either one if 30 + 60p = 60 - 40p.

So the receiver should equate 30 + 60p and 60 - 40p to maximize the payoff:

30 + 60p = 60 - 40p so 100p = 30 and p = 0.3

The receiver should move forehands 30% of the time to maximize the payoff and backhands 70% of time. In this case the receiver's payoff will be 30 + 60*0.3 = 60 - 40*0.3 = 48. The server's payoff will be 100 - 48 = 52.

After calculating the payoff for both the server and the receiver, the mixed strategy came out to be:

Receiver: 0.3F + 0.7B

Server: 0.4F + 0.6B

And this is the only strategy that cannot be "exploited" by either player.

Important Observation

There is an important observation that found by many mathematicians who study mixed-strategy of tennis: If a player is using a mixed strategy at equilibrium, then he/she should have the same expected payoff from the strategies he/she is mixing. Based on this observation, we can easily find mixed-strategy Nash Equilibrium in a 2*2 game.

Let's find the mixed-strategy Nash Equilibrium of the following game which has no pure strategy Nash Equilibrium.

In this example, let p be the probability of player 1 playing U and q be the probability of player 2 playing L at mixed-strategy Nash Equilibrium. We want to find p and q. Do you know how to do it? We will go over this example in class on Tuesday.

Conclusion

Although research has shown that players in a tennis match should constantly adapting his/her mixed-strategy Nash Equilibrium to gain more payoff, it is much harder to find a mixed-strategy Nash Equilibrium in because of the mental portion of the game. But it is important for us to get the idea that either player can do better by choose a mixed-strategy than pure strategy. In this post, I only focus on how mixed-strategy Nash Equilibrium applied to tennis match. Game theory can also applied to whether or not to challenge the line call during a tennis game. I provided some useful links for further reading. All in all, many people have mixed opinions about tennis, but for me it is just a purely enjoyable sport!

Resource

https://econ.duke.edu/uploads/assets/dje/2006_Symp/Wiles.pdf
http://blogs.cornell.edu/info2040/2011/11/11/game-set-match-game-theory-in-tennis/
http://www.researchgate.net/publication/46554892_Using_Game_Theory_to_Optimize_Performance_in_a_Best-of-N_Set_Match
http://www.tayfunsonmez.net/wp-content/uploads/2013/10/E308SL7.pdf
http://web.stanford.edu/~ranabr/Tennis.pdf