Meditation on Tennis

After finishing a match of tennis, I often hear similar stories from my opponents; If I win, they’ll be quick to tell me how awful of a game they’re having, and if I lose, they’ll think they are merely playing average. No matter the outcome, my opponent will never consider if I am playing well or poorly. I find this strange for two reasons. If my opponent thinks their performance is variable, why do they think I am so consistent? More puzzling, why do my opponents prefer to think that they are playing poorly, than consider the possibility that I am playing really well. Isn’t it more meaningful to play well, but lose to an opponent at the top of their game, than play poorly and lose to someone playing mediocre?

This raised another question for me – how can someone even know if they are playing a good match of tennis. Consider a tennis match between two identical players. Sure, over the long run, each player will win roughly half the games. However, if you look at the scores of each game, they will typically not be 7-5, but range from 6-1, 6-2, 6-3, 6-4 with varying degrees of probabilities. This is not because either player is playing well on any given day, but because for every tennis match of equal skilled players, there is tons of randomness.

To see if a player is performing well, the superficial thought is just look at who wins the match. I think too often people look at the outcome of a match (they won by a lot, they lost by a lot), and make an inference as to how they played. If the outcome of a match is guided by randomness, I think it is wrong to judge the process (how you played) by the outcome (the final score of the game).

To illustrate this point, let’s jump to basketball for a moment. Imagine an 80% free throw shooter who never gets fatigued. They go in the gym tomorrow and take 1000 free throws, but only make 790 of them. Is it fair to say they performed poorly? What about if they did make 800 of them, but for shots 401 through 410, they missed 6 shots; did they suck during those 10 shots? No, they are an 80% FT shooter, and their outcome was guided by randomness.

(Of course players can improve overtime, but for any given day, they are at a fixed level of skill.)

When I hit a serve in tennis, I tell my body the exact same command every time. As far as I can tell, my mind and body are acting the exact same way for every serve. Despite this, my serve only goes in roughly 80% of the time. I have a set skill level that dictates the accuracy (and power) of my serve, but beyond that, I view it as a mechanical action guided by the laws of probability. No matter how hard I focus, I will never get my serve in 100% of the time. When the ball goes in, I don’t think I performed well, and when my serve misses, I don’t think I did anything wrong. I hit the ball, and it either goes in or it doesn’t; I don’t think I have any further control over this action.

But I don’t just think that my serve is a mechanical action, I think nearly every shot I take in tennis is. I wind up to hit the ball, my brain speaks to my body, and the ball has a certain probability of being hit in/out, or in various good/bad locations on the court. If you apply this to an entire tennis match, the outcome will involve large amounts of randomness.

Monte Carlo simulations are mathematical models used to predict future events based on assigned probabilities. The model uses a random number generator to see how often different combinations of probabilistic events take place by simulating randomness. If each shot you take in tennis has a set probability of going in/out, being hit into the net, being hit into the corner etc, you can simulate an entire tennis match based on the cumulative probabilities of each shot.

I used this model to simulate playing ten tennis matches against a player at my exact same skill level. Since this is a computer simulation, there is no playing well or poorly, just playing at our normal skill level and probabilistic luck. Despite both players being at an identical skill level, the results were:

  1.  A 6-2 3-6 6-1
  2.  A 3-6 6-2 6-2
  3.  A 6-1 6-3
  4.  A 6-2 7-6
  5.  B 5-7 6-7
  6.  B 1-6 6-4 3-6
  7.  B 6-1 1-6 2-6
  8.  A 6-3 6-4
  9.  B 5-7 4-6
  10.  B 5-7 3-6

Even though I ended up winning 5 games, the game to game results and scores varied widely. Think about if this series happened in real life. Both players would say that they played well at certain points, struggled at others, persevered, collapsed etc. but we know this is wrong. There was no struggling, no collapsing or being clutch, there was merely probabilistic events unfolding due to pure randomness.

If outcomes are a good indicator of playing well, we should expect to find athlete’s performance in the first half of a game carry on to the second half. In baseball if a pitcher is going into the 5th inning with a no hitter, we should expect them to enter the 6th inning pitching better than average. Similarly, if they gave up a ton of runs by the 5th inning, we should expect them to pitch worse than average in 6th. Phil Birnbaum and others have shown that no matter how good or bad a pitcher has played thus far, their next inning will default to what their season averages are. In other words, good outcomes are largely just a factor of randomness, not playing well.

We know intuitively that one can have a bad tennis match; If a player is sick, or if they don’t try hard, they can certainly perform worse. But barring these circumstances, so long as a player is trying hard and playing strategically, I don’t think they can know if they are playing well or not. I think people make too strong an inference about how they play based on the final score of the game, and as shown above, looking at the score tells you close to nothing about the quality of play. When a player is in a match against an equally talented opponent, the score sometimes might be a blowout in their favour or for their opponents, despite the fact they are both playing average. My thesis is that without outside information about performance (ie being fatigued, not having played the sport in years etc.) it is difficult to distinguish between playing poorly, and losing due to a random distribution of luck.

I’ll translate this for my friends who play golf.  If it’s the first time playing all summer and you shoot +20 you normally shoot, you are likely playing poorly. However, If you normally shoot 90 at your home course, shooting an 85 or a 95 mid-summer isn’t an indication of a good/bad game, but of the random distribution of probabilities. 

I think this has an important implication for playing sports. If you missed a few shots in a row, it is not a sign of playing poorly. If you hit four great shots in a row, it is not a sign you are playing great. It just means that the balls had a chance of going in or out, and that’s how luck played out. Embrace the joy of playing. We need to shut up and stop blaming ourselves every time we miss.  Put a smile on your face and have fun. We aren’t playing poorly and we aren’t playing well– we are just playing.

I want to thank Phil Birnbaum for his feedback and help with ideas for this article.

This article was inspired by The Inner Game of Tennis, not only the best book about tennis I’ve ever read, but one of the best books about life.

If my writing doesn’t do it for you, I recommend you read David Foster Wallace article on Roger Federer.

If you want to simulate your own tennis matches, you can do so here: