Playing the big points

[Chris Lloyd]

Do some tennis players play better on “big points”? Come to think of it, what are the big points and can this be measured? Recent work by Olivier Gossner, Brian Rogers and Julio Gonzalez-Diaz has established a very neat framework for answering both questions.

The basic idea of measuring importance of a point is simple – it is defined to be the probability of you winning the match if you win that point minus the probability of winning the match if you lose that point. We can also talk about the importance of a point within a game – it is the probability of winning the game if you win this point minus the probability of winning if you lose.

These seems like sensible definitions to me. In fact, I am hard pressed to think of a reasonable alternative. You will expect points to become more important towards the end of a game, when the score is close. So match points in tie-breakers in the fifth are most important. Similarly, games towards the end of an evenly balanced set will be important. All obvious, but the value add is that we have an objective numerical measure. What is not immediately clear is whether you can easily calculate this.

Tennis has a scoring hierarchy – points win games, games win sets and sets win matches. And it is reasonably easy to show that the importance of a point in the match is the product of three terms – the importance of the point in the game, times the importance of the game in the set, times the importance of the set in the match. And these three numbers are reasonably easy to calculate by backwards recursion.

All these quantities obviously depend on the probability of you winning a point – which the authors assume to depend only on whether you are serving or not but otherwise constant for each player over the game.

It turns out, not surprisingly, that 30-40 on serve is*usually the most important point in a game (for a range of sensible serving probabilities). The most important game is the 10 th at 4-5 on serve, followed by 4-all or 5-all and the most important set… not surprisingly the fifth. Here is a typical table for the importance of games in sets (for two players with serving probabilities .75 and .63).

Does anybody remember former Aussie great Fred Stolle’s theory that the seventh game of a set was the most important? He used to drive*us all*nuts. At the end of the sixth game (regardless of whether it was 3-3, 4-2 or 5-1)* you just knew he was going to say it. Well in the table above the seventh game nothing special. Even at 3-3 the importance is only 0.57 which is not as important as 3-4 or 4-4 or 4-5 or 5-5. Some one please tell Fred.

Now, to the second part of the research. We have identified what we mean by a big point. How do we then measure player’s ability to play the big points? This is pretty easy. At the end of a match, we go through the game and calculate the importance of each point. We then fit a logistic regression with importance of point as a predictor, as well as dummies for serving and returning abilities. The coefficients for the last two are not of interest in themselves. They*adjust for how balanced the game is. In a very one-sided game there will not be many important points.*The coefficient for importance of points is the measure of your ability to play the big points. The probabilities of winning on serve or receiving can be*estimated empirically*from the data for that match, of from a more general model of the players ability against different player on different surfaces. Indeed, it could also vary through the match.

The paper has not been published yet. Unfortunately, the authors have not yet given us what we would really love to see – a ranking of well known players by their ability to play the big points.

Hat tip to Sven Feldmann for leting me know of this research.



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