The next myth in our look back at Klaassen and Magnus’s takedown of 22 tennis myths is Myth 3. This myth concerns the importance of points and whether each point is equally important to the server and returner.
Myth 3: “Every point (game, set) is equally important to both players”
In Analyzing Wimbledon Klaassen and Magnus make the logically argument that a point is always equally important to the server and receiver.
Let me back up a bit and first explain what is meant by importance here. In its statistical sense, as used by K&M, importance is the definition proposed by sports statistician Carl Morris. This definition says that a point’s importance is equal to the change in the probability of winning a game when that point is won versus win it is lost. In other words, point importance is how much winning that point puts a game “in the bag” versus losing the point.
So, K&M say that, on this basis, however much a point increases the chance that a server wins the game when the server wins the point, it necessarily causes an equal drop in the chance the returner wins the game, just as each rubber in a Davis Cup round (like what is playing out between Australia and the USA this weekend) is one team’s loss and another team’s gain.
This does not mean all points are equally important, which is definitely not the case.
What it means is that, whatever the importance of a point to the server, it is also that important for the returner.
Which are the most important points in tennis?
Since Myth 3 really hinges on the symmetry of tennis and not any statistical argument, I thought I would take this topic a bit further and ask which points are the most important for the current game?
Figure 1 below shows the breakdown of point importance for the ATP in 2015, using the same definition of importance as Morris and K&M. The observed importance is highlighted in blue and the point score shows the score for the server on the left, and the receiver on the right. Tiebreaks are excluded.
One thing that is striking when plotted out in this way is how much range their is in importance. The most important point at 30-40 has an influence of 70% on the probability of winning the game. Contrast this with the least important point of 40-0 that has an influence of just 4%.
Readers might be particularly surprised by the low importance of any point that is a game-deciding one, 40-0 in this case. While it is true that if the server wins the point at 40-0, he or she wins the game; it is also true that the chance of winning the game when the point is lost is still quite high. The reason is that the server has so many chances to comeback from a point or two down and, being the server, this player has the upper hand on each additional point played.
In general, break points are the most critical situations for the server to get out of if he or she is going to have any hope of winning the game.
Note that in this summary, 30-40 points are equivalent to a 40-AD situation and 40-30 points are equivalent to a AD-40 situation.
Point importance on the women’s side basically agrees with the men’s (Figure 2) though the range in importance is narrower because of the lower dominance of the serve.
Are all points as important as predicted?
You’ll notice that each plot has an “expected” value. This corresponds to the importance of points that would be expected if the server played every point with equal probability. The probability assumed for the men is 64% and for the women is 57%, which K&M obtained from average serve performance at the 2010 Grand Slams. These averages have not changed considerably in 2015.
Although the actual importance of points in recent tour play lines up well with expectation, there are a few interesting cases where we see discrepancies, flagged by expectations that are outside of the margin of error (noted by the error bars) of the estimated importance in 2015. For the men, 30-40, 15-40 and 0-40 points have been notably less important in actual play than would be expected if servers always served with 64% effectiveness. (The same result was actually shown back in 2001 in a paper by Peter G. O’Donoghue, though the focus of that paper was on gender differences rather than discrepancies from predicted importance.)
There is a similar pattern on the women’s side, though the magnitude of the differences is somewhat smaller.
What is the implication of these deviations from what is predicted by K&M?
The basic answer is that players do not always play each point with equal effectiveness and that the dynamics result in a narrower range of importance than would be predicted under the “equally effective model”. Other work shows that the chance of a server winning a point is lower under pressure, like the break point situations where we see the biggest discrepancies in importance. This could be because servers choke or returners raise their game, on average.
Whatever the underlying cause, the results of this is to lower the chance of a server making a comeback when behind in the score and, thereby, lowering the importance of those point situations.
Yet, even with these deviations from predicted, we can still make the interesting conclusion that not all break points are equally important.