Rafael Nadal said that he plays “each point like my life depends on it," but does that make sense? If you have a limited amount of energy, is it sensible to put the same level of effort in every point, when some might matter more than others?
This raises the question: how do you measure the importance of points in a tennis match? Morris (1977) defined importance based on how pivotal the point is. This can be calculated as the probability of winning the match if the first player wins the point minus the probability of winning the match if the second player wins the point. The definition is intuitive. Losing important points seriously dents your chance of winning the match, whereas losing unimportant points leave it relatively unchanged.
I built a win probability model to explore how important different points are. The win probability model is based on the assumption that each player has a fixed chance of winning each point (depending on whether they serve.) I calibrated it based on two average male players playing each other on hard courts. On average serving players won 66% of points in men’s tennis matches on hard courts (source). So the model assumes the serving player has a 66% chance of winning each time.
The model is not perfect. Results will differ based on player ability and playing surface. It also assumes players perform at the same level each point irrespective of the scenario. This is not the case, players tend to perform worse following a previous error. However, the model provides a good enough estimate of win probability to estimate how important each point is.
So, which points are important? The graph below shows the distribution of point importance across all possible scores in a three set match, broken down by the type of point and set. It shows that most points don’t matter that much and a few matter a lot. This is referred to the distribution of point importance having a right skew. This is true for all types of points.
Break points, set points, tiebreak points all on average matter more than other points. Interestingly match points are often not that important. In many matches by the time the players get to match point it doesn’t matter. Sometimes though, they can be very important.
The most important points are in a third set tiebreak once the score has reached 5-5. All points past this point (e.g. 5-6, 7-6 or 7-7) have a pivot score of 0.5, which is much higher than the median point (0.06). For instance at 5-5, if player 1 wins the score is 6-5 in the final tiebreak and they have an 83% chance of winning the match. If they lose, the score is 5-6 and they only have a 33% chance of winning the match. Winning or losing the point changes their win probability by 50 percentage points which gives the point a 0.5 pivot score.
Point importance varies based on both the game situation and the broader match situation. The most important point in a given game is 30:40 (or Advantage to the returning player). Within that game that point is more than 4 times more important than the point at 0:0 in the game. The table below shows the importance of different points in the first game of a match where player 1 serves first.
The importance of a given point also depends on the wider match situation. The table below shows the importance of a point at 40:40 in the first set for different game scores. So even at a high-stakes game score of 40:40, the point could matter little (if one player is up 5-0 in the set) or matter a lot (if the set is tied at 4-4 or 5-5.)
It is clear that some points matter much more than others. How do tennis player respond to this? Do the top players raise their level when it matters most? Or is this model of point importance good in theory, but difficult for players to use in practice? Trying to calculate the importance of each point might distract a tennis player from their most important task: playing well. I hope to explore these issues further in future blogs.