Should I have left that chase?
An interrogation of the mathematics and statistics of deliberately conceding chases in real tennis
For observers and players new to real tennis, one of the bizarre concepts to learn early on is that, in some circumstances, the best strategic decision is to not attempt to play at the ball at all, contrary to almost all other racket and ball sports. Chases are an endlessly fascinating part of the game, essential to real tennis’s strategic thinking and quintessential charm. But when is actually the best time to leave a ball to fall to a chase, instead of throwing one’s body at the ball in the hope that the marker calls “stroke”?
To answer this question, I look at the ball-by-ball match data available through the scoring app CourtMarker, filtered down to include only singles matches where both player’s handicaps are less than 20. Using this data, we can extract three key pieces of information. First, there are three ways a point can end in general play, either being won by the server (49% of the time), the receiver (36% of the time) or a chase (15% of the time). Second, if a chase has been laid, we know how often it is likely to be called by each measure. Third, given a chase length, we know the likelihood it is won or lost (or called a chase off).
Using this data, we can construct a mathematical graph that maps all possible permutations of the score within a game and how likely each player is to win the game from each position. There are approximately 100,000 different scores that could occur in any given game, meaning there are approximately 1 million different scenarios over the course of a six game set. Here, we are assuming we have two real tennis robots with equal ability, so we are ignoring factors like momentum, serving strength, playing style and so on. Let’s call our real tennis robots Albert and Bob1, and assume they’re playing a single set to six games with no handicap.
Some straight-forward numbers to begin. If Albert is serving at the start of any game, the output of the model suggests that he has a 58% chance of winning that particular game. We can break this down further, depending on whether or not Albert has conceded the service end over the course of the game (that is, the number of end changes is odd). The likely outcomes are therefore:
Albert wins, holds (or returns to) the service end: 33% (or 1 in 3)
Albert wins, concedes the service end: 25% (or 1 in 4)
Bob wins, but remains at (or returns to) the hazard end: 16% (or 1 in 6)
Bob wins, and won the service end: 26% (or 1 in 4)
The likelihood that the players will be on different ends in the next games is therefore close to evens (51%). We can also propogate this forward to compute the effect of serving first at the start of the set gives a 52-48 advantage to whomever called “rough or smooth” correctly.
Now let’s introduce chases. Here, we’re going to give the real tennis robots a choice: they can either let the ball go for a chase or they can play it but with a 0% chance they will put it back in play — think of a tight backhand jammed underneath the galleries or a ball dropping flush with the back wall. In another way of thinking, at any given point in the match, which is more valuable, holding the service end at the expense of giving away a point, or conceding the service end at the risk that you may not beat the chase?
Let’s first consider the match to be at 5-all, so the winner of the game wins overall, and that Albert is serving. When Bob is two points away from the match (deuce, advantage or 30-something up), it is never advantageous for Albert to concede a point, as doing so would give up a match point. Instead, he should always opt to leave a chase and try and survive to fight it out from the receiving end. On the contrary, when Albert has match points, there are now chase lengths where he should give up on a point rather than concede the serve as he should have the service advantage on future match points. At 40-0, he should not concede chases better than 4, at 40-15 he should not concede chases better than 2 and at 40-30 (or advantage) he should not concede chases better than 1 and 2. The chase length decreases as the impact of conceding a point increases.
The relevant data here is shown in the figure below. The regions below 0% are the scenarios for which conceding a point is better than leaving a chase. Note that a score of 30-all is equivalent to deuce and that 40-30 is equivalent to advantage. Scorelines where the opponent has a game point are not shown.
If the match was at an earlier stage, the calculus changes as we have to consider the advantage of holding the serve going into the next game. Say the match is at 3-all, with Albert leading 40-0. Albert should be more willing to risk losing the game and holding the serve — especially given that his odds of winning the game are already high — so should try and prevent any chase, including hazards, from being laid. Again, the length of chase he is willing to bet on decreases as he concedes further points, being somewhere between second gallery and hazard second gallery at 40-15, and between 3 and 4 and worse than 6 at 40-30 (or advantage). The exact numbers depend on how close to the end of the set the game score is.
Again, the data is shown in the figure below for a match at 3-all. The scale on the y-axis is much smaller, as at 3-all, each individual reste has a much smaller impact on the final result than at 5-all. However, a much larger portion of the scenarios are below the 0% line, showing that at this stage in a set, maintaining the service end is much more important than the loss of any particular point.
There are plenty more scenarios here — we haven’t even considered examples where a chase has already been laid, and one is debating whether to lay a second one. And of course, in any real match there are many, many other factors to consider, especially if one gives themselves a non-zero but still small chance of making a particularly difficult shot. There are also many different analyses that can be done with this dataset, so if there is anything you would like to see, get in touch. But at the very least, next time you are running forward to chase down a high ball only to dump it into the net, you’ll be able to claim you’re making a strategic decision — at least insofar as not conceding a chase.
Our real tennis robot maker was inspired by World Champions of yesteryear.