Because the French Open and Wimbledon have been available on the BBC website I’ve been watching a lot of tennis recently. And as I do so I can’t help thinking about whether mathematics has anything to say about the tactics that the players should adopt in various situations. And the more I think (or rather, idly muse) about this question, the more it becomes clear that the modelling problem it presents is a pretty hard one. Most of this post will be a discussion of questions rather than a serious attempt to supply answers.
Just to make the discussion more concrete, here are a couple of more specific questions, which I’ll come back to later. The first one is fairly simple.
1. It is generally held to be a slight advantage to serve first in a set. The reasoning goes like this. Let’s suppose (for simplicity) that the game goes with serve till 4-4. If you are serving first, then you will be in a very dangerous position if your serve is broken, since you will then have to break back immediately or lose the set. However, at least you won’t have lost. By contrast, if you are serving second and the score is 4-5, then you can’t afford to be broken — if you are broken then you lose the set and do not get even a small chance to redeem yourself. And if you have just broken your opponent so that it’s 5-4, then you still have the task of serving for the set.
However, a simple model would suggest that this reasoning is flawed. If you have a probability of winning a game on your serve and a probability of winning it on your opponent’s serve, then over the next two games you have a probability of winning both, of winning one, and of losing both, and the order the games are played in makes no difference.
I’ll come back to the (not particularly interesting) solution of this “paradox” in a moment, but before that let me turn to a second and rather general piece of conventional tennis wisdom.
2. It seems pretty obvious that different circumstances call for different styles of play. For example, if you are a break up and 40-0 on your serve, then it feels as though you can afford to take a few risks, whereas if you are serving a second serve at match point down, then you should play it a little bit safe, since a double fault would cost you the match (and similarly, if you are going for a winner, you shouldn’t aim too close to the line, and so on).
But a very simple argument suggests that a different piece of tennis wisdom, the one that says, “Forget where you are in the game and just take things point by point” must be correct. After all, the best strategy cannot be anything other than to maximize the probability that you win the point, so if it ever makes sense to serve a reasonably powerful second serve and risk serving a double fault, then it makes sense even if you are match point down.
I have more mathematics-of-tennis questions that I want to mention, but first let’s dispose of these two, by thinking about what the model is that underlies the second argument in each case and what important factors it fails to take into account.
It is a model much loved by setters of questions in elementary probability: on each point, if player A is serving then there is a probability that A will win and a probability that B will win, whereas if player B is serving then these probabilities are and . Moreover, the outcomes of all the points are independent.
If that were a realistic model, then there would indeed be no advantage in serving first, and one should play the same tactics on every point. However, there are two (at least) very important things that it does not take into account. One is that the probabilities change according to how nervous a player is (and also, it has to be said, appear to ebb and flow during a match for no apparent reason). So for example it might be a good strategy to serve a rather lame second serve when you are match point down, since if you do a normal one then nerves are more likely to cause you to mess it up, and your opponent’s nerves may cause them to mess up their return even if under normal circumstances they could hit a winner off it. Similarly, the pressure of serving at 4-5 down is greater than the pressure of serving at 5-4 up, so perhaps there really is some justification for the belief that serving first is an advantage.
Unfortunately, there doesn’t seem to be an easy way of realistically incorporating players’ psychological states into a model, so let’s set that question aside (unless someone feels like having a go at it). But there is a second consideration that does lend itself to a mathematical analysis, and that is the fact that if you play precisely the same tactic every point, then you will be handing an advantage to your opponent. For instance, you may win the point with probability if you serve out wide, and with probability if you serve near the T, but if you serve out wide every time, then your opponent can take advantage of this by standing wider, at which point the probability of winning the point will no longer be but something lower that may well be less than .
I think (without having checked) that one can deal with the second consideration by defining a strategy to be something that tells you not what shot to play at each stage, but what probability distribution to choose your shot from: e.g., it might tell you to serve out wide 50% of the time, for the T 30% of the time, and straight at the body 20% of the time. And the optimal strategy would be the probability distribution that maximized your overall probability of winning the point. That sounds to me like fairly standard game theory.
A third factor I won't discuss here is conservation of energy — not the basic law of physics, but the more homely physical principle that says that if you run around too much then you will get tired later on. That might tell you that if you can just get a shot back by running all the way across the court, but can't do anything with it when you do except gently pat it over the net straight at your opponent, then it might be better not to bother, even if it increases your chances of winning the point from 0 to 1/100.
There's one more question I want to mention, and it's the main one I wondered about. It is how one should model the tactical decisions that are made during a rally. As a warm-up, here is a special case. It's your first serve. Is it better to go all out for an ace, or should you try for a serve that has a slightly higher chance of going in and still leaves you in a very strong position? Under suitable assumptions, that question is pretty easy once one puts some numbers in. Let's suppose that if you go for an ace, then you'll serve one with probability and you'll serve a fault with probability . (That ignores the possibility that your opponent might occasionally be able to get your serve back, but let us indeed ignore that.) And let's suppose that if you go for a serve that is merely very hard to return, then you'll succeed with probability and serve a fault with probability , and furthermore that if you succeed then your opponent will give you the opportunity to smash, and that your smash will win the point with probability (which is very close to ) and will go out with probability . Then if you go for an ace, you win with probability , and if you go for a win on your second shot then you win with probability .
After that trivial calculation, the question that remains is whether is likely to be higher or lower than , and rather boringly I don't have any idea. By roughly how much do professional tennis players increase their first-serve percentage if they make their serves very slightly slower, or very slightly less close to the lines? And how much can they afford to do that before their serves become easy to return? Obviously it depends a lot on who is playing, so a different question might be whether it is in fact the case that there are professional tennis players who are trying for aces when it is clearly not the best strategy. (Of course, the mixing-it-up principle could complicate this question and lead to the conclusion that one should sometimes try for an ace and sometimes go for a good first serve that leaves one in a strong position in the ensuing rally.)
Another simple question: is it conceivable that it might sometimes be good tactics to attempt first-serve-type serves all the time? For instance, if you are Ivo Karlovich on a good day and your first-serve percentage is 75% (as happened at times during Wimbledon), and if you are winning 95% of the points when your first serve goes in, then your probability of losing the point if you go for first serves every time is , which is , which is 19/20 times 15/16, which is certainly at least 7/8. That gives your opponent almost no chance of winning a game.
Maybe I should change that question to a related one: why is it that nobody ever plays that tactic, except perhaps on the occasional point?
This is a fairly simple question. Let's suppose that with my best strategy, if I have just one serve left then I have a probability of winning the point. And now suppose that it is in fact my first serve. Let's suppose that I can choose the probability that my first serve goes in, and that there is some function , which tells me the probability that I will win the point if I do a probability- serve. So for example, if I go for an ace, then may be fairly low, but , whereas if I go for a safer first serve then is higher and correspondingly lower.
If I have just one serve left, then I have a very simple optimization problem: I just need to maximize , and we are assuming that its maximal value is . But if I have two serves left, then the quantity I am trying to maximize is . What difference does that make? Well, we are assuming that is a decreasing function of . If is chosen to maximize , then, at least if is differentiable, decreasing by a small will decrease by (since the derivative of at is zero) but will also increase by . Therefore, the maximum of will occur at a lower value of , which corresponds to a riskier serve. (I've made various assumptions there that I can't be bothered to make explicit.)
Finally, here is the general case of the question that intrigued me most. It is to try to find a model that would justify the following piece of reasonable-sounding tennis advice: that if you are in the middle of trading groundstrokes with your opponent, then you should not go directly for a winner, but should instead try to play yourself into a stronger and stronger position until you can hit a winner with less risk. That advice may not be universally applicable, but it does appear to be the case that some players sometimes go for winners when they would have been better advised to be patient. (Equally, if you are playing Roger Federer, you may realize that you have no hope of winning a long rally so you are more or less forced to go for a risky shot that will be a winner if it comes off.)
How does one even think about such questions mathematically? I'm more concerned about that than about the actual answers that a model might yield. One might try something like this. In any given situation, you have a range of shots you can attempt. They have various probabilities of success, and if they do succeed then your opponent has a range of possibilities that depends on the shot you do.
It is natural to simplify the model as follows. In any given situation, you have a function like the one discussed earlier. That is, you can choose a shot that will go in with probability , and if it goes in you will win the rally with probability . But hang on, how do we have any idea what is? Well, by hitting your probability- shot you give your opponent a choice similar to yours: a function that says "If I hit a probability- shot then I will win the rally with probability ." But things are more complicated, because depends not just on but also on how risky your shot was — that is, on .
In other words, your aim is clearly to maximize , but this can be broken down further as a wish to maximize .
We can take this further. Let measure the riskiness of the shots and let denote my probability of winning if I chose to play a -shot, you chose to play a -shot, and so on. (This is a slight change because it focuses on my probability of winning and saves me having to write "".) Then it looks as though my best strategy is to maximize , which is the same as maximizing , where , which is the same as maximizing , and so on. Let me just write out the th version of this expression in full gory detail. My probability of winning appears to be (I haven't checked this carefully) if is even and a similar expression ending with if is odd.
But isn't bounded, so it seems as though we need to take some kind of limit as tends to infinity. Does anyone have any idea what sort of thing that limit is? It makes my head spin like the ball on a heavily sliced second serve. And that is just the question of modelling the situation in the first place: actually solving the optimization problem given the function looks pretty unpleasant. So are there simplifying assumptions that would at least allow one to justify some of the advice given to players about when to attack, when to go for winners, and so on?
One final remark: I do not for one moment think that anything in this post, even when fully developed, would be of the slightest use to a tennis player. I just thought I'd better make that clear.