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Late last year I wrote a blog about Ace:Error ratio. I've updated my thinking since then. I’ll talk today about my updated thoughts on ace:error ratio and serving strategy more broadly. I continue the main idea from that blog: a higher ace:error ratio is not always better, but update my thought process about how to figure out the optimal ace:error ratio. In this blog you'll find out why a 1:1 ace:error ratio means you're doing something wrong at the service line and why a 2:1 ace:error ratio is even worse.
Total serving efficiency is equal to:
(in-play average xP)*(in-play proportion) + (terminal average xP)*(terminal proportion)
Choosing a serving approach is a maximization problem, we’re trying to get that number as big as possible. We do that, in part, by choosing an aggression level. As aggression increases, we would expect in-play average xP to increase, in-play proportion to decrease, terminal average xP to decrease, and terminal proportion to increase. To put it colloquially, as a server really lets it rip on their serve, our opponents pass worse, we miss more serves, get more aces, and our ace:error ratio becomes more skewed to errors.
Some of those are good and some are bad, so we’re trying to find when the tradeoffs make us better and when the tradeoffs make us worse to find the best approach.
This is an example graph created using made up numbers to demonstrate this concept. As you can see, there is one maxima.
I made this curve by fabricating a terminal serve xP curve and an in-play serve xP curve and then putting them together with the formula for total xP from above. The graph of all 3 looks like this:
A piece of vocabulary: “Breakeven success rate.” Breakeven success rate is the rate at which you need to do something such that if you do it, your odds of winning are the same as if you had not. Let’s explore this concept with an example:
Note that because I was playing with numbers to get these two approaches to come out with equivalent total xP, these are not points from the graph. This is a sample server with a .14 terminal serve proportion and a 1:1 ace:error ratio. In the xP column are sample numbers for how often the serving team wins after a given outcome, or sample xP values for each outcome. By multiplying those numbers by the frequency numbers, I got the weighted xP in the right column, and then added all those up to get the total xP. This is how often the serving team should expect to win the point when this server is serving.
From this table, we can separate out terminal and in-play serves. Here the sums of the weighted efficiencies show what percentage of terminal or in-play serves the team win:
As you can see, the terminal serves are more efficient than the total and the in-play serves are less efficient than the total.
Here is a sample server with roughly equivalent total xP, but has a .22 terminal serve proportion and a 1:1.2 ace:error ratio.
Here are their terminal and in-play serves broken out:
I made this by fiddling with the numbers to make the efficiencies work out, so this isn’t a claim that any server who serves like the first example could change and serve like the second example instead. Rather, I think it’s illustrative to talk about what you have to do to make an approach change worth it.
Here are the really important numbers:
So, to compensate for the .0455 decrease in terminal serving xP, we only needed a .0061 increase in in-play xP. This is a breakeven success rate. So, when we’re judging approach changes we should ask: how much do I expect each xP and proportion to change? Because such a large proportion of serves remain in-play, even a small change in the xP of that population can outweigh significant decreases in terminal serving xP.
I suspect that a 1:1 ace:error ratio likely means you’re doing something wrong because you could probably be more aggressive and the tradeoff would be worth it. Let’s look a little more into why.
I think it makes sense to think of marginal changes between serving approaches with what I’m dubbing a “batch” framework. Here’s what I mean:
Think of servers as changing their approach by adding batches of terminal serves to their total. Said conversationally, “By becoming more aggressive, this server will have 3 more aces and 2 more errors over the course of their next 100 serves.” These 5 new terminal serves are a batch. By thinking this way, we can make decisions based on the values of terminal serve total xP, in-play total xP, and batch xP and the relative sizes of those groups. One important mathematical certainty: If batch xP is greater than in-play total xP, always add the batch. Total xP will always increase by adding the batch in this case. If batch xP is less than in-play total xP, we will need to see in-play total xP increase by some amount to compensate and raise total xP. Let’s look at an example:
Imagine a server who, with a conservative approach, can have a terminal serve % of 6% and a terminal xP of .6707%, which is a 2:1 ace:error ratio. If this server served 100 balls, they’d have 4 aces and 2 errors. Let’s say that this server can add a batch of 8 terminal serves, 3 will be aces and 5 will be errors. The new batch would have an xP of .375. The new total would have a terminal serve % of 14% and a terminal serve xP of .5, which is a 1:1 ace:error ratio.
Both of these approaches are points on the graph from above. Let’s compare these two.
This hypothetical shows what my made up graph says would happen if a server made this change. Maybe that’s not persuasive. Let’s instead look at the minimum in-play xP we would need to see to have a breakeven success rate:
This answers the question: “If I think my 2:1 6% terminal server could add a batch of 3 aces and 5 errors by becoming more aggressive, how much better do their in-play serves need to get to make it worth it?” The answer is we need to go from winning in-play serves 40.49% of the time to 40.75% of the time. That’s such a small shift that I’d say this is an approach change worth doing.
This graph’s total xP peaks at 21.4% terminal serves. At this point the terminal xP is .4164 and the in-play xP is .4702. The total xP is .4587.
If we were to add another batch to get us to our maximum xP on this graph, we’d add 7 more terminal serves to get to 21%, 2 of which are aces and 5 of which are errors. That will get us to a .429 terminal xP by adding a batch with a .286 xP. So at 21% terminal, in 100 serves our server would have 9 aces and 12 errors.
Here the breakeven success rate is higher, the server would need to improve their in-play xP by 1.45 percentage points. To give a sense of what that means, in 100 serves, 79 of which stay in play, a 1.45 percentage point increase in in-play xP is taking 12 3-passes and changing them to 5 1-passes and 7 2-passes. If you think that’s a realistic shift, then this server should go past a 1:1 ace:error ratio.
Finally, to go past our maximum xP to being too aggressive, we can add another batch to get us to 30% terminal. We’d add 9 more terminal serves, 2 of which are aces and 7 of which are errors. That will get us to a .367% terminal xP by adding a batch with a .222% xP. The total xP of this approach would be .4508, which is worse than the maximum xP of .4587.
The shape of the terminal xP curve on the graph reflects the shape of adding progressively worsening batches of serves as you get more aggressive.
At the top of the column I said I would explain why a 2:1 ace:error ratio is worse than a 1:1 ace:error ratio. First, I will say that anyone who has a 2:1 ace:error ratio is probably a good server. My contention is that they are not using their talent to its full potential, if they were to really dial up the aggression, they could be even better.
If the server whose production is described by this graph were to have a 2:1 ace:error ratio, at 6.1% terminal serves, their terminal xP would be .6677. Their in-play xP would be .4054. Their total xP would be .4214.
In contrast, if they had a 1:1 ace:error ratio, at 14.2% terminal serves, their terminal xP would be .5002. Their in-play xP would be .4437. Their total xP would be .4517.
This is what I mean by a 2:1 ace:error ratio being worse than a 1:1 ace:error ratio. If the most efficient serving approach for the server is closer to 1:1 than 2:1, a server serving 2:1 needs a bigger approach change to unlock their full potential.
Let’s talk about the shape of these curves. The terminal xP curve is shaped to reflect the following intuition: 1) as terminal serve % increases, terminal eff decreases, 2) the next batch of terminal serves will be much less efficient than the current total.
The in-play xP curve begins at the xP of a 3 pass by your opponent, reflecting the intuition that if you literally never miss, your opponent will likely pass a 3 pass almost always. It then increases at a decreasing rate and slows to almost a halt at slightly more efficient than always generating a 1 pass, reflecting the intuition that if you’re really letting it rip, your opponents will pass bad.
The challenge for coaches is estimating where an athlete is on these curves and advising them accordingly. My suspicion is that most float servers are to the left of their maximum, and that they can become more aggressive, see relatively small decreases in terminal serve xP and reap the rewards in total xP.
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