If you have problems winning close games the following lines might help you winning them most of the time.

After reading this article you should be able to answer the following questions:

- What is the score tree?
- How can it help us to judge whether we should serve or receive first?

Do you need any prior knowledge from an other blog article to understand this? **No**.

## Introduction

We start with the assumption that we expect a close match(set) because we don’t need such advanced deliberations if we are clearly losing or winning.

Additionally we assume that it’s an advantage to serve at critical scores.

## The score tree

If we collect all possible point scores we get the tree on the left side. The green fields are scores where we have won and the red where we have lost.

The yellow rectangles are scores where we’re even with the opponent. Because of this the left side of the tree consists of all scores where we have an advantage in points and the right side consists of scores which favor the enemy.

Picture two shows four examples of possible set paths. Please click on the images to enlarge them, they open in a new tab.

We now add another border, the five point border ( Based on this article, where it was stated that in 85 % of all cases the player who first reaches the five point mark will win the set.

Finally we point out all scores which lie within a two point frame above the given borders. We choose this two point frame, because the server is changed after two points played. This means that once we’re in this range we or our opponent can reach the five point score or win the set within one service period.

The tree within a tree effect can be seen quite easily in the above advanced score tree and shows us the important mini set inside every set.

## Serve and receive

The two pictures below highlight the scores where we serve if we start to serve ( green ) and if we start to receive ( red ).

If we combine the advanced score tree and the serve and receive trees we get this:

Since we just want to consider close sets we rule scores out, which have a difference of three or greater. The result can be seen below.

Let’s look a little close at our final trees. The serve first green tree consists of less important points (8) than our receive first tree in red (10). If we receive first, we have one more serve which we can control to achieve the five point goal first or regain a realistic chance to do so. This also holds true for the last points to win the set. The red tree covers points which are closer to the end of the set then the green one, which means that in really tight sets the red tree gives us a big advantage.

This helped us to understand, why we should always choose to receive first if we expect close sets and matches.

## Psychological side note

Like all aspects of the game, the serve or receive question also has psychological aspects.

Reasons to receive first:

- opponent might lack the needed fine coordination for serves at the beginning
- signals the opponent that we aren’t afraid of him
- less pressure on us since the server is usually expected to win his serving points (since the server has an advantage there)
- the enemy might even lack the coordination for his third ball attacks and miss them
- even if we fall back to 0:2 we have still enough time to get to five points first or win the set

Reasons to serve first:

- we are mentally weak and can’t hold the pressure of an early 0:2
- we can dominate the game from the beginning
- we serve at 8:8
- we might give the enemy a change to come into a rhythm if he scores with his first third ball attacks if we don’t start serving

We saw that if we are mentally stable there is hardly a reason to serve first. If we aren’t mentally stable to hold the pressure, we should work on our mental weaknesses to enjoy the advantages of receiving first.

## Mathematical side note

You can leave this part out if you don’ want to deal with the mathematics and this simplified model, it just provides another argument why receiving first isn’t a big disavantage.

For the remaining readers we go on with the simple model.

By looking at the picture on top we see a probability tree for our serve. We assume that we have a constant chance to win the point (x) if we are serving. If we don’t win the point then our opponents gets the point, therefore his chance is 1-x. Finally, our chance to win two points during our serve is x^2, the chance that we play 1:1 is 2*(x-x²) and the probability to play 0:2 is (1-x)². Common values for x are greater than 0.5 but smaller than 0.6 which means we win more than 50% of our serves but less than 60%.

A summary for all possible values can be seen below. The horizontal axis displays the advantage you think you have if you serve. A value of 0.1 means you think you have a probability of 0.5+0.1=0.6 to win the point, which means 60%. The vertical axis displays the probability for certain scores depending on the value of the horizontal axis. The black line is the score 0:2, the green line is the score 1:1 and the red line displays the score of 2:0.

Let’s consider an interesting application. The probability for at least achieving an 1:1 ( this means 1:1 or 2:1 ) during our serve is given by x^2+(2*(x-x²)). On the contrary, the probability for the opponent to gain the same result is (1-x)^2+(2*(x-x²)). If we consider the difference between these two we gain -1+2x. This term describes our advantage in percent to achieve a score better or equal than 1:1 compared to the chance of the enemy. This means if we assume to have a maximal chance to win 60% of our serves we have a 20% higher chance to score 1:1 or 2:1. If we score 50% of our serves we have no advantage over the receiver. This little example showed us that we maximally have a 20% higher chance to score at least 1:1. We see that we don’t lose too much if we give the serve away and start receiving.

## Summary

Chose to receive first!

EasterEgg: The score tree is hidden in the picture of the racket.

Hahaha! 🙂 I am not sure how to interpret this article. Is it a light-hearted analysis ? Or something really serious cooked up with GameTheory or something to explain the “human” factor in rallypoint scoring play systems ?

Anecdotally I request very weak players to always start serving in games against me. My reasoning is the serve advantage allows the player to “play” one extra ball. (1p) serve, (2p) push, (1p) attack/push, (2p) attack/push, etc.

Even against strong players, I tend to request them to start the serve since it gives me the ability to try to read their serve and level. But in matches where I want to win, I tend to start the service.

I am not sure, does service receive really give so much advantage for players on equal footing ? If I have “trick” serves early in the game, doesn’t the point lead benefit me ? Secret techniques could skew the results in favour of the server ! And similarly for the receiver!