How does knowledge of mathematics help us gain insight into the world of tennis?

As the players emerge to the cheers of the crowd and stroll onto the court to confront each other—fated to meet by a tournament structure prepared weeks ago—commentators draw upon a wealth of statistics to discuss the upcoming match for the watching fans. Blisteringly fast serves turn into thrilling rallies where every motion of the ball is tracked in real time by analysing radar signals. After the tournament is over, the world rankings are adjusted.

A thrilling Grand Slam tennis final appears to be a long way from the world of mathematics. Yet there are many ways in which mathematics sits quietly underneath the surface.

Planning the play-offs

Long before players set foot on the court, every match in their tournament needs to be scheduled. To decide how best to match up the players in the first round, the entrants to the tournament must first be seeded . The easiest way to seed the players is to look at their world rankings—but how are these determined in the first place?

World rankings

Every week the men's Association of Tennis Professionals (ATP) and the Women’s Tennis Association (WTA) compute and update the world rankings for professional tennis players. They do this by awarding ranking points to every player (separately for singles —and doubles —matches) then ranking players in order from most to fewest points. In order to rank doubles teams, the ranking points from both players in the team are simply added together.

How are the ranking points calculated? Each player’s best tournament results over the last 52 weeks are selected (though some important tournaments, such as the four Grand Slam tournaments, must be included in the calculation), and points are awarded for each tournament based on the type of tournament and how far the player progressed before being eliminated.

Points awarded for men’s and women’s tournaments in 2016 Listed below are the points awarded for tournaments in 2016. Tournament Win Finals Semi-finals Quarter-finals Round of 16 Round of 32 Round of 64 Round of 128 Men’s Grand Slams 2000 1200 720 360 180 90 45 10 Women’s Grand Slams 2000 1300 780 430 240 130 70 10 Men’s ATP World Tour Masters 1000 1000 600 360 180 90 45 25 10 Men’s ATP World Tour 500 500 300 180 90 45 20 - - Men’s ATP World Tour 250 250 150 90 45 20 10 - - Women’s Premier Mandatory 1000 650 390 215 120 65 35 10 Women’s Premier 5 900 585 350 190 105 60 1 - Women’s Premier 470 305 185 100 55 30 1 -

The WTA currently awards players in a 2016 Grand Slam tournament 780 points for being eliminated in the semi-finals, 1,300 points for being eliminated in the final and 2,000 points for winning the tournament; the points almost double for every extra round that a player progresses. This is an example of exponential growth, a mathematical process that causes numbers to grow rapidly.

At the time of writing, Samantha Stosur is the highest-ranked Australian women’s tennis player. In 2015, Stosur was eliminated in the round of 64 in the Australian open (earning 70 points), the round of 32 in the French Open (earning 130 points) and Wimbledon (another 130 points), and the round of 16 in the US Open (earning 240 points). So, at the very start of 2016, the Grand Slam tournaments contributed 70 + 130 + 130 + 240 = 570 points towards Stosur’s combined ranking points (for singles tennis) for the previous 52 weeks.

Samantha Stosur, the highest-ranked Australian women’s tennis player at the time of writing. Image source: Yann Caradec / Wikimedia Commons.

Seeding a tournament

Once the players are ranked, seeding a tournament can just be a matter of listing the top 32 entrants in order of ATP or WTA rankings. This is how the Australian Open, French Open and US Open tournaments are seeded. Wimbledon, however, chooses to do things a little differently. Instead of using the ATP ranking points, they use their own modified points calculated using the formula:

$$\text{Ranking points} = \text{ATP} + G_{\text{all}} + \left(75 \% \times G_{\text{best}} \right)$$

where:

ATP is the total ATP points accumulated by the player until the Monday before the tournament starts

G all is the total number of points earned for all grass court tournaments in the past 12 months

is the total number of points earned for all grass court tournaments in the past 12 months G best is the number of points awarded for the player’s best performance at a grass court tournament in the 12 months before that.

As Wimbledon is played on grass courts, the formula is designed to give more weight to ranking points that players earned on grass courts. Interestingly, Wimbledon bases their seeding for the women’s tournament on the unmodified WTA rankings, in line with the other Grand Slams.

Structuring a tournament

Once the top players have been seeded, there are two main considerations that determine how tennis tournaments are structured.

First, the number of matches needs to be kept to a manageable level. Grand Slam tournaments start with a field of 128 players. If we design a tournament in which every player plays a match against every other player, then every single player would have to play at least 127 matches throughout the tournament (exhausting!) and there would be at least 8,128 matches in total. The tournament would have to run over most of the year!

Where did the number 8,128 come from? A branch of mathematicians called combinatorics is used by mathematicians to quickly count things without actually having to count them one by one. For instance, how many tennis matches would need to be played before every player in the field of 128 has played every other player? This is the same as asking how many possible pairs of players we can make by choosing 2 players out of the 128. Let’s imagine the two players standing next to each other… $$\_\_\_ \; \_\_\_$$ Now we have to select the players. We start by choosing the first player in the pair. How many choices do we have? There are 128 players in the tournament, so we have 128 options for the first player … $$128 \; \_\_\_$$ Next, we choose the second player in the pair. How many choices do we have now? Well, we’ve already picked out 1 player, so that leaves 127 players left to choose from … $$128 \; 127$$ Now for each of the 128 choices of the first player, we can make 127 selections of the second player. So the total number of possible pairs is $$128 \times 127$$ But wait, we counted too many pairs! Here’s the problem: choosing Federer first and Djokovic second gives us the same pair of players as choosing Djokovic first and Federer second. So we’ve actually counted each pair twice. To reach the correct number of pairs, we have to halve our count … $$\frac{128 \times 127}{2}$$ which is 8,128. With this method we can find the answer to our question without having to count each combination individually.

Can we do better? What is the fewest number of matches that we need in the tournament?

If we assume that a player has to lose at least one match in order to be knocked out of the tournament, then no matter how we structure the tournament we’re going to need at least 127 matches. That is because we’re starting with a field of 128 players, and 127 of them have to be knocked out before we can determine the winner.

Can we design a tournament that determines a winner after precisely 127 matches? Yes, we can, and it’s surprisingly straightforward. We use what is called a ‘single-elimination’ tournament structure (or what mathematicians who study graph theory might call a ‘binary tree’).

This tournament features the mathematical process of exponential decay—half the players that start any given round are knocked out by the end of that round, which causes the number of players left in the tournament to drop off rapidly as the tournament progresses. The winner and runner-up of the whole tournament only need to play 7 matches, and every other player in the tournament players fewer matches. With the minimum number of 127 matches played, the Grand Slam tournaments can be held in a very reasonable timeframe of 2-3 weeks.

Why do Grand Slam tournaments have 128 entrants? If we use a single-elimination tournament structure and want to avoid byes (having some players sit out the first round), then the number of tournament entrants must be a power of 2 . To see this, note that we have exactly 1 winner, so the last round involves 1 final match involving 2 players—and those 2 players must have won their respective semi-finals. So there must have been 2 semi-final matches played in the previous round, involving 4 players in total—and those 4 players must have won their respective quarter-finals. So there must have been 4 quarter-final matches played in the previous round, with 8 players in total … and the pattern emerges! Since the number of players halves as we progress through the rounds (the winners keep playing, while the runners-up are eliminated), the number of players must double as we move backwards from the final to the first round. All that doubling means that our number of starting players must be 2, 4, 8, 16, 32, 64, 128, 256 or larger powers of 2. Grand Slams have 128 players because it’s the ‘Goldilocks’ option—not too few players (as 64 would be), not too many (as 256 would be), it’s ‘just right’.

The second thing to consider is that, ideally, the tournament should become more difficult for the players as the rounds progress—players should be more likely to encounter higher-ranking players later in the tournament. This will naturally happen as stronger players triumph in their rounds, but we need to be careful in our tournament design: we want to make sure that the top seeds don’t meet each other (and knock each other out) at the start of the tournament, so we place the seeded players carefully in tournament brackets. The first and second seeds are placed at the very ends of the first round, like so:

This ensures that, if the first and second seeds win all of their matches, they can’t face each other until the finals. A similar idea is used to make sure the top 4 seeds can’t meet each other until the semi-finals—seeds 3 and 4 are placed (randomly) in the following two positions.

Likewise, none of the top 8 seeds can meet until the quarter-finals, the top 16 won’t meet until the fourth round and the top 32 seeded players won’t meet until the third round. The remaining 96 unseeded players (8 of which are ‘wild card’ entries to the tournament, selected on other grounds than their world ranking) are then placed randomly in the remaining slots. This careful setup of the tournament draw is why you never see two big-names clashing until the later stages of the tournament.