Not all points are created equal: the true value of tennis points

Tags: sports, tennis
Date: 2025-04-10

One of the most fascinating aspects of tennis is that a player can win a match by winning fewer points and/or fewer games than their opponent. Consider, for example, a match between A and B that ended 6-4 0-6 6-4: player A won the match by winning 12 games, while B lost despite winning 14 games. The same concept applies to the total number of points.

This concept is also valid for other sports, such as volleyball, padel, table tennis and similar sports, but it’s my blog and I like tennis more.

A trait of a good tennis player is the ability to treat every point in the match as if it were the most important one, and then manage to forget about it before the next point begins.

"In the 1,526 singles matches I played in my career, I won almost 80%. What percentage of points do you think I won in those matches? Only 54% ... When you lose every second point, on average, you learn not to dwell on every shot."

Roger Federer

While thinking that all points have the same value helps maintain high concentration, this is intuitively false. Consider two very different scenarios: 15-15 and 40-40. Even intuitively, it's clear that a point at 40-40 has a greater influence on the outcome of the match. The same concept can be applied to games (think of 5-4 versus 2-1), and even to sets. The closer you get to the end of a game, the more the points weigh; the closer you get to the end of a set, the more the games weigh; and finally, the closer you get to the end of the match, the more the sets weigh.

By the way, this reminds me of a famous quiz that is supposedly asked in job interviews. A match is played by two players, A and B. You have a sequence like "AABABAAB...", which represents the sequence of points won by one player and the other. How can you quickly determine who won the match?

A clumsy model

One might wonder if there's a way to effectively estimate the value of a point considering the score situation. I don't think there is one, so I'll try to create one myself.

Three elements influence the value of a point:
- the score situation in the game;
- the number of games won in the set by both players;
- the number of sets won by both players.

Since the closer we get to the end of the game/set/match, the more relevant the point becomes, we could take for each of these three elements the minimum distance to win the game/set/match for each of the two players, and then sum their inverses.

For example, at 30-15, the minimum number of points to win the game is two for player A, while it's three for player B. If the score in a set is 3-2, the minimum number of games to win to take the set is 3 for player A and 4 for player B. If the score is 1 set all, the minimum number of sets to win the match is 1 for both (assuming a best of three sets match). The inverse of these minimum distances would be:
- 30-15: 1/2 + 1/3;
- 3-2: 1/3 + 1/4;
- 1 set all: 1/2 + 1/2.

Trying to formalize, we would have a formula like this:

Point_Value = (1/Dist_game_A + 1/Dist_game_B) × (1/Dist_set_A + 1/Dist_set_B) × (1/Dist_match_A + 1/Dist_match_B)

This way we would have results like:
- 1 set all, 5-3, 30-15 -> (1/2 + 1/2) x (1 + 1/3) x (1/2 + 1/3) ‎ = 1.111
- 1 set to 0, 2-2, 40-40 -> (1 + 1/2) x (1/4 + 1/4) x (1/2 + 1/2) ‎ = 0.75
- first point of the match -> (1/2 + 1/2) x (1/6 + 1/6) x (1/4 + 1/4)‎ = 0.833
- 1 set all, 4-4, 30-40 -> (1/2 + 1/2) x (1/2 + 1/2) x (1/2 + 1)‎ = 2.5

One of the many limitations of this model is that it treats points, games, and sets in the same way, when in reality this is not the case. But the purpose of this model is not to represent realit, rather it is to convey a concept.

"The analysis is severely limited by my lack of understanding of what I am doing."

Unknown student

If it's true that in football the team that scores the most goals always wins, it's also true that some goals count more than others. A goal at 0-0 influences the match more than a goal at 4-0, just as a goal in stoppage time weighs more than a goal in the opening minutes (simply because there's less time to recover). In general, we could say that there are moments more relevant than others to the outcome of the match.

I'll spare you the philosophical conclusion along the lines of "this doesn’t apply to sports only”, even though it's true and I would have liked to say it.