Random Walks and Tennis Scoring

[GameSetAndMath]

We have heard this statment many times; "It is a game of small margins". What exactly
does that mean? Intuitively, it means that the difference between winner and loser is not
that much.

Here is a complete illustration of that idea by British Mathematician Ian Stewart.

Suppose there are two players A and B, A being slightly better than B. What do
I mean by slightly better, say A has a 60% chance of winning a random point played
between them and B has 40% chance of winning it (if they are equal calibre it
would be 50-50). Then, you would naively expect that A would win 60% of all the
matches played between A and B.

However, the intricacies of tennis scoring is such that, if A has 60% chance of
winning a random point, A has 99.61% chance of winning a three set match between
A and B in a three set match.

To win a game, one must cross 40 and have difference of at least two points. To win
a set without tie breaker, one must reach 6 games and have difference of at least two
games. All of these intricate rules of scoring have the effect of amplifying the chances
that the better player (he may be better by a small percentage in winning a random
point played between the two players) wins the match. A small advantage in winning
a random point gets amplified to a huge advantage in winning the match by the rules
of tennis scoring.

There is a beautiful analysis of this done by IAN STEWART using the
theory of RANDOM WALKS and I am attaching it as a pdf file. Some of you
may be interested in taking a closer look at it.

Even if you cannot follow the details of the Math involved, the article
would still make an interesting reading as it is from a recreational math book
and not from a research paper.

 

[GameSetAndMath]

RE: It is a Game of Small Margins

Here is an article from tennis. com that discusses the psychological aspects
of tennis scoring on the players. This is a completely non-mathematical article.

http://www.tennis.com/your-game/2013/11/mental-edge-keeping-score/49853/#.UrdQPmB3uM8

While I am not denying that there is a psychological element in the big points, as you
can see from the article that I posted (just read first 2 pages), the reason the better player
has huge chances of winning is not just that the better player plays the big points well.
It is because the tennis scoring system AMPLIFIES the difference in the quality of the players
to such an extent that if one player is even slightly better than other player, the better
player is almost sure of winning the match.

 

[Kieran]

RE: It is a Game of Small Margins

Interesting points, GS&M. I read this the other day but didn't have time to reply. Remember, the big points can play tricks on any man's mind. The shots he played routinely in the first set can be made more difficult at 30-40, 4-4 in the fifth. At that stage, the racket becomes heavy and hitting the ball becomes an alien, fearful experience.

The guys who win the big points are the ones mentally tuned to do so. They block it out, focus on the details and come out the other end intact. So often we've seen gifted players tighten up simply because they were playing a slam final.

It's a great point about the scoring system, but there are intricacies involved. I'll read the article by Ian Stewart over the next few days, but wanted to keep this topic up near the top of the page because it deserves to be discussed in detail...