## Is There a Way to Understand Results in Tennis?

### Introduction

In this article, I will discuss the merits of two alternative theories for understanding what is happening in a tennis match: Momentum and The Fractal Theory of Tennis. I will show that momentum is an overused cliché. When pared down to the point where it actually generates a testable hypothesis, momentum doesn’t matter more than a little bit. On the other hand, an alternative theory, The Fractal Theory of Tennis is grounded in the fundamental psychological relationship between favorites and underdogs in a tennis match: the underdog’s fear of victory.

### Momentum

Sportscasters frequently attribute to Momentum what is more accurately called The Unremarkable Hypothesis: teams in the lead tend to win. It should come as no surprise that The Unremarkable Hypothesis holds in tennis. Our data from the Men’s Pro Tour in 2005 shows that players with the first set in hand win 83.8% of the time. This can either be a straight set or 3-set victory. By way of comparison, in 2005, NFL teams, which had the lead at halftime, won 78% of their games.

An alternative definition of momentum in tennis is given by Chuck Kriese, the men’s coach at Clemson. He defines it as ‘winning three points in a row’. He says that a player’s goal is to build momentum whenever possible and to prevent his opponent from building momentum. In other words, win as many three point sequences as possible, while limiting the three point sequences of one’s opponent. If you think about this for a little bit, you will realize that this simply means winning more points than one’s opponent. This is the Corollary of the Unremarkable Hypothesis: if we score more than they do, we’ll win.

The starting point of a coherent discussion of momentum is to recognize that two players are precisely equal at two points during a match. The first is trivial. It is right at the start of the match. The second is much more interesting. It occurs when the players split sets in a 2-out-of-3 set match. *However, not only are the players tied, but one of them, the one who won the 2**nd set, has momentum.*

The testable hypothesis here is: does momentum matter when players are evenly matched, i.e. tied after 2 sets? The data used to investigate the momentum hypothesis is from the 2005 Men’s Professional Tour with the exception of the Grand Slams which play a 3/5 format. The number of first and second set winners will be compared to the expected results from the repeated toss of a fair coin. A high probability suggests that the coin is not fair, i.e. that momentum exists.

There were 741 3-set matches on the Men’s Tour in 2005. Of these, 386 (52.1%) were won by the player who won the 2nd set, and 355 were won by the player who won the first set. The prob-value that this percentage of 2nd set winners would be greater than the observed number (386) strictly by chance is .135. While this probability is quite low, it falls just outside of the standard confidence interval most widely used in the social sciences to analyze statistical significance (.1). I conclude that, while the data are suggestive of a small momentum effect (2.1%), I cannot reject the hypothesis that these results were generated by the repeated toss of a fair coin.

This result also appears to hold for juniors. I looked at all main draw matches from L1+ B14 Eastern tournaments in 2005. Of the 106 3-set matches played, 55, or 51.9%, were won by the player who won the second set.

### The Fractal Theory of Tennis

The Fractal Theory of Tennis embodies four basic principles:

1) Tennis is characterized by a ‘pecking order’. Most matches are played with both a distinct ‘favorite’ and an ‘underdog’, i.e. players have expectations about their likelihood of winning given their relative positions in the pecking order. Chuck Kriese, writes in *Total Tennis Training*: ‘The pecking order is so rigid that I firmly believe if a player is not favored to win at least in his own mind, he usually cannot and will not win’.

2) The underdog is comfortable where he is. An upset win will force him to change his/her mind about his position in the pecking order. The theory hypothesizes that, at a subconscious level, individuals do not like to change. Therefore, there is a built-in inertia that favors the higher-ranked player. Concretely, this inertia manifests itself with the lower-ranked player making unforced errors whenever she is about to take the lead.

3) If the higher-ranked player fails to capitalize on these unforced errors, the lower player gradually begins to build confidence and he takes the lead.

4) This battle with fear, the fear of change, is re-enacted whenever the ‘threat of victory’ is replicated. It is this replication within the game, within the set, and within the match that gives the Fractal Theory of Tennis its name.

Here is an example of what the Fractal Theory allows the tennis fan to observe. Let’s go back to last year’s US Open where Andre Agassi met James Blake in the quarterfinals. Blake had already defeated Rafael Nadal earlier in the tournament and was playing extremely well, but the pecking order still had Agassi as the favorite (#1). The match began with brilliant play from Blake, who nevertheless made silly errors whenever he was about to take the lead early in the first set (#2). However, Agassi failed to capitalize on these opportunities and Blake gradually began to feel comfortable with winning (#3) and took the first set. When Blake had taken a two-sets-to-love lead he was again confronted with the possibility of winning (#4: what he had confronted at the game and set level now reappeared with respect to winning the match). Andre raised his game and Blake played a lackluster 3rd and 4th sets. The match was soon tied at 2 sets apiece.

Momentum would suggest the likelihood of an Agassi victory at this point since he had won the two previous sets before the fifth. And, in fact, Agassi did win. However, Fractal Theory allowed me to make a far more interesting prediction which I made at the start of the fifth set: Blake would get an early lead and confront the victory demon one more time. This is because Fractal Theory hypothesizes that the fifth set will take the form of the match up until that point. In fact, Blake served for the match at 5-4 in the fifth, but couldn’t close it out.

Statistical testing of the Fractal Theory is impossible. It would require watching a large number of matches and categorizing points according to their ‘fear value’. For example, simply looking at the number or percentage of break points converted during a match would go nowhere in investigating the Fractal Theory. This is so because we don’t know whether or not break points were saved by the favorite with aces or with unforced errors on the part of the underdog.

In the end, the best test for the Fractal Theory is for all of you tennis fans to try it out while watching your next tennis match. And I believe that you will see what I see: fear, fear, everywhere! For all of his skills (soft hands, etc.), John McEnroe had radar for fear. When his opponent showed it, McEnroe didn’t delay in going for the kill.