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. Furthermore, when I adjust for the fact that some players are better than others (more highly ranked), momentum is truly nonexistent. On the other hand, an alternative theory, The Fractal Theory of Tennis, which is grounded in the fundamental psychological relationship between favorites and underdogs in a tennis match, generates interesting (though largely untestable) results. This relationship between the favorite and the underdog is fundamentally characterized by the underdog’s fear of victory.
Momentum
If you watch sports on television, you will often hear ‘experts’ using the term ‘momentum’. Changes in momentum are often used in order to explain outcomes in many sports. For example, a fumbled kickoff return in a closely contested football game will often be labeled as a ‘change of momentum’. However, that same fumble by a team leading by 6 touchdowns never is labeled as such because it will have no possible effect on the outcome of the game. Momentum is clearly context dependent: the score must be ‘close’ for momentum to come into play.
The fact is, that the team with the ball, in an even game with time running out, will most likely win because they have a higher probability of scoring next. In other words, they have the lead in a probability sense. This mistaken use of the term ‘momentum’ is actually what I call 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 3set victory. By way of comparison, in 2005, NFL teams, which had the lead at halftime, won 78% of their games.
There are similar circular arguments about what constitutes momentum in tennis. One leading coach, Chuck Kriese, the men’s coach at Clemson, 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. It’s like the basketball coach whose team is trailing at halftime who remarks: ‘we need to make our shots and make some stops on defense’. This is the Corollary of the Unremarkable Hypothesis: if we score more than they do, we’ll win. Thus, momentum, in most discussions, is nothing more than circular reasoning. (Aside: Despite my critique of one of his momentum concepts, Kriese’s book, Total Tennis Training, is a MUST read for all tournament players. In fact, his discussion of the ‘pecking order’ in tennis is a key component of The Fractal Theory).
So, given that it’s difficult (if not impossible) to analyze momentum at the micro level of ‘within the game’, I propose to analyze momentum at the macro level of the match. The starting point 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 2outof3 set match. However, not only are the players tied, but one of them, the one who won the 2nd set, has momentum.
The testable hypothesis here is: does momentum matter when players are evenly matched, i.e. tied after 2 sets? The answer to this question also implies certain conclusions. For example, if momentum is found to matter, then this suggests that either the winning player is ‘learning’ as the match progresses or the emotional balance has tipped in his favor. Conversely, if it turns out that momentum has no effect on the outcome of matches, then this would suggest that the scoring sequence has no impact on the outcome. Both players would focus and perform equally well during the 3rd set. This would mean that the third set would literally be like the toss of a fair coin, i.e. a 5050 chance for both players.
The data used to investigate the momentum hypothesis is from the 2005 Men’s Professional Tour. All of the data was gathered from Stevegtennis, a popular website with the results of all of the professional tournaments. All tournaments in 2005 are included in the data with the exception of the Grand Slams which play a 3/5 format. As mentioned above, the number of first and second set winners will be compared to the expected results from the repeated toss of a fair coin. Using the binomial theorem, I calculate the probability that a fair coin would produce the observed result. A high probability suggests that the coin is not fair, i.e. that momentum exists.
The results are reported in the Table below. There were 741 3set 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 probvalue 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.
Table
Sample  # of matches  first  second  % first  % second  prob 
Full  741  355  386  47.9%  52.1%  0.135 
Grass  61  41  20  67.2%  32.8%  0.998 
Hard  372  177  195  47.6%  52.4%  0.189 
Clay  261  114  147  43.7%  56.3%  0.024 
Carpet  47  23  24  48.9%  51.1%  0.500 
Outdoor  565  272  293  48.1%  51.9%  0.200 
Indoor  166  77  89  46.4%  53.6%  0.198 







When broken down further by court surface, the momentum hypothesis seems to hold only on clay. Without getting into an involved discussion, my guess is that this is the result of ‘learning’. There is less room for luck on clay once an opponent ‘figures you out’ because the points last longer.
If you think about it a bit more, then the test that I conducted above to detect momentum suffers from a potential defect: two players are not really equal at the start of a match. One player is always ranked higher than the other. I adjust the momentum data with the following rules: 1) if a seeded player plays against an unseeded player, he is ‘seeded’; 2) if a higher seed plays a lower seed, he is ‘seeded’; 3) if two unseeded players play each other, they can be regarded as ‘equals’.
When the data is adjusted to take account of 3set matches between ‘equals’, the results are truly astounding. Of the 288 3 set matches between equals, the first set winner wins 143 matches (49.7%) and the second set winner 145 matches (50.3%). Momentum is clearly a chimera with the exception of matches played on clay. On clay, 2nd set winners win 57% of all 3set matches between ‘equals’.
So, momentum can really only be said to exist on clay, where 5657% of the 3rd set winners exhibit momentum depending on the model used to estimate momentum. Yet, this effect is still fairly small (67%). As suggested above, the absence of momentum could mean a high level of competency in dealing with adversity. If this is true then perhaps results from junior tennis, where a player’s mental skills are not as well developed as those of the pros, would show that momentum does, in fact, exist. However, this is not the case. I looked at all main draw matches from L1+ B14 Eastern tournaments from 2005. Of the 106 3set matches played, 55, or 51.9%, were won by the player who won the second set.
Momentum doesn’t explain much when it’s stripped down to its essentials. The fact that the player with momentum wins about 52% of the time is not very different from a coin flip. The small difference declines to zero when I account for the relative quality of players in a match. The fact that momentum is often confused with the fact that the team (or player) with the lead tends to win, or the team that scores more than the opposition ends up victorious, is circular thinking at its worst.
So, all things considered, momentum is nearly as uneventful as the Unremarkable Hypothesis and it’s Corollary. Therefore, if momentum is not a meaningful lens through which to understand tennis matches, the natural question is: is there a better lens? An alternative theory, ‘The Fractal Theory of Tennis’, provides a better alternative.
The Fractal Theory of Tennis
What are Fractals[1]?
Fractals model complex physical processes and dynamical systems. The underlying principle of fractals is that a simple process that goes through infinitely many iterations becomes a very complex process. Fractals attempt to model the complex process by searching for the simple process underneath. Almost all fractals are at least partially selfsimilar. This means that a part of the fractal is identical to the entire fractal itself except smaller.
Fractals are used to investigate many things. For example, the NY Times (Feb. 9, 2006) explained how a physics professor employed fractals to examine the authenticity of several Jackson Pollock paintings. He looked for ‘patterns that recur on finer and finer magnifications, like those in snowflakes’.
What is The Fractal Theory of Tennis?
The Fractal Theory of Tennis models a ‘complex process’, a complete tennis match. Certainly a tennis match is a complex process, given all of the variables in play at any one time: court surface, shot selection, type of ball, stroking technique, etc. The ‘simple process’ that goes through ‘many iterations’ is the repeated confrontation of the underdog player with his root fear: the fear of victory. The selfsimilar aspect suggests that this fear must first be confronted at the level of the game, then the set, and ultimately, the match.
The Fractal Theory of Tennis embodies four basic principles. First, 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, the men’s coach at Clemson, 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’.
Secondly, 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 builtin inertia that favors the higherranked player. Concretely, this inertia manifests itself with the lowerranked player making unforced errors whenever she is about to take the lead.
Thirdly, if the higherranked player fails to capitalize on these unforced errors, the lower player gradually begins to build confidence and he takes the lead.
And finally, this battle with fear, the fear of change, is reenacted 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.
The elements of the Fractal Theory are summarized in this chart:
Fractal Theory of Tennis
 Tennis has a ‘pecking order’. Most matches begin with a clear favorite and an underdog.
 The underdog is content to lose to the favorite (at the subconscious level).
 The favorite must capitalize on the underdog’s willingness to lose.
 Elements #2 and #3 confront each other at the levels of the game, the set, and the match.
Here are two concrete examples of what the Fractal Theory allows the tennis fan to observe. First, 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 twosetstolove lead he was again confronted with the possibility of winning (#4: what he had confronted at the game and set level now reared its head 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 Theory would suggest the likelihood of an Agassi victory at this point since he had won the two previous sets before the fifth (52.4% if I use the percentage of 2nd set winners from the hard court category in the momentum table and apply it to a 3/5 set match). 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 54 in the fifth, but couldn’t close it out.
To be fair, the Fractal Theory predicts neither the winner of the match nor the fact that the favorite will always win the match. In fact, upsets do occasionally happen. What it does say is that Blake would once again confront the victory demon in the fifth set and that he was more likely to fail. This was the same demon that had scared him from victory in the early games of the match and again in the 3rd and 4th sets. It was still possible for Blake to slay the demon in the fifth set.
A second example of using the Fractal Theory was the 2006 Australian Open semifinal between David Nalbandian and Marcos Baghdatis. Nalbandian was the clear favorite (#1) and he raced to a 63, 51 lead. At this point, his game stalled (he acted like an underdog!) and Baghdatis climbed back to 5all. At 5all, ESPN commentator Patrick McEnroe declared: ’Watch out, the momentum is clearly with Baghdatis’. He promptly lost the next two games giving Nalbandian a seemingly insurmountable 2 sets to love lead.
However, the story is actually a bit more complicated than this. Baghdatis was actually serving at 5all, 4030 in the 2nd set. At this moment (confronting his subconscious fear of victory), he promptly dumped three straight backhands into the net to lose the game (#2)! Baghdatis was ready to lose this match, but Nalbandian refused to take advantage of his errors, and soon the match was tied at 2 sets all (#3). At this point in the match, McEnroe abandoned his momentum speak and was definitely in touch with the fear factor: ‘I’ll take the guy with no fear in the 5th set versus a guy who’s fresher and has fear’. (Aside: what McEnroe is missing here is that Baghdatis also had fear! But, it was hidden (subconscious) and would resurface at the start of the 5th set!)
The Fractal Theory hypothesizes that Nalbandian would take the lead in the 5th set because the 5th set will take the form of the match up until that point. In fact, he led both 20 and 42, but was unable to close out the match. The Fractal Theory again did not predict a winner of the match. It only said that Nalbandian would lead and the drama would be whether or not he could close out the match. In this case, the favorite lost because he could not overcome his own fear, i.e. he acted like an underdog!
Statistical Testing of the Fractal Theory
Statistically testing 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.
However, the Fractal Theory has several implications. If the barrier to defeating a favorite is as significant as the theory hypothesizes, I should observe a higher percentage of overall wins, easy wins (straight set wins), and 3set wins when a seed plays a lower ranked player than when any two random players meet. In fact, this is the case. Seeded players win 69.3% of all matches when playing unseeded players. They win in straight sets 71.8% of the time versus 68.9% for all match types. And, they win 61.8% of all 3set matches against unseeded opponents.
Certainly, these numbers do not prove the existence of the Fractal Theory. A simple hypothesis such as ‘better players usually defeat weaker players’ would do equally as well. However, if these numbers were not as they are, it would enable me to reject the Fractal Theory as a plausible way of understanding a tennis match.
In the end, the best test for the Fractal Theory is for all of you tennis fans to use it 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.
Originally posted on tennisweek.com, July 2006.
[1] From Eric Green’s website: www.cs.wisc.edu/~ergreen. Eric has an M.S. in Computer Science from the University of Wisconsin, Madison.