Monday, November 01, 2010

The Bateman Impossibility Theorem

Anyone who's put together tennis clinics has heard the following:

"My child needs to play with better players to get better."

The implication of this statement is far-reaching. It means simultaneous improvement of two or more players is impossible. Don't believe me? I'll prove it.

1. Let A and B be any two elements of the set of all tennis players.
2. Both A and B want to improve.
3. To improve, A and B must play with better players:
For A to improve playing with B, A > B,
For B to improve playing with A, B > A.

4. Since both A > B and B > A cannot simultaneously be true, A and B cannot improve playing together.
5. What holds for A and B holds for all tennis players.
6. Therefore, simultaneous improvement of two (or more) players is impossible.

I call statement 6 The Bateman Impossibility Theorem -- simultaneous improvement of two (or more) players is impossible.

Since we know that tennis players do improve, we know that something in the proof is wrong. Perhaps condition 2 is wrong and not all players want to improve. But why would such a player want to be in a clinic, and especially pay for a clinic? Perhaps if the parent of inferior player A pays for superior and indifferent players C, D, and E, then they'll join the clinic even though they don't want to get better. I suggest parents make this offer to the coach setting up groups. "Since my child needs to play with better players, I will pay the better players to be in the group with my child." Seems the right thing to do.

Perhaps condition 3 is not true. Do we have any evidence that condition 3 is not true? To that end I offer up the improvement of Rafael Nadal over the past few years. Nadal did not practice with Roger Federer, the only player arguably better than he over the last few years, and only rarely played against him. If we accept that Nadal has improved, then we must be very suspicious of condition 3. In fact, it seems to me that the continued improvement of most of the top players in the history of tennis falsifies condition 3.

In addition to the world's top players, I think we observe that the top player on a college team can and does improve, despite not having anyone on the team better than he or she to practice with.

In fact, I think we observe players getting better all the time practicing with equal or lesser players.

Therefore, I think we can safely say that condition 3 above, the assertion that playing with better players is a necessary condition for improvement is false.

Much as I hate to admit it, The Bateman Impossibility Theorem does not hold. Good thing. I'd hate to be stuck paying better players to play with me all the time.

I encourage coaches to share this proof with any parents who request that their child play with better players. It won't make any difference, I'm afraid. But you can try it.

Saturday, July 31, 2010

Soft Courts

I've been looking into affordable soft tennis courts lately. Clay in Colorado, and other dry climates, is an expensive pain. Grass is a pain everywhere. The soft Rebound Ace type courts are not what I'm looking for, too high a coefficient of friction. I want soft courts that allow sliding, like clay.

Omni and Classic Clay are two that I've tried recently. The Omni has sand, almost dirt, on top of an artificial grass surface. Pretty good. It gets my socks and legs dirty, but the playability is good, good traction with sliding to stop at high speeds. The ball bounces are pretty true and of decent height. Classic Clay is good, too. That's a less dirty product. The surface seems softer than the Omni and has a lower bounce, I thought.

On the USTA's tournaments web site I saw that a high school in Minnesota, Brooklyn Center High School, has artificial grass courts. They look to be Omni. I'm curious if there are any other high schools in the US that have soft tennis courts -- grass, clay, carpet, artificial grass or clay.

Sunday, February 28, 2010

The Trouble with Slice

I told a kid the other day, "If you master that slice forehand you can be a very good bad player. But if you keep working on that topspin forehand maybe you can become a good player."

It's not that good players can't hit slice forehands, of course. Good players can hit all the shots. Turn on your TV, though, or head to a local Division I college match and you'll see mostly topspin shots.

We tennis players cannot escape from physical reality and physical reality dictates that for a ball to be hit hard, over the net, and within the boundaries of the court, topspin is necessary. Unfortunately it's much more difficult to master topspin. Why? Because almost every ball that bounces comes off the court toward you with the top of the ball spinning in the direction of travel. That means if you want to hit a shot the other direction with topspin you must reverse the ball's spin direction. It's much easier to simply redirect the ball back the other direction without changing the ball's spin.

Also, for little kids, underspin, or slice, gives the ball lift which maximizes the carry of the ball. That's great if you're small and weak and have trouble getting the ball over the net and deep into your opponent's court. Slice is also great for kids because it bounces funny. Funny is fun and funny is difficult for beginning players to handle. They don't have much experience with extreme spin and they are surprised by the bounce of the ball.

All those advantages of slice disappear fairly quickly as players grow in size, strength and experience. But habits once learned can be hard to break. Kids who spend a few years hitting a lot of slice struggle mightily to master topspin.

This allure of slice makes life tough for coaches. Coaches have to intervene and get kids to forego the immediate reward of slice in favor of the short-term pain of learning to hit topspin. Smaller courts, lower nets, smaller racquets, and bigger, lighter balls can help get kids to hit with topspin. But even in the scaled down environment of Quick Start tennis, slice has many advantages. It's up to coaches to do the difficult job of selling kids on topspin. Kids who like to watch professional tennis on TV are much easier to sell on topspin since that's what they see on TV. The kids who come out for tennis but who are not fans of the professional or college games prove much tougher. If I come up with a way to solve this problem of slice, I'll post it here.

Thursday, February 04, 2010

Practice Tired

While watching the Kansas at Colorado basketball game last night I remembered how important it is to practice tired. Execution and decision-making both suffer when players become fatigued. In tennis this generally happens late in long points and late in long matches. Therefore it's important to engage in drills that extend beyond the length of most points to train fatigued shot-making and decision-making.

Since most points are not very long (six hits total or less) most of a player's practice will be, and should be, focused on the early stages of points. After all, even long points start at the beginning! But it's important not to totally neglect the longer points. I think one general strategy to point play may be to play more conservatively the longer the point goes. Since at most levels the vast majority of points are lost and not won, playing conservatively all the time is probably a winning strategy. But supposing you don't play that way all the time, at least consider refraining from going big late in points for two reasons. You don't have as much control and as precise timing late in points so you're more likely to miss. Your opponent is more likely to miss as s/he tires. I guess if you both adopt my strategy here the point will never end as you both turtle and just get the ball back. Maybe pull the trigger on your 50th shot!


As bad as my driver impact is my iron impact is more pathetic. No wonder I tear up short courses and get torn up by long courses.

Wednesday, February 03, 2010

Cheating and Tennis

I have a project idea for an enterprising psychology graduate student. Here's the hypothesis: competitive tennis players are more likely than average to cheat in school. This can be tested at the high school or college level. Players can be identified by their membership on their high school or college teams. Cheating can be measured by students subject to academic discipline. Unfortunately privacy rules probably prevent such a study.

My theory, unfortunately, is that tennis encourages cheating. Players make their own line calls in tournaments. We see cheating all the time and we see immense gains from cheating. It is very difficult to catch or stop a player from cheating. Opponents can request line judges, but the judges seldom stay on court for long. Even where umpires are on court, in college matches for instance, cheating still takes place.

So we have a situation where cheating is easy, rewarding, and has few negative consequences. My thesis is that this creates a habit of cheating that carries over into other aspects of tennis players lives.

I hope my theory is wrong.

Tuesday, February 02, 2010

My Sorry Swing

Based upon what we see below, it's amazing I can hit the ball at all.

The Importance of Impact

This is cruel, but it shows how important the impact position is in golf. This could just as easily be a video of me, but in this case it's not. Maybe soon I'll put my own sorry impact up here for comparison. But for now, watch and weep.

Sunday, January 31, 2010

Skill Level and Modern Tennis

So I just finished watching Murray and Federer in the Australian Open final. I was struck by how darn much skill it takes to win the modern game of tennis. The guys move so well and strike the ball so well that winning a point is ridiculously difficult. The windows the players have to hit through are tiny and close quickly. If you don't have the skill and courage to rip a ball to a corner when that window opens, you cannot win.

Back in the old days, my day, there seemed to be more ways to win. You could simply wait for your opponent to miss. That can work today, but it won't work in the finals of the majors, I don't think. You could serve and volley. That tactic often produced return errors from your opponent, passing errors from your opponent, or relatively easy volley winners angled to the open court if played behind your opponent. You could also chip and charge. A well struck, deep sliced approach shot put the defender at enough of a disadvantage that the points often ended like the serve and volley points did. If you were quick enough, you could also succeed defending against the serve and volleyers and the chip and chargers. It was possible to get into position to hit dipping passing shots or lobs that neutralized the attacker.

Now it really only works to serve big, return big, move your opponent off the court a bit, and go big into the small openings you create. Those finishing shots struck from around or just inside the baseline against an opponent who moves well and rips the ball are just way tougher to execute than the old finishing volleys or overheads were.

It's truly amazing to see the game played so well.

Friday, January 29, 2010

Australian Open Series, French Open Series, Wimbledon Series

I think the US Open Series has been a hit. The summer hardcourt US tournaments draw more attention from TV and from the fans due to the publicity of the US Open Series which began a couple of years ago. The top players still don't play that many tournaments. It's too tough on them since they play so many matches in all the tournaments. But a coherent leadup to the US Open is good for TV and good for the fans. It also motivates the marginal players to participate a little more for the chance to increase their winnings and exposure.

So why not define an Australian Open series, too. If the Aussie Open were moved back a couple of weeks into February, there would be ample time for an Asia/Pacific based Australian Open Series. The series could start just after the new year and run for six weeks or so up to the start of the Australian Open. Tournaments in Japan, South Korea, China, New Zealand, and Australia could make up the swing.

The same could be true for the French Open. The calendar already lends itself to this with a well-defined European clay court season. All that's necessary is to line up a sponsor for the French Open series and off you go.

The Wimbledon Series is a bit tougher since Wimbledon begins just two weeks after the French Open ends. Maybe we could push Wimbledon back one week, just long enough to have a little three-week Wimbledon Series. There aren't many grass court tournament venues left, and fewer players who like playing on it, so a shorter Wimbledon Series makes sense.

With this way of setting up the tennis calendar, all the tournaments would have a coherent marketing plan. Monte Carlo, the first stop in the French Open Series. Tokyo, the first stop in the Australian Open Series.


During the Tsonga vs Federer Aussie Open semi final Chris Fowler saw a Shot Spot replay of a ball 2 mm wide and said "We often hear that tennis matches are decided by millimeters. There's an example."

So is it true that tennis is a game of inches? Are matches decided by the slimmest of margins? Do matches turn on a crucial point?

The accepted wisdom is that yes, tennis is a game of inches and that matches hinge on pivotal points or moments. I'm suspicious.

I think the vast majority of matches are not decided by such things. Several years ago I wrote a little program to play simulated tennis matches. I used no-ad scoring to simplify my programming (I'm not a programmer). I played simulated 2 out of 3 tie-break set matches between players of varying skill. I varied the skill by varying the probability of each player winning a point. The greater the relative probability of winning any point, the greater the relative skill. Simple enough. Objective.

So I played matchees between evenly matched opponents, each having a 50-50 chance of winning any given point. Then I gave one player a 51-49 edge and ran a hundred matches or so. I did this up to a 60-40 edge where it was no longer interesting. Why not? Because the 60% player won every set easily.

My take on this after doing all my simulations is that tennis is not a game of inches and does not turn on big points most of the time. I think a better, though less dramatic, way to think of tennis is as a game where small advantages accumulate. The better player is more likely to win the longer the match. That's why Federer is so hard to beat in best of 5 set matches. He's better than his opponents so his advantage grows the longer the contest.

If you're going to beat Federer in a match, make it as short as possible. If you're going to beat Tiger Woods in golf, play him over one hole, not 72.

I did one other thing, too, to test the turning points theory. I charted a few matches, juniors and professionals. I had point-by-point data. I compared the accumulation of points to a random-walk and couldn't tell the difference. The random-walk point patterns looked indistinguishable from the randomly generated point sequences. So much for dramatic turning points in matches.

Wednesday, January 27, 2010

Randomization Strategy

How do you decide where to serve? How do you decide whether to poach in doubles? I suggest a coin or dice.

This could even work for deciding how many times you bounce the ball before you serve. Yesterday I experimented with randomizing how long I took between being ready to serve and tossing the ball. That is, I finished my routine of stepping to the line and bouncing the ball. Most players, myself included, serve with the same rhythm every time. That's generally a good thing. But it allows the returner to return in rhythm, too. By varying my pause just before I tossed to serve, I was able to keep my opponent from getting into a rhythm.

That's where a coin or dice come in. An opponent can only catch on to your routine if there's a reason behind your routine. If you allow a coin or some other randomizing device to determine where you serve, for instance, then by definition there is no pattern to catch on to.

The most convenient randomization device I use is my wristwatch. I play with a watch, so it's reasonable to expect that the seconds digit on my watch will come up randomly odd or even as I glance at my watch. Even means serve wide and odd means serve down the middle. Simple. Unpredictable.

If your opponent is stronger on one side, you can bias the "coin" in favor of that side. I can switch to a stopwatch running on my watch and then I can use the hundredths place for randomizing. If I want to serve 2/3rds of the time to my opponent's backhand, then I'll serve there when the hundredths place shows 1-6 and serve to the forehand when it shows 7-9. If it comes up 0 I can serve to the body or just click start and stop again. None of this takes more than a couple of seconds to do, so it does not delay play.

Give it a try. It seems odd, but if you don't want an opponent to pick up your patterns, don't have any patterns. That's the definition of randomness.

Tuesday, January 26, 2010


We're always looking for drills and games to help tennis players improve. The drills must represent the match environment to have the most benefit. So what does that mean exactly? It means that the strokes used in the drill must appear with the frequency they appear in matches. The exertion levels and work/rest ratios must be those found in matches. The movement patterns and tactical opportunities must be the same as those in matches. The stress levels must rise and fall like in a match. You get the idea. A drill where a player stands in line, then hits one forehand and one backhand to the same target, and returns to the back of the line fails in all respects, for example.

Enough preamble, on to the new (to me) game that we've been messing around lately. This can either be played with a groundstroke feed or with a serve. Of course, I prefer the serve since all tennis points begin with a serve and, depending upon your level, 85-95% of points include a return. The key to the game is the scoring system. A player wins a point (+1) by hitting either a clean winner or forcing his/her opponent into an error. A player loses a point (-1) by committing an unforced error. Winners are obvious, but the forcing shot vs unforced error distinction is gray. That's a feature, not a bug, in my mind. The players will frequently confer quickly after a point to come to an agreement about how to score a point. "That may have been unforced, but you did have to run pretty far to hit a low passing shot, so let's say I forced your error." There are no correct answers and the coming to judgment is part of the learning process. The game proceeds until either player gets +3 points or -3 points. In other words, either player can "win" the game (+3) or either player can "lose" (-3) the game. If you play with a groundstroke feed, you roughly take turns feeding. If you play with serves, the preferred method, then each player serves one game to +3 /-3 and then the other player serves the next game.

I like to play sets of these games to +3,-3 using the same scoring. So if I serve and "win" a mini game, meaning I get to +3, then I get +1 game. If I "lose" my service game, meaning I got -3 points, then I lose a game, going to -1 game. We alternate serving games until one player loses by getting -3 games or wins by getting +3 games.

One of the first times I played this game, we were tied in the set at +2 to +2. It was my opponent's turn to serve. We decided that he should just serve, rather than doing some tie-breaker or something. That way the pressure was on him to hold and on me to break, just like tennis.

We've experimented with more or less points. Playing to +5 /-5, and to +2/-2. We find that the +3/ -3 lasts about the right amount of time. There is just the right amount of pressure of game ending points.

Obviously you could do this with doubles, too. If you have a team situation, a coach could offer input on the scoring. That way the players and the coach could each learn what the other thinks is an unforced error compared to a forcing shot.

One warning. Low skill level players will simply be in a race to avoid -3. That's a problem, but it also shows them vividly that at their level, tennis points and matches are lost, not won.

I found that my tactical awareness jumped while playing this game. Recently I'd become fond of just pounding balls, not playing tennis, even when I was playing games or sets. It's fun to rip the ball and I don't play competitively any more. Further, when I did play I was mostly a serve and volleyer, or a guy who approached the net a lot. Now, with a modern game, I like to just pound groundstrokes. This game showed me that while it's fun to hit big groundies, I don't get them in often enough to actually win a groundie contest that way. In order to win at this game, and to avoid losing at this game, I need to be aware of when I've hurt my opponent and move forward to end the point. I became much more opportunistic in my court positioning and movement. That was an eye opener for me. Very valuable.

Give this game a try. I think you'll learn a lot and have a lot of fun doing it.

Monday, January 25, 2010

Continuous Play

Building on my last post regarding the drama of tennis, I'd like to make another move that would enhance spectator appeal.

I'm not sure when the rules for changing sides changed from "continuous play" to "have a seat, get a drink, towel off, and get a breather", but I'd like to go back to the old rule. My guess is that TV wanted regularly scheduled commercial breaks. I like televised tennis, so if commercial breaks need to happen every two games, then so be it.

But from a spectator's perspective, the delays on change of ends are ridiculous. The newish rule of not taking a break when changing ends following the first game of a set is a step in the right direction. I'd go further. I'd give players a longer break between sets, but not allow them to do more than to towel off and get a drink as they change ends within a set. The end change during a set would resemble the first game and within tiebreak side changes we see now. Give the players a sit-down break between sets. Three or four minutes seems reasonable to me.

If TV needs more time, then TV matches could have their own rules. How many of us play on TV anyway? Juniors, high school, college, and adult tournament players don't need to sit so much. Tennis points aren't that long in the first place. Players recover plenty between points. They don't need a whole minute to sit around every two games. Let's get on with it and keep moving.

It was OK for Laver and Rosewall in the 1969 French Open. It's OK for the rest of us now.

Tennis Drama

I just received an email from John Yandell at Not a personal email, just the one he sends out to everyone regarding his latest monthly offering. One of the articles he mentions is by Alan Fox. In the article Alan calls the tennis scoring system diabolical. I haven't read Alan's take, yet, but I think the scoring system is diabolical, too.

The tennis scoring system, points accumulating into games, games accumulating into sets, and sets accumulating into matches makes for interested waves of stress and drama. Tension builds within games and then falls as a new game starts. Tension builds within sets and then falls when a new set starts.

Among the changes I would like to see in tennis is the elimination of the second serve. I won't go into all the reasons for such a move, but I think that the tennis scoring system and the pattern of drama that it causes makes me think twice about the elimination of second serves. With only one serve, the probability of winning a point is much more equal between server and returner. This means that holding serve will be much less important. If a break of serve is not so important, then the drama of each game will decrease early in a set. Winning a game will still be significant, of course, but much less so than when breaking serve is much more difficult.

When I watch Wimbledon I'm much more interested in the first few games of a set than I am when I'm watching the French Open. On the grass at Wimbledon, breaking serve is very difficult. Therefore a break of serve at any time could determine the winner of the set. On the clay of Paris, a break of serve is more common, so an early break of serve is not as likely to determine the winner of a set. Therefore, the drama of the early games of a set is much lower on clay than on grass.

By analogy then, if players have only one serve instead of two, breaks of serve would be much more common and the drama of the early games of a set would be much lower.

That's good if you want a chance to go to the bathroom, get a drink, or talk to your friends. But it's bad if you're looking to keep the interest of the fans and keep the tension high for the players.

Sunday, January 24, 2010

Entropy and Making a Change

Ever wonder why great players hesitate to make changes in their games?

One reason, I think, is that once you're very good at something, any change you make is far more likely to make you worse than make you better. There is no single best way to play any game, but there are certainly far, far more ways to play poorly than to play well. So once you play well, the odds dramatically favor a change being for the worse, not for the better.

Pull Phase and the Inside Out Forehand

Take a look at the video lesson below that I did for a student. It deals with the Pull Phase of her forehand, specifically the direction of the pull.

In the lesson I'm encouraging her to pull the butt of the racquet in a line more parallel to the intended target line. You can see that she pulls the racquet in a line away from her body. This is a common pull direction and may explain why so many great players today prefer the inside out forehand and struggle with the forehand when they have to move to their left. A forehand struck on the move to a righties right means that the racquet will have to rotate a lot in order to hit the ball straight ahead or to the left. I wonder of this pull phase is responsible for the preference that modern players demonstrate for the inside out forehand.