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.