If you play tennis then you know the coordination problem. Fumbling in your pocket to grab a ball and your rallying partner doing the same and then the kabuki dance of who’s gonna pocket the ball and who’s going to hit first? Sometimes you coordinate, but seemingly just as often the balls are simultaneously repocketed or they cross each other at the net after you both hit.
Rallying with an odd number of balls gives you a simple coordination device. You will always start with an unequal number of balls, and it will always be common knowledge how many each has even if the balls are in your pockets.
I used to think that the person holding 2 or more should hit first. That’s a bad convention because after the first rally you are back to a position of symmetry. (And a convention based on who started with two will fail the common knowledge test due to imperfect memory, especially when the rally was a long one.)
Instead, the person holding 1 ball should hit first. Then the subgame following that first rally is trivially solved because there is only one feasible convention.
By the way, this observation is a key lemma in any solution to Tyler Cowen’s tennis ball problem.
Of course this works with any odd number of balls. But five is worse. It becomes too hard to keep track of so many balls and eventually you will lose common knowledge of the total number of balls in rotation.
Cheap Talk 
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July 8, 2010 at 11:44 pm
JC
If you’re actually playing a game, it’s obvious who’s going to serve, right? And if you’re just knocking the ball around you should be able to shout/wave to inform the other person you want to serve. Unless you’ve taken a vow of silence or something.
Here’s an alternative explanation: For any point being played in the game, having 3 balls available with the person who is serving should cover almost all possibilities without them having to stop to ‘recycle’ balls:
1) correct first serve, point gets decided – only 1 ball needed
2) serve replayed – first serve was a fault or a let or needed to be replayed, second serve is correct and the point gets decided – 2 balls needed
2 a) first serve was a fault ,second serve is also a fault, point gets decided – 2 balls needed
3) first serve is a fault/serve/replay, second serve also needs to be replayed (replay/let/fault-after-let etc), third serve is correct or is a fault, so the point gets decided – 3 balls needed
This would go on to need 4 serves for the point to be resolved only if 2 of the 3 previous serves were lets or replays, and that’s a very rare scenario. Most players after hitting 2 lets or a let and a fault would gently lob the ball over at the third attempt and the point would get resolved.
Once the point is over, there is a natural break when the balls can be recycled. Ergo, 3 balls allows for smooth play.