The tennis ball problem

Natasha often says we should open more cans of tennis balls.  Last night we were playing with eight balls and she wanted to play with twelve.  Of course once all eight have been plowed into the net, you have to go collect at least some of them.

How many tennis balls should you play with?

Let’s say you had many, many balls and you could open the cans for free and never run out.  Opening a new can every four points (four balls fit in a can) would lead to a massive clean-up and carry problem at the end.  Furthermore how much help is it having more balls?  Once they hit the net you still have to deal with getting another ball into play.  In other words, the real trick is to manage your stock well (read: aim for good volleys), not to just to speed up the flow of balls into the court.

Just one ball is not efficient, because when it falls out of play it is probably far from you.  The greater the number of balls, the more likely at least one will be close.

Many problems in life, including those of dating, the number of children you should have, and optimal inventory management, resemble the tennis ball problem. 

I do not know how to solve the tennis ball problem, but I feel that twelve balls is too many.


Twelve is far too many. You'll end up with a bunch of balls strewn around the court, with the consequent risk of serious injury. Also, some balls will get a lot more play than others, leading to dramatic pressure variations.

I'd say four is the optimal number, but your mileage may vary.

Retrieving tennis balls is also exercise, no? I'd say you want about as many tennis balls as you can load in both your pockets without setting any on the ground to trip over or being uncomfortable. Probably 5 then (4 in the pockets and one to hit). Maybe a couple more if you have especially large pockets.

"I do not know how to solve the tennis ball problem..."

Simple: Hire a ball boy.

Twelve is ridiculous, unless you are in training and someone has a hopper and is consistently firing shots at you. In a match, 3 is the correct answer. Four is acceptable.
Without ball boys (or girls), the tendency for some balls to get used more than others, thus wearing out quicker leaving you with inconsistent bounce, is the bigger problem. Balls significantly wear out in a single set.
Given your premise, open new balls every game, and throw the old ones out.

Any balls not in or very close to the net or in a corner should be picked up on each point.

You don't have infinitely large pockets. Four is I think ideal. That means the greatest number of balls in your pockets at one time is 3. This is the question I want the answer to: how come there are only two tennis shops in the WashDC area, and one is run by incompetents and the other is run by price gougers?

Tennis coaches over here usually have a large net of balls, maybe 50, so they don't have to interrupt instruction frequently to collect balls. It's quite a workable strategy.

Thanks for your support. Tyler, please stop on the way home to buy more balls.

Well, if we want to think of it economically, tennis balls usually (or at least used to) come in packs of three. So playing with four requires opening another canister, at which point you might as well play with six.

I'm not a tennis player, but I am a mother, so I would suggest that what you need is only one ball and a small toddler (18 months to 2 years old) to happily run after it saying "Ball! Ball!" The toddler will fetch your ball, and you will get a workout from running after the toddler, and everyone involved will have a good time.

Play badminton instead of tennis. The ball never gets very far away and the quicker pace makes for a more enjoyable game.

What the implications are for dating, number of children and inventory managment is left as an exercise for the reader.

I find the standard 3 balls per can to work well. You can start off with two in your pocket, while you serve with the third. Assuming you miss the first serve, that ball will end up in a corner or at the net, and you can pull out one of the other two. Use that for the second serve and to play out the point, and then you have one left in your pocket. Then, you just need to get one of the other two balls from the court, in case you need to serve twice. The third ball can be picked up when it's convenient to do so, with less chance of posing a hindrance than if you had a lot more balls on the court.

Yikes. If you need 12 balls let's organize a match and play for money, Tyler. Three is optimal, four is workable. Anything more than that is annoying. Anything less leads to too many delays, particularly on let services.

Thought experiment variation: Let's say you had a magic hopper, up against the fence behind the baseline, that you could pull tennis balls out of whenever you wanted. In fact, let's say you had two, one on each side of the court. You can also put balls back in the hopper whenever you've collected them (just roll them towards it and it will eat them up). Now, the question is: Optimally, how many balls you pull out of the hopper during course of the match?

One thing that should be clear on this version of the thought experiment is that, as long as you have someplace out of the way to put your extra balls (like the magic hopper), there's no real disadvantage during the match to having more balls. The "more balls means more clean-up" view is false, since having more balls waiting in the hopper doesn't stop you from picking up loose balls during the match. There will only be more clean-up at the end if you judge, during the match, that it would be preferable to wait to pick up the balls at the end rather than going to get them then. The only reason why you might want the total number of balls to constrain you is if you have some failure of rationality - either bad judgment about when to pick up balls or (more likely) a self-control problem (procrastination in ball pick-up). Or, you may want to constrain your partner, if you disagree with the in-match decisions that they make about ball pick-up. I'll let you decide if you think that it would be beneficial to impose this restriction on you and your tennis partners.

The other problems with having more balls, I think, are:

1. Transporting them to and from the courts
2. The balls getting in the way, if you don't have someplace out of the way to keep them (like the magic hopper)
3. Feeling foolish, embarrassed, or annoyed about having so many tennis balls
4. Making a bad impression on other tennis players for having so many balls
5. The financial cost of buying more balls
6. The added difficulty of keeping track of which old balls should be taken out of rotation

In many cases, costs 2 and 3, which result from social norms and internalized norms, may be the most significant.

Looking at the other side of things, there are clearly diminishing benefits as you add more balls. What are the benefits, and why do they diminish? The two main benefits that I see are flow and efficiency:

1. When you need a ball, more balls means that the nearest ball is usually closer, so there is less of a disruption in the flow of the match.
2. Ball pick-up is more efficient (involving less total distance traveled) if you pick up several balls from the same area at once, and fetching more balls less often increases this efficiency

Both benefits, I think, diminish fairly rapidly. My view is that flow is more important than efficiency, and that there are minimal gains in flow to be had once you have enough balls to play through a game without having to go very far to shag balls. If you want, I suppose you could increase that aspiration to two games without serious ball pick-up, so that you're only chasing down balls when you change court. The number of tennis balls required depends on how you play, but my guess is that eight would be enough for most people.

It should be clear by this point that there is not one solution to the Tennis Ball Problem, as different strategies are appropriate for different people.

The answer lies in "economic order quantity" which tells how much inventory to order when you order. This is an economics blog, isn't it?

Inventory is the total number of balls out.

Carrying cost is the cost of the balls on the ground, plus those carried in your pocket.

Order cost is the fixed cost of stopping play to pick up the balls. The cost of picking up each ball is ignored.

Look at the graph on the wikipedia entry. Total cost starts high, drops, levels out for a while, then rises. I suspect when you plug in numbers and test for sensitivity that 1-4 balls are low, 5-10 will be in the flat part, and 11+ will start to rise.

you don't need balls to play tennis. all you have to do is put the disc into the xbox. i've never had to retrieve a ball.

By making an invention to bring the balls back to you, you never have to run because the balls will caome back within reasonable time that you hit them. Concluding, you really only have to by 4-8 balls and you keep playing tennis until your done. Forget hiring a ball boy because the ball boy will always be there.

I say dont be lazy and deal with it and pick up the balls!!!

The answer is the most number of tennis balls you (and the person playing against you on the other side of the net) can comfortably hold in your pockets. If you can put 3 tennis balls in each pocket, then 6 for each player, 12 total.

This was always my solution. The pain in the ass is bending down to pick up the balls and having loose balls on the court in your way. The most balls you can pick up and hold at once, and still have handy (i.e. on your person) for the next serve is how man you should play with.

Some might find it uncomfortable to have 3 balls in each pocket... i always had really big, deep pockets. If you can only fit one ball in each pocket, then 4 balls is all you need assuming 2 pockets on each of the 2 players. Although I suppose the server can hold on to one more as he'll always have one hand free for a ball anyway. So 5....

Haven't read all the posts, but the standard professionally is 6 new balls every 7-8 games, depending upon the surface and tournament.

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