In orther words, with steel balls, bot a 13 inch and a 15 inch string are potentially equally as powerful depending on the nature of the collision. The longer string will deflect more, however. The reason is when a string is pulled or impacted, it stretches the same everywhere. A long string isn't more stretchy, it just has more inches to stretch and thus deflects more on impact. For the tennis ball, the results are the same as in the loose vs tight situation.
More deflection means more cushioning, less ball deformation, and more energy return.
All things being equal, longer strings convert more incoming energy into outgoing energy. Fat and thin strings Imagine a 15 guage monofilament string under 50 pounds of tension at equilibrium. The entire cross section is being pulled to 50 pounds and is being stretched by this tension. If you could maintain this tension but melt away the outside layers such that only the 17 guage string was left behind, the string will get longer as you strip the outer layers away. That is - the thinner string is under more stress to support the identical tension.