Serving strategy in tennis: Accuracy versus power

In tennis, the server has an advantage—the opportunity to serve again without penalty, if the attempt results in a fault. A common strategy is to hit a powerful or “tricky” first serve followed, if necessary, by a weaker second serve that has a lower probability of faulting even if it is easier to return. Recently, commentators have argued that this standard strategy is flawed and that the second serve should be as difficult to return as the first. This advice contradicts Gale’s theorem, which we reformulate and provide with a new (analytic) proof. Then we extend it with a model of the rally that follows the successful return of a serve, providing additional insight into the relative effectiveness of the “two first serves” strategy. The only tools we use are basic probability and introductory calculus.