In tennis, a server should win a larger percentage of second-serve points compared to first-serve points; that much we know. Partly that’s because a server optimally serves more faults (serves that land out) on first serve than second serve. But what if we condition on the event that the first serve goes in? Here’s a flawed logic that takes a bit of thinking to see through:

Even conditional on a first serve going in, the probability that the server wins the point must be no larger than the *total* win probability for second serves. Because suppose it were larger. Then the server wins with a higher probability when his first serve goes in. So he should ease off just a bit on his first serve so that a larger percentage lands in, raising the total probability that he wins the point. Even though the slightly slower first serve wins with a slightly reduced probability (conditional on going in) he still has a net gain as long as he eases off just slightly so that it is still larger than the second serve percentage. Indeed the lower probability of a fault could even raise the total probability that he wins on the first serve.

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May 3, 2011 at 11:26 pm

Jonathan WeinsteinWell, there are two marginal effects: the one you mention is that he shifts second-serve points into the first-serve-in category, which is a gain. But also, he has just decreased his winning chances on all the other first-serve points, which is also a 1st-order effect. So the net effect is ambiguous (the optimum will be where the effects exactly cancel.)

As you probably had in mind, this feels a lot like a pricing problem where, in order to attract the marginal customer, you have to cut prices for all of the inframarginal customers, so that the net effect is ambiguous. The fact that price must be above cost at the optimum corresponds to the fact that 1st-serve winning percentage conditional on 1st-serve-in must be greater than conditional on 1st-serve-out.

I thought about whether there is an analogue to perfect price discrimination where you avoid the tradeoff. The interpretation winds up being kind of weird…it would mean that you can hit a serve which simultaneously achieves the following for every p: whatever strength serve (about the min level, which is your 2nd serve quality) you are capable of getting in p of the time, you in fact hit it at least that well (and in) p of the time. Another way to say this is that your distribution function of serve quality achieves the lower envelope of all available distribution functions. About as likely to happen as perfect price discrimination.

May 4, 2011 at 8:56 am

jeffJonathan Weinstein again wins the Jonathan Weinstein Prize for better-than-perfect Cheap Talk comment.