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.

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