In a famous paper, Mark Walker and John Wooders tested a central hypothesis of game theory using data on serving strategy at Wimbledon.  The probability of winning a point conditional on serving out wide should equal the probability conditional on serving down the middle.  They find support for this in the data.

A second hypothesis doesn’t fare so well. Walker and Wooders suggest that the location of the serve should be statistically independent over time, and this is not borne out in the data.  The reason for the theoretical prediction is straightforward and follows from the usual zero-sum logic.  The server is trying to be unpredictable.  Any serial correlation will allow the returner to improve his prediction where the serve is coming and prepare.

But this assumes there are no payoff spillovers from point to point.  However it’s probably true that having served to the left on the first serve (and say faulted) is effectively “practice” and this makes the server momentarily better than average at serving to the left again.  If this is important in practice, what effect would it have on the time series of serves?

It has two effects.  To understand the effects it is important to remember that optimal play in these zero-sum games is equivalent to choosing a random strategy that makes your opponent indifferent between his two strategies.  For the returner this means randomly favoring the forehand or backhand side in order to equalize the server’s payoffs from the two serving directions.  Since the server now has a boost from serving, say, out wide again, the returner must increase his probability of guessing that direction in order to balance that out. This is a change in the returner’s behavior, but not yet any change in the serving probabilities.

The boost for the server is a temporary disadvantage for the returner.  For example, if he guesses down the line, he is more likely to lose the point now than before.  He may also be more likely to lose the point even if he guesses out wide, but lets say the first outweighs the second.  Then the returner now prefers to guess out wide. The server has to adjust his randomization in order to restore indifference for the returner.  He does this by increasing the probability of serving down the line.

Thus, a first serve fault out wide increases the probability that the next serve is down the line.  In fact, this kind of “excessive negative correlation” is just what Walker and Wooders found.  (Although I am not sure how things break down within-points versus across-points and things are more complicated when we consider ad-court serves to deuce-court serves.)

(lunchtime conversation with NU faculty acknowledged, especially a comment by Alessandro Pavan.)