During our first few days of studying game theory, Professor Easley alluded to a study done on professional tennis matches that was similar to the Palacios-Huerta paper we read about the von Neumann Minimax Theory in professional soccer. You can find it at http://www.u.arizona.edu/~jwooders/WimbledonAER.pdf – “Minimax Play at Wimbledon” by Mark Walker and John Wooders. This is another situation in which the players of the game are highly motivated and rational.

Walker and Wooders explain that every point of a tennis match can be turned into a 2 x 2 game just like the ones we have been studying. The server of each point has two options: to serve to the right side or the left side of the returner. The returner, they claim, guesses to which side the serve will go. One could debate that a significant portion of service return involves reaction rather than guessing. Though they don’t state it explicitly, the writers appear to recognize this by approaching their analysis from the server’s point of view and neglecting to mention the returner even once after their initial explanation of the game. (Plus, of course, it is impossible to watch a serve and know which way the returner guessed). They also distinguish between four different types of games within each match. In tennis there is a deuce-side serve and an ad-side serve and the strategy and mental processes behind each is different. Then two more games can be made by analyzing the game of both serving sides by reversing the roles of the server and returner.

Anyhow, what they did was take ten high profile grand slam matches and computed (for each of the four games) the frequencies with which the server served to each side of the returner, the probability of winning a point on each side, and of course the success rates of serving to each side. As I understand it, these success rates are what the Minimax Theory says should be equal on each side. In many cases they are, but this data is not nearly as convincing as that of Palacios-Huerta. Have a look at page 1526 (6 of 18 in the pdf) for all of this data. That is the most important page you could look at. The data looks decent (the matching of the Win Rates column).

This brings up a discussion that Dr. Easley touched on that there are many more variables in this tennis situation than in the soccer one, even though the players of each game might be equally rational. First of all, even discounting the direction of a serve, it still has many variables such as the speed and spin. Also, serves and returns happen all throughout a tennis match (penalty kicks in soccer are only at the end and are all equally crucial). So the players may gamble more at certain times compared to others (in other words there’s a lot more involved mentally at any given time; then again perhaps all of the extra thinking helps the players achieve Minimax values). Walker and Wooders try to compensate for variability by creating as much isolation as possible. This is why they looked at the four types of games separately and looked only at matches between players who knew each other’s games well.

On the other hand, the extra data we get by looking at tennis can be an advantage. These serving games can be looked at among players of all levels since serving is highly integral to the sport. It would be easy to conduct this same study among high school or college players to see how well their win rates match each other. The skills of non-professional players tend to be skewed: they have more exploitable vulnerabilities and fewer strengths (shots and tactics at which they are much better compared to other shots and tactics).

This same study might be interesting if conducted among female professionals, who serve slower than the men do (due to wing- and shoulder-span, meaning that it would be less of a game involving simultaneous strategy selection and depend more on reaction. It would still be a similar game from the server’s perspective because there would still be different probabilities of winning the point when serving in each direction.

A final study that might be interesting is a longitudinal examination of one or few rivalries between players. Andre Agassi and Pete Sampras faced one another about 30 times (professionally; even more as juniors), many of which had implications just as great as those of the matches Walker and Wooders use. The logic behind the longitudinal study is that the players become even more rational and motivated as they learn the best ways to play against one another.