Experts Test Game Theory on World's Top Tennis Players

Hawk-Eye data collected from 500,000 tennis serves at 3,000 matches

John Wooders, professor of economics at NYU Abu Dhabi, doesn’t play tennis, and as a fan he most enjoys watching his son on the court. But he does have a professional interest in the sport: Wooders has found a way to use the performance of the world’s top tennis players to test game theory.

Wooders, a game-theory expert, is one of three authors of a scholarly paper examining one element of tennis strategy: the choice of direction of serve.

We’re interested in seeing if the players behave in a way that’s consistent with theory and we find that largely they do.

John Wooders, professor of economics

Modern game theory, the mathematical modelling of interactive decision-making, began with the genius of John von Neumann (1903-1957). His work was extended by John F. Nash Jr. (1928-2015), subject of the 2001 Russell Crowe film A Beautiful Mind.

Within the fast-growing field of experimental economics, however, game theory has a problem: behavior – often involving undergraduates participating in laboratory experiments – does not always conform to theory.

Wooders, with economists Romain Gauriot of the University of Sydney and Lionel Page of Queensland University of Technology, found a better data-set to test Nash’s theory, their paper explains: “the precise trajectory and bounce points of the tennis ball for nearly 500,000 serves from over 3,000 professional tennis matches.” The data comes from Hawk-Eye, a computer-and-cameras technology widely used in pro tennis and other sports. The company promises “sophisticated millimeter-accurate ball tracking."

Wooders notes that “in undergraduate experiments the stakes aren’t very high, and the participants don’t have a lot of experience” while in high-level tennis the opposite is true. That, plus the size of the data-set and a new statistical test, gives the authors confidence in their two conclusions.

The first of their conclusions is that pro players, men more than women, do behave in accordance with “Nash equilibrium” – the game-theory advance that won Nash a Nobel Prize in 1994.

“We’re interested in seeing if the players behave in a way that’s consistent with theory,” Wooders says, “and we find that largely they do.” Further, higher-ranked players adhere to the theory more closely than other tournament participants. The draft paper, Nash at Wimbledon, is now being prepared for submission to a major scientific journal.

One prediction of Nash’s theory is that if the server is randomizing the direction of serves, his payoff – his probability of winning the point – should be the same from either side.

“Think about the receiver choosing a position on the baseline,” Wooders says. “If he stands too far to the left, the server will do well serving to the right, and vice-versa. So the optimum position for the receiver is a place that makes the server indifferent between left and right.”

The Hawk-Eye data showed that male pros match the theory more closely than females. Wooders believes that’s because men’s serves average 160 kilometers per hour, compared to 135 for women’s. Therefore a woman receiver who doesn’t play optimally has more time than a man to react, making the disadvantage less important.

We think that people (players) are trying to be random but they’re just not very good at it.

John Wooders
Science could eventually affect game strategy

The paper’s second conclusion has to do with “serial independence” – the idea that the direction of the next serve does not depend on the direction of previous ones. Tournament players, Wooders explains, “switch direction of serve too often to be random … we think that people are trying to be random but they’re just not very good at it.”

So why aren’t predictable players being taken advantage of? Perhaps they are. “Higher-ranked male players display less serial coordination in their serves,” Wooders observes. In other words, more-nearly-random serving direction correlates with more success."

However, usually only rigorous statistical analysis reveals digressions from randomness; who can assess randomness accurately while in the middle of a match?

But if a player does detect a pattern in an opponent’s serves, he must hide that knowledge. Wooders points to a YouTube video in which one former world number-one discusses another: Andre Agassi, explaining a “tell” that helped him predict Boris Becker’s direction of serve, says the hard part was hard concealing his discovery from his opponent.

In baseball, advances in data collection led to the phenomenon known as Moneyball, after Michael Lewis’s 2003 book; alert general managers began using new statistical tools to find undervalued players. Will the work Wooders is doing lead to Serveball? It’s possible, he says, that players are already using the information, in a less scholarly way. “One reason the theory works, probably, is that players and coaches do study tapes of their opponents.

It’s also possible that with the data we have one could conceivably do consulting work for these guys … That’s not something that we’re going to do, but … if this kind of research gets better known among the players themselves … it could actually affect the way that they play.”