# Professional tennis is probably one of the toughest sports. As the game has shifted into the realm of high speeds and relentless rallies, equipment has gone through immense developments and players are reaching the peaks of their physical form, the hidden key to winning may now lie with game theory. Economic research over the last few years suggests that the likes of Djokovic, Nadal and Federer not only spend hours mastering their serve, forehand and backhand, but also seem to spend a significant amount of time honing their game theory skills. Game theory looks at decision-making in circumstances where a competitive analysis is needed to set the optimal course of action for a player. Using a large collection of data aggregated through the Hawk-Eye technology, a computerised ball-tracking system, economists now have an opportunity to test game theory assumptions in tennis.

### IN GAME THEORY, BEING UNPREDICTABLE IS A WINNING STRATEGY

In strategic interactions between multiple agents, it is often preferable to be as unpredictable as possible. In football penalty shootouts for example, a goalie cannot react to the kicker’s decision in time, as it takes a fraction of a second for the ball to reach the goalpost. The goalie should make an educated guess on where the kicker will place the ball and make a jump before he sees the direction of the flying ball. The kicker must therefore randomise the direction of the kick and choose to go left, right or down the middle. The result of the randomisation is a so-called mixed strategy equilibrium, where every action played is a best response to the other players’ mixed strategies. In tennis, Nash’s theory of mixed strategy equilibrium is also often used to analyse strategic interactions requiring unpredictability.

Walker and Wooders found that the serve and return games of John McEnroe, Bjorn Borg, Boris Becker, Pete Sampras and other professional tennis players at Wimbledon is highly consistent with the theory of mixed strategy play, where a positive probability is assigned to all strategy choices in a set. The theory implies that a player’s strategy is a best response to the strategy of the opponent, so that a player is indifferent to the serving directions (i.e. payoffs are the same). A player should randomise between different sides of the serve and would choose a probability distribution over the set of actions, yielding the equilibrium play solution. The server’s winning probabilities should be the same, either when serving to the left or the right.

It turned out the servers’ win rates – frequencies of points won serving to the right or left – are nearly the same, which is consistent with the mixed strategy equilibrium. Interestingly, however, the players’ choices do not seem to reflect a completely random pattern. The server must choose the direction as if it is picked randomly, to make any guesses from the receiver near impossible. In simulating a random patter, however, tennis players tend to avoid repetition, as doing so intuitively seems less consistent with randomly generated results. This is in line with the research in psychology, which suggests that agents who try to generate random sequences tend to “switch too frequently”.

### OPTIMAL DECISION MAKING WITHOUT SYSTEMATIC BIASES IS KEY TO VICTORY

In professional tennis, players exchange balls at high speeds, accurately placing the balls inside the edges of the court. It is therefore not particularly uncommon for the ball to land controversially close to the lines. In such cases, line umpires call out balls that are out and players face the dilemma of whether to challenge the call. There is an opportunity cost associated with challenging, as the number of challenge calls a player can make is limited. Each player is allowed three incorrect challenges per set, and an extra challenge if the set goes to a tie-break.

To test the optimality of players’ challenge calls, economists at Stanford University used a trade-off model between players’ payoff by reversing the umpires’ decision and the opportunity cost of making the challenge. According to the theory, players should be more likely to resort to the challenge during the more vital points and when the value of having the option to challenge in the future is lower (the value of keeping the challenge for later).

Players’ behaviour is largely consistent with the optimal play suggested by economic theory and can increase a players’ probability of winning by 1.55 percentage points. They are indeed more likely to challenge the more crucial points and when the value of saving the challenge call is lower. This suggests that tennis players might have developed a heuristic to assist them in optimal decision making when faced with a complicated problem through feedback rounds after matches. This mental shortcut allows them to increase their chances of winning and is close to the near rational behaviour predicted by economics. A conclusion of the study that most players’ don’t challenge frequently enough is a potential bias; a slight deviation from rationality suggested by economic theory.

### MOMENTUM EFFECT IN A DYNAMIC CONTEST IMPLIES A CHANGE OF STRATEGY

When watching professional matches, we often wonder whether being just ahead in the game (leading by a point within a given game) gives any advantage to a player. A so-called momentum effect in tennis is highly controversial, the equivalent to a concept of “hot-hand” in basketball, and is widely debated. “Hot-hand” phenomenon states that a person who experiences a successful outcome in a random event is more likely to succeed in a future attempt. Gilovich, Tversky and Vallone have famously debunked the theory in basketball, concluding that having a “hot-hand” does not predict hits or misses. But is there a “hot-hand” in tennis?

Gauriot and Page explore whether a change in incentives during dynamic contests leads to an optimal reaction from tennis players. In contest theory, agents need to take into account the actions of the opponents in the past and future. Tennis is a game that takes place over time and may be modelled as a dynamic contest. Unfortunately, past performance is usually correlated with unobserved characteristics of a tennis player, which in turn influences their future performance. To separate the effect of the momentum on the probability of winning the point from the unobserved characteristics, Gauriot and Page model the situation based on points where a ball lands just inside the lines of the court.

The economists find significant proof of momentum effect for male tennis players, showing that players that won a previous point are significantly more likely to win the next one (5-10 percentage points). This goes against the common assumption in tennis that each point is played in separation and does not influence the outcome of the next point. The finding of the study implies that tennis players should choose the right strategy to take the momentum effect into account.

The athletic performance of the tennis players can be easily seen. The decision aspect of the game, however, is less obvious, but by means of large volumes of data backed up by economics, we can get a glimpse into the importance of game theory in tennis. Next time you watch a grand slam match, do pay attention to the way Djokovic applies game theory on court.

##### SOURCES:

Abramitzky, R., Einav, L., Kolkowitz, S. and Mill, R. (2012), On the optimality of line call challenges in professional tennis. International Economic Review, 53: 939-964.

Gilovich, T., Tversky, A., Vallone, R. (1985), The Hot Hand in Basketball: On the Misperception of Random Sequences. Cognitive Psychology, 17 (3): 295–314.

Gauriot, R. and Page, L. (2014), Does success breed success? A quasi-experiment on strategic momentum in dynamic contests. QuBE Working Papers 028, QUT Business School.

Walker, M. and Wooders, J. (2001), Minimax Play at Wimbledon. American Economic Review, 91 (5): 1521-1538.