The Grand Slam: how hard should you hit?

The trade-off between performance and accuracy is a problem faced by a lot of different animals in a variety of situations. For example, consider a squirrel running along a bare branch to get from one tree to another; the faster it runs, the less time it spends exposed to predators. However, as the squirrel runs faster, it also increases its chances of mis-stepping and falling to its potential doom. So, to get the best of both worlds, the squirrel needs to optimise its running speed depending on its chance of slipping (the width of the branch) and the cost of falling off (the height from the ground).

Squirrels know what’s going down (or do they)? Image source: Wikimedia commons.

These sort of performance/accuracy trade-offs are also commonplace in the human world. How fast should you smash out a text message to your supervisor asking him (politely) to email back your latest draft before the number of typos makes the whole thing unintelligible?  In particular, these trade-offs are of a great deal of interest in elite sports. An awesome example of a sport where this trade-off is of utmost importance is in singles tennis. 

Serving hard: Heather Watson, Roger Federer and David Ferrer. Image source: Wikimedia commons.

In tennis, it’s pretty well accepted that if you serve really hard, it’s more difficult for your opponent to return the ball. But the harder you serve, the more likely it is that you’ll miss the service area and fault. So, players will usually belt it out on their first serve, but if they miss the first serve they’ll hedge their bets and serve softer the second time round to make sure they don’t double fault.

A/Prof Robbie Wilson, Dr Chris Brown and I have been testing this idea about performance trade-offs and optimal strategies using data from the men’s singles in the 2013 Australian Open. We’ve found this observation to be generally true: the probability of winning the point increases as the serve speed approaches its maximum, but the probability of faulting increases as well (for most players – some players are really consistent at getting it in regardless of how fast they serve). This was reflected in the frequency of high serve speeds in the first and second serves.

Jérémy Chardy, Andy Murray and Janko Tipsarevic. Image source: Wikimedia commons.

We’ve also constructed an optimality model which predicts the optimal serve speed taking into account the probability of faulting and the cost of a fault. An optimality model is, in essence, a mathematical model where you input the risks and rewards of a specific situation for a given individual, and it will tell you the optimal response for that individual if it wants to both minimise the risks and maximise the rewards. Optimality modelling is useful because it allows us to calculate the optimal response of specific individuals to any situation. We are looking at whether their opponent’s world ranking (ability to return a fast serve) and the point they’re going for or defending against (normal, game, set or match) affects their serve speed in relation to their optimum, but more on those results later.

Rafael Nadal, Caroline Wozniacki and Jérémy Chardy. Image source: Wikimedia commons.

We hope that our research can teach us more about how animals optimise their behaviour and physical efforts to improve their chances of successfully performing a given task. Depending on what we find, we might even be able to offer specific recommendations to tennis players wanting to improve their service game – who knows what the future might hold!

Andrew Hunter, a PhD student in our lab, is looking at performance/accuracy trade-offs in soccer. Will the results be similar between an individual and a team sport? We don’t know yet, but it will be interesting to find out.

Novak Djokovic, Agnieszka Radwańska and Venus Williams. Image source: Wikimedia commons.