Game theory and Wimbledon

Wimbledon is now underway, so it is time to have a little think about how game theory might apply to tennis.

Let’s assume that Roger Federer is playing Raphael Nadal in the final.

And let’s say that Nadal is serving. When he serves Federer has to make a guess as to which side the serve is going, if he gets it right he has a better chance of winning the point than if he gets it wrong.

If we assume that Federer’s backhand is slightly weaker than his forehand then where should Nadal aim?

The obvious first thought is that he should aim to Federer’s backhand as that is his weaker side. However, if he just aims to that side then Federer will be able to predict it and do better. Remember, Nadal does best if Federer guesses wrong, so he needs to mix up his forehand and backhand serves. He needs a mixed strategy.

The more that Federer guesses the backhand side the more Nadal will go to the forehand in response.

The equilibrium point is where Federer will go to the backhand side just enough to mean that it doesn’t matter to Nadal which side he serves.

It obviously depends on the exact success rates for the different options but the surprising conclusion is that Federer will go to his weaker side more often than not and in response Nadal will serve to Federer’s stronger side more often.

Think about it this way, if Federer doesn’t favour his weaker side then he will lose too many points on that side. He needs to protect himself by having a bias to that side.  Nadal’s response is then to put more serves to Federer’s strong side.

The other interesting thing is what happens if during the match Nadal gets into a groove with his serve to Federer’s backhand and he starts to be more successful with it. In response to this Federer has to predict the backhand side more often to counteract the improved serve from Nadal on that side. Because Federer is now going to the backhand side more often, it is a better strategy for Nadal to favour the forehand.

In fact, Nadal improving his serve to Federer’s backhand results in him serving more often to Federer’s forehand. For Nadal just having a stronger serve changes Federer’s response, the threat of using it is enough without having to actually use it.

Today’s takeaway:  Sometime just having a strength can be enough to change your opponents behavior as they try to compensate, you might not actually need to use the strength much to win.

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