Is it better to play your strength or your weakness? Finding the right mix

Top ranked tennis player Rafael Nadal seems to win by surprising opponents with his unexpected left-handed spin..but perhaps even more surprising is that Nadal is naturally right-handed!

It might be fair to say that Nadal’s success stems not just from his bulging muscles, but also from his ability to surprise opponents. The key to success in tennis, and many competitive games, is finding the right mix.

At the heart of the problem is understanding randomness and knowing when to play your strength or your weakness. This is easier said than done, however, as today’s problem will illustrate.

Finding the right mix

I’ve come up with a hypothetical situation between Rafael Nadal and Roger Federer to analyze. Here’s the problem*

Rafael Nadal is serving against Roger Federer in the championship match. Nadal is deciding whether to aim towards Federer’s forehand or backhand side. His success depends on Federer’s expectation.

If Nadal serves to where Federer expects, then Federer can give a good return and nullify Nadal’s serving advantage. In such situations, we’ll venture that Nadal wins just half of the ensuing rallies–no better than a coin toss.

The situation is different, however, if Nadal can surprise Federer. In this case Nadal will gain a serving advantage. Based on player history, we estimate Nadal will win 80 percent of unexpected serves to the backhand side (his strength in this problem) and 70 percent to the forehand side.

*This is an updated version of a tennis problem in Avinash Dixit and Barry Nalebuff’s Thinking Strategically

There are three interesting questions to investigate:

1. If Nadal and Federer play optimally, what is the right mix?
2. Will Nadal aim more to the backhand side (his strength) or to the forehand?
3. What happens if Nadal improves his spin and winning percentage to the backhand side?

The interesting part is that these questions have counterintuitive answers. To begin with, the right mix will have Nadal favor serving to the forehand side–in this setup, that is his less successful serve!

Things get even more strange if Nadal were to improve his already powerful serve to the backhand side. As we will see, his best move in this case will be to aim to the backhand side even less frequently.

It’s an interesting game theory result that improving a strength may secretly make your weakness better. This can have vast implications for your own career and business choices. Let’s go through the math to see why.

If Nadal and Federer play optimally, what is the right mix?

The following figure summarizes the payoffs to the game:

The numbers in the cells represent Nadal’s winning percentage for a point. Nadal’s goal is to choose a strategy that maximizes the winning percentage, given that Federer will try to minimize the winning percentage.

Notice that since this is a zero-sum game, we don’t need to write another set of numbers for Federer. Since Nadal’s gain is Federer’s loss, we know that Federer is playing with the same payoffs values but his goal is the opposite–he wants to minimize the winning percentage.

To solve this game, we need to compute the probabilities of choices for each player. The right mix of strategies–the equilibrium–will be the strategy set that makes the other player indifferent among his choices. I’ve alluded to how to solve randomization problems before, so I won’t go through the laborious details here.

When all is said and done, we find that Nadal will end up winning 62 percent of all points by playing a 60:40 mix of forehand:backhand against Federer’s 40:60 mix (in which he loses 62 percent of Nadal’s serves).

Will Nadal aim more to the backhand side strength or to the forehand?

The interesting part is that Nadal ends up aiming more to the less favorable side–the forehand–in the equilibrium.

Similarly, Federer defends more to the backhand side even though it’s the forehand where he loses most of his points. What is going on?

The answer has to do with the counterplay of random strategies. As Nadal aims more to the backhand side, Federer anticipates better and defends the backhand side more. This makes Nadal’s winning percentage get closer to 50 percent, so Nadal then starts mixing in serves to the forehand side. Ultimately Federer will defend the backhand just enough so that Nadal is indifferent between where to aim, and this is the equilibrium we have derived. The only way Nadal gains the advantage of the backhand surprise is by playing the forehand more frequently. We can visualize the ebb and flow through the following diagram:

The same logic explains why Federer favors his backhand defense, even though he ends up losing more points when he guesses wrong with the forehand.

What happens if Nadal improves his strength of aiming to the backhand?

Suppose Nadal takes the off-season to improve his serve to the backhand side. Assume now that if he catches Federer off-guard, he will win 85 percent of such points. What will happen?

We can visualize the change in the diagram as an increase to the slope of the backhand serve line:

As we expect, Nadal increases his winning percentage to 63 percent. But the interesting part is how he does it.

Nadal now has a mix of 64:36 of forehand:backhand compared to 60:40 before. In other words, as he serves better to the backhand, he aims to the forehand side more!

The reason has to do with the adjustment of random strategies. As Nadal improves his strength to aim toward the backhand, Federer will defend to the backhand side even more. In response, Nadal will serve to the forehand side and take Federer off-guard.

As Dixit and Nalebuff explain in Thinking Strategically, “A better backhand unlocks the power of your forehand. Similarly for Larry Bird, an improvement in his left-handed shooting changes the way he is defended and allows him to shoot right-handed more often” (p. 182).

Conclusion

When choosing the right mix, it is not just about playing to your strength. It is important to play to your advantage, which may require you play your weakness more often. This becomes even more important as your strength improves!