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!



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  • Scott

    Should the mix (while remaining 60:40) be completely random, or should pains be taken to remove random mixes that resemble predictable patterns?

    Simply having a 60:40 mix isn’t enough. The focus here is not so much on randomness as it is unpredictability (the difference between the two is subtle and arguable).

    Nadal, obviously, can’t use the same 60:40 mix each game, so how should he go about determining the mix for each game? Leaving it up to in-the-moment decisions would be unwise as we are unconsciously geared towards seeing and developing patterns. His best bet would be to leave it to an emotionaless, non-human algorithm.

    But should that algorithm be completely randomn? There is a non-zero chance that the mix will resemble a predictable pattern (such as FFFFFFBBBB, FFFBBFFFBB, etc). In such cases, Federer will assume it actually is a pattern and respond accordingly. In the cases where he is right, Nadal’s win percentage goes down.

    So rather than being completely randomn, should he select mixes that are not only random, but reveal no obvious pattern?

    That is, is it simply important that there be a random 60:40 mix, or should we consider how that 60:40 is derived?

  • Pingback: Tennis, Game Theory, & Learning To Balance Strengths & Weaknesses | Simoleon Sense

  • Drobviousso

    This is very similar to the run/pass balance problem in the NFL, which has bedeviled both of the participants of the superbowl this year.

    For ‘zona, they have a fantastic passing game and a comparatively abysmal running game. They have made a conscious effort to run a lot, in order to keep the opposing team’s defense honest. This is a big problem against very good run stuffing teams that can stop the run from the nickle (the set with more resources to stop the pass than the run). Their Superbowl opponent Pittsburgh is capable of doing this.

    For Pittsburgh, their offensive coordinator wouldn’t know a random play call if it put his cat in box that may or may not release poison gas. I believe (but don’t have proof) that they run more Run-Run-Pass pattern of plays than any other team in the league. Thus, opposing teams can stack the box (the set with more resources to stop the run) on first and second, and then focus on the pass on 3rd down.

    http://smartfootball.blogspot.com/2008/06/runpass-balance-game-theory-and-passing.html

    (go Stillers!)

  • http://ravig.com Ravi Grover

    Rnadomness is important — I remember reading a study that found that the top tennis players had serve patterns that were indistinguishable from randomness, while worse players did not

  • http://www.mindyourdecisions.com/blog/ Presh Talwalkar

    The mix must be random, as you point out Scott, or else it will lose out as opponents learn the patterns and respond in kind.

    Thanks for the excellent analog in football Drobviouso

  • Ivar Källblad

    In some matches the server wins a higher percentage of the points after the first serve included the faults of that serve than he does on the second serve that is 100 percent in. What does game theory say of the solution of only hitting first serve strokes on the serve.

  • Frank

    Presh,
    Glad to have found your blog. I am reading the Dixit book right now and am very interested in his tennis example. You did a nice job recreating it, although I want to point out a very convenient calculation method to determine the optimal mix strategy.

    For the server’s perspective, take the difference between the two possible payoffs he receives when aiming to the forehand, and do the same for when he aims to the backhand. Make a ratio out of these two values, this ratio is the optimal serve strategy to the backhand:forehand side. Note that you have to reverse the ratio if you want the forehand:backhand.

    For this example, to get the forehand:backhand ratio you would see that for forehands nadal gets 70 or 50, difference of 20, and for backhands he gets 80 or 50, difference of 30. So his ratio of forehands:backhands is 30:20 or 60:40.

    I think this is a really cool method that has illustrates the mechanism of whats going on. It shows us that the larger the difference is between the payouts of our opponents options in response to our strategy, the less often we should use that strategy.





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