Balanced Strategy – The Mathy Version

diagram of the example laid on in this post; with the specs given, our attacker's optimal frequencies are 2/3 inside-out, 1/3 cross-court

For every probabilistic decision in a non-cooperative game, there exists a balanced (typically mixed) strategy which creates indifference in the opponent – it doesn’t matter how the opponent defends, their equity will be the same.

As a probabilistic game itself, tennis is no different.

We’re going to do a quick example where we derive a mathematically unexploitable forehand approach shot strategy.

Before we move on, understand that knowing how to actually calculate the game theoretical equilibrium for a particular situation is obviously not essential to playing correct tennis strategy – it’s not like we’re out there crunching numbers on the court.

However, the math is still instructive, if for no other reason than to both prove that there’s more than just gut feel behind correct strategy, and to examine the implications of the results.

Our Example – A Forehand Approach Shot

The attacker is hitting an approach shot. They will hit it either inside-out or cross-court. The defender is going to guess one way or another as the attacker swings. This attacker can hit both directions well, while the defender defends far better out of the forehand corner than the backhand corner (and both players are right handed).

Before any strategic adjustment, the attacker’s highest equity shot is the approach shot into the defender’s weaker backhand. As such, the attacker’s HESBA is the inside-out forehand.

Here’s how often the attacker wins the point, given the defender’s guess.

The attacker hits inside-out, and wins:

  • 75% of points when the defender guesses inside-out
  • 100% of points when the defender guesses cross-court

The attacker hits cross-court, and wins:

  • 100% of points when the defender guesses inside-out
  • 50% of points when the defender guesses cross-court
The payoff matrix for our two players in this approach shot situation; each box contains the attacker’s win percentage after the approach shot has been struck, and the defender’s guess made.

A few things to note.

  1. When the defender guesses wrong, he loses 100% of the points. (We’re not modelling in the few approach shot misses here). This means that defensive strategies like “always guess cross-court” are easily exploitable by always hitting the opposite way.
  2. Note the effect of the attacker’s HESBA. Even when the defender guesses correctly to the backhand side, he still only has 25% equity in the point, while the attacker has 75%. On his forehand side, where he defends well, a correct guess yields 50% equity, despite the defensive situation.

Before we move on, I’m going to spoil the solution for you so that you have a reference as we discuss various strategies for the attacker moving forward – the attacker can guarantee an 83.3% winrate if he uses an accurate strategy. The proof of that will come later.

A Simple, Exploitable Strategy

First, we’re going to analyze an obviously exploitable strategy for the attacker – 100% inside-out forehand. This is the optimal strategy against an opponent who isn’t willing to guess – if it wins 75% against an opponent who guesses that way, it probably wins over 90% against an opponent who isn’t guessing – but against this defender, it’s leaving points on the table.

The defender, to exploit this, will choose a 100% guess backhand side strategy.

The result – Attacker 75%, Defender 25%. Not as good as the 83.3% I promised we could get.

So how can we do better?

Well, the defender’s “100% guess the backhand side” strategy leaves the forehand side wide open. When the defender guesses backhand, and we hit to the forehand, we win 100% of the points. We’d love to utilize that fact, but not so much that the defender can start profitably guessing that way instead.

Let’s have the attacker try throwing in just a few cross-court forehands and see if it improves their equity. Their new strategy:

  • 80% inside-out
  • 20% cross-court

Now, if the defender employs his same “always guess backhand side” strategy, the attacker wins 20%*(1) + 80%*(.75) = 80% of points. That’s 5% better than the 75% from before, but still 3.3% worse than the optimal 83.3%.

The Nash Equilibrium

We can do even better. There exists a point in every non-cooperative game called a Nash Equilibrium – the point at which neither player can improve their performance by altering their strategy. This point is reached when both players are playing their respective unexploitable strategies.

In tennis, our unexploitable strategy (as an attacker) is the mix of frequencies such that our opponent’s decision is indifferent – it doesn’t matter how they defend, they can’t improve their equity in the point. They could always guess one way, always guess the other way, or mix the frequencies of their guesses however they wished, and their win percentage would be the same. (The principles for a defender making their attacker indifferent are the same, but we’re sticking with the attacker’s perspective for this article.)

Our goal is to find this mix of frequencies.

At Fault Tolerant Tennis, we routinely refer to a strategy using these frequencies as a “balanced strategy,” a “balanced mixed strategy,” or an “unexploitable strategy.” Note that a balanced strategy is not always optimal – if your opponent is playing a clearly exploitable strategy, using an exploitative strategy will yield better results than using a balanced strategy, but balanced strategies are far more stable, and better under pressure, so we strongly prefer them.

Our Attacker’s Unexploitable Strategy

To solve for our attacker’s unexploitable mix of frequencies, we find the frequencies such that all defensive strategies for our opponent produce identical results.

Here are the possible defensive positions, and how many points the attacker wins against each.

Defender Defends Inside-Out

  • Attacker wins: 75%*(frequency hitting inside-out) + 100%*(frequency hitting cross-court)

Defender Defends Cross-Court

  • Attacker wins: 100%*(frequency hitting inside-out) + 50%*(frequency hitting cross-court)

Now, we need to set these expressions equal to each other – we need to find the frequencies such that our defender is indifferent; whether he defends inside-out or cross-court, (or, by extension, any mix between the two), his win percentage will be the same.

Let’s call our frequency cross-court “Fc” and our frequency inside-out “Fi”.

75Fi + 100Fc = 100Fi + 50Fc

50Fc = 25Fi

2Fc = Fi

And, since these are frequencies, they must sum to 100%.

Fi + Fc = 1.

Solving this system, we see Fi = 2/3 and Fc = 1/3.

Our unexploitable strategy is:

  • 2/3 inside-out
  • 1/3 cross-court

The Attacker’s Unexploitable Win Percentage

When the defender guesses inside-out, the attacker wins:

(2/3)*(75%) + (1/3)*(100%) = 5/6 = 83.3% of rallies

When the defender guesses cross-court, the attacker wins:

(2/3)*(100%) + (1/3)*(50%) = 5/6 = 83.3%

It’s the same!

It doesn’t matter how the defender defends. If the attacker uses 2/3 inside-out and 1/3 cross-court, they will always win 83.3% of the rallies regardless of the defender’s guess.

In Defense of Balanced Strategy

The above analysis means that, as our attacker steps up in the court and prepares for an approach shot forehand, he has 83.3% equity in the point. If the attacker adopts the balanced strategy above, that equity is a mathematical guarantee; there’s nothing the defender can do about it. Notice that the attacker has a significant edge here – 83.3% is high, because the attacker is in a very good spot, and according to the numbers in our example, quite effective at converting short balls.

Trying to win to win more than 83.3%, by trying to exploit an apparent suboptimal defensive strategy from the defender, constitutes a risk. It might work, perhaps the defender really is defending incorrectly, and the attacker can abuse that and win 90% of these points, but there’s no guarantee.

In general, exploitative strategies are better when you’re worse. If your guaranteed equity isn’t enough to win the match, you need to hope your opponent makes strategic mistakes, and then attempt to exploit those mistakes. Once your opponent catches onto your exploit, of course, they’ll exploit you back, and then you’ll have to adjust again. The game of chicken begins. If you’re the worse player in the match – if your equity with a balanced strategy is simply too low to compete – you may not have a choice but to try and win it.

On the contrary, if you’re the better, then simply play an accurate balanced strategy, and you can’t lose.

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