Mixed Strategy – An Antidote to Guessing

example of a forehand from the forehand corner mixed strategy

A mixed strategy is a strategy in which we mix our shot selection decisions for a given situation; we won’t hit any shot 100% of the time.

Consider an extremely common situation – the forehand from the forehand corner. Against a player with far inferior forehand to your own, a pure strategy of 100% cross-court may be appropriate. Most of the time, though, your opponent will be able to defend effectively off of both wings if they know what’s coming.

In a perfect world, you’d like to hit every shot to their forehand, but if they know that you’re going to do that, they’ll start to cheat over to the forehand side as you swing, and subsequently the advantage you gain from hitting to that side vastly diminishes.

Most players are players are both competent enough and willing to make strategic adjustments to prevent you from using your initial highest equity option 100% of the time. In order to maintain your edge then, you need a way to take advantage of that adjustment.

This is why we play a mixed strategy – a strategy that mixes in second best shots which are worse in a vacuum, but improve against your opponent’s adjustment.

Why Not Just Always Counter Their Guess?

If your opponent is guessing to the cross-court side, why not always go down-the-line? And similarly, if they are guessing over to the down-the-line side, why not always go cross court?

If your opponent is guessing poorly, go for it.

Sometimes, your opponent will commit to their guess early enough that you can observe their guess before you swing. If that’s the case, then yes, just hit it away from them. Your opponent is guessing improperly – from a strategic standpoint, what they are doing doesn’t even constitute guessing; they’re really just altering their defensive positioning.

When you hit it away from your opponent in this case, you are leveraging their improper guessing technique. That’s great – mixed strategy isn’t meant to defend against opponents who transparently change their defensive positioning. If it’s obvious, in the moment, how to hit the ball to take advantage of your opponent’s adjustment, go for it.

But let’s give our opponent credit and assume that their defensive adjustments will be made late enough in our swing that we can’t look up and check what they’re up to – to do so at that point would ruin our fault tolerance.

In this case, we don’t know how they will adjust before we select our shot.

But still, you ask, why play a randomized strategy? Isn’t it better to choose a particular shot each time to counter what my opponent is doing?

Leveling Doesn’t Work

If my opponent guesses left, I want to hit right, and if they guess right, I want to hit left. So shouldn’t my strategy still be to figure out how they’re guessing, and then do the opposite? Why would I play a strategy where I randomly select my shots, like 65% cross-court 35% down-the-line, when, depending on what my opponent does, there’s always one of those two I’d rather do?

Because you don’t know what they’re going to do.

Trying to approach your decisions this way leads to absurdity:

  • I want to go cross-court, because my forehand is better than theirs
  • They know that, so they’re going to guess cross-court, so I’m going to go down-the-line
  • But they know that I know that they’re going to guess cross-court, and then I’ll go down-the-line instead, so I’ll actually still go cross-court
  • But they also know that I know that they know that I know that they’re going to guess cross-court, and so go-down-the-line instead, and then because of that still go cross-court… so I’ll go down-the-line
  • But they ALSO know that I know that they know that I know that…

One of the MAJOR casualties of this line of thinking is the nonsensical reasoning it produces on important points:

  • Now it’s match point. They KNOW that I like cross-court the best, and that I’ll use it on match point, so I have to go down-the-line
  • Ah, but of course I would think that, it’s match point after all, so I should actually go cross-court because surely they’ll guess down the line

It’s impossible to execute under these conditions.

It looks like you’re doing analysis, like you’re making progress towards some objective truth about what the right shot to hit is, but really you’re just pointlessly flailing around in an infinitely recursive mental loop, producing no strategic insight whatsoever. This kind of reasoning, often called “leveling,” is often mis-applied in the above way. The infinite recurse is not useful thinking in and of itself – it only leads to an unproductive guessing game.

Luckily, this leveling death spiral can be avoided, so you can just relax and execute on match point.

Mixed Strategy to the Rescue

We know that quality opponents will make strategic adjustments to prevent us from playing our best shot 100% of the time. These adjustments will improve their performance against our best shot, at the cost of worsening their performance against other shots.

All adjustments are tradeoffs. Consider our case from earlier where our best shot is our cross-court forehand, so our opponent often cheats over to the cross-court side to cover it. This positioning is better against the cross-court forehand, but worse against the down the line forehand.

When our opponent is adjusting late enough, we have to make our shot selection decisions under uncertainty.

In order to maintain our equity against a particular adjustment, we need to start hitting those initially second best shots which are improved against that adjustment.

But remember, we have to select our shot without knowing whether our opponent has chosen to hold their ground, or to cheat over and cover the cross-court. If we knew he was holding his ground, we’d want to go cross-court, and likewise if we knew he was cheating over, we’d like to go down the line.

But we don’t. We can’t use a 100% strategy, because we have no idea which 100% strategy to use. When our opponent is adjusting late enough, we have to make our shot selection decisions under uncertainty. This means playing our shots with frequencies that are effective regardless of how they decide to defend.

So how often do we hit each shot?

The answer – it depends on how good you are at everything.

How to Find Your Frequencies

There are actually game theoretical rules that determine exactly how much we can press our advantage without leaving ourselves open to a counter-strategy. Mathematically, there exists an unexploitable, balanced, mixed strategy which, when adopted, makes it irrelevant how our opponent defends – no matter how they choose to guess, our equity remains unchanged. In the case of tennis, the mathematical explanation, tends to be more academic than practical (we still provide it, of course).

From a practical standpoint, when deciding how to mix your frequencies to counter an opponent’s adjustment, your highest equity shot before that adjustment was made is still your starting point. It’s the shot where your greatest advantage lies, and as such the one you’ll try to use most.

Titrate With A Second Best Shot

As your opponent adjusts to your HESBA, you’ll see its effectiveness start to decrease. You’ll notice that, before, you were winning 4/5 points every time you selected it, but now, for some reason, you’re only winning half. Changes are, your opponent is cheating over while you aren’t looking.

To combat that, start mixing in another shot with a low frequency, a second best shot in a vacuum, but a shot which improves against the adjustment your opponent is making to your HESBA. This shot will likely be very effective at first, as your opponent has probably adopted something close to a pure “counter the HESBA” strategy.

After the defender commits to guessing, the attacker will notice the cross-court forehand becoming less effective, and should employ a mixed strategy, mixing cross-court and down-the-line shots, to maximize his equity against the guesses.

Increase the second best shot’s frequency until you feel like your HESBA is effective again, and you’ve arrived at a relatively balanced mixed strategy. You might be tempted to go farther than this, but I’d recommend against it. You don’t want to get stuck in a leveling game. You’re not trying to make the best decision on this particular shot, but rather trying to select the best strategy overall.

You want to put your opponent in an impossible position with your frequencies. If they try guessing cross-court more, they’ll just start losing harder to your down-the-line shots, and if they start guessing down-the-line more, the cross-court shots will be too punishing. Like I mentioned earlier, there’s actually an equilibrium point at which it mathematically doesn’t matter how your opponent defends, your equity is always the same.

That’s what we’re aiming at, and qualitatively, titrating up your second best shot which punishes their adjustment until your HESBA feels effective again is how you get there.

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