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A striker who always goes to their stronger side scores well, until the keeper notices. Then the strong side is covered every time and the advantage is gone. Being good is not enough. You also have to be hard to predict.

Mixed strategies in sport

This is the mixed-strategy lesson applied where the stakes are real and the data exist. Penalty kicks, tennis serves, pitch selection: in each, one side wants to be unpredictable at the frequencies that keep the other side guessing. The right mix is usually not 50/50, and a keeper who reads you turns any pattern into a save.

From earlier lessons

Two quick ones before we start.

You mix your play in order to...

In a zero-sum game, the mix that guarantees the most is...

Predict, then play

You take penalties against a keeper that watches your history and dives where it expects the ball. The scoring chances are not symmetric: your natural side is more dangerous, but the keeper knows that too. Take a run of shots and watch the meters.

Why did that happen?

Take a guess, then reveal the reason

Your natural side scores more, so favoring it looks smart, until the keeper adjusts and covers it. Then the whole point of the mix is defeated. The unexploitable mix is the one where the keeper cannot do better by favoring either side, and because the sides are not equally dangerous, that mix is tilted rather than even. Any predictable habit, always going natural, alternating, following a streak, hands the keeper a read.

The math

Take the scoring rates estimated from professional kicks (Palacios-Huerta). Each entry is the kicker’s chance of scoring, given the side kicked and the side the keeper dives.

Exhibit 2 Scoring probability (kicker's chance)

Kick \ Keeper dives	Left	Right
Left	0.58	0.95
Right	0.93	0.70

This is (almost) zero-sum: what the kicker gains in scoring, the keeper loses in save rate. Let the kicker shoot Left with probability $p$ . The keeper compares diving Left against diving Right, and the kicker picks $p$ to make the keeper indifferent:

\underbrace{0.58\,p + 0.93(1-p)}_{\text{keeper dives Left}} = \underbrace{0.95\,p + 0.70(1-p)}_{\text{keeper dives Right}}.

Solving, $0.93 - 0.35p = 0.70 + 0.25p$ , so $p = \tfrac{0.23}{0.60} \approx 0.38$ . At that mix the scoring rate is about $0.80$ , and the keeper cannot raise its save rate by leaning either way. Shoot Left far more or far less than $38\%$ of the time and a sharp keeper cuts you below $0.80$ .

Definition, the exploitable pattern

Unpredictable does not mean even. It means no strategy the opponent can adopt beats their equilibrium response. A fixed bias is exploitable, and so is any deterministic rule (alternating, copying your last make), because a rule can be learned. Only a mix at the right frequencies, drawn fresh each time, is safe.

Work it out

Find the mix, then see the cost of a habit

Set the keeper’s two options equal: $0.58p + 0.93(1-p) = 0.95p + 0.70(1-p)$ . The left side falls in $p$ (favoring Left helps the Left-diving keeper), the right side rises. They meet at $p \approx 0.38$ , where scoring is $0.58(0.38) + 0.93(0.62) \approx 0.80$ .

Now suppose you always take your natural side (say Right). The keeper learns this and always dives Right, holding you to $0.70$ instead of $0.80$ . The predictable habit costs you ten goals in a hundred. Randomizing at the right rate is worth more than picking your best-looking shot every time.

A note from the real world

Reality check

The theory holds up outside the classroom. Studying professional penalties, Chiappori, Levitt and Groseclose (2002) and Palacios-Huerta (2003) found that top players mix at close to the equilibrium frequencies and show no serial pattern a keeper could exploit. The best in the world really do play minimax. The Behavioral unit later asks where this discipline holds and where it breaks down.

Key idea

The winning mix equalizes the opponent’s options, which is usually not 50/50, and it has to be genuinely random. Any pattern, however clever, is a pattern the other side can learn.

Check yourself

Check yourselfapply

Your two sides score at different rates. Why is the unexploitable mix not 50/50?

Check yourselfanalyse

A striker alternates strictly: Left, Right, Left, Right. Is this a good way to be unpredictable?

Check yourselfbrings back: minimax

At the equilibrium mix the kicker scores about 80%. What does the minimax logic say this number is?

Sources

Chiappori, P.-A., Levitt, S., and Groseclose, T. (2002). Testing mixed-strategy equilibria when players are heterogeneous: the case of penalty kicks in soccer. American Economic Review, 92(4), 1138-1151.
Palacios-Huerta, I. (2003). Professionals play minimax. Review of Economic Studies, 70(2), 395-415.