Where should he put it? Penalties and the Nash equilibrium

The football question

A penalty taker has a favourite side. So does every goalkeeper who has watched the footage. If the keeper knows where you like to put it, where should you put it?

Stick to your favourite side and the keeper goes there too. Switch every time and he learns that instead. Any pattern he can spot, he can beat. That's why a penalty isn't really about the kick. It's a guessing game between two people who both know the other is guessing, and it's the textbook example of game theory: the maths of decisions when someone else's decision matters to yours.

The concept

A game here means two or more players, the choices open to each, and what each outcome is worth to them. Game theory asks what each player should do, knowing the others are asking the same question.

At a penalty the choices are simple. The taker aims at his natural side, the one he's stronger at (for a right-footer, usually the keeper's right), or his weaker side. The keeper dives one way or the other. What happens depends on both choices at once, so it can be laid out as a table: a payoff matrix.

A penalty is also a zero-sum game: every goal the taker gains is one the keeper loses. One table of scoring chances describes both sides. The taker wants it high; the keeper wants it low.

A football example

These are real scoring chances from a well-known study by the economist Ignacio Palacios-Huerta, of 1,417 penalties taken between 1995 and 2000, more than nine in ten of them in league matches in Italy, Spain and England.

Why any habit gets punished

Suppose the taker always goes to his natural side. The keeper soon knows, dives that way every time, and the taker scores 69.9%. Always going to the weaker side is worse: 58.3%.

The keeper has the same problem. A keeper who always dives towards the natural side lets a taker who's noticed go the other way and score 95.0%.

Mixing it up beats either habit by ten points or more. But it has to be the right mix.

The equilibrium

Take the keeper first. Say he dives to the taker's weaker side a share g of the time. Then the taker's chance of scoring, for each place he can aim, is

$$\begin{aligned} \text{aim weaker:}\;\; &58.3g + 95.0(1 - g) \\ \text{aim natural:}\;\; &92.9g + 69.9(1 - g) \end{aligned}$$

The keeper's best plan is the g that makes the two lines equal, so the taker gains nothing whichever way he goes. Setting them equal and solving:

$$\begin{aligned} a &= 95.0 - 69.9 = 25.1 \\ b &= 92.9 - 58.3 = 34.6 \\ g &= \frac{a}{a + b} = 42.0\% \end{aligned}$$

The same reasoning, from the other side, gives the taker's mix. He should go to his weaker side a share k of the time that leaves the keeper nothing to gain from either dive:

$$\begin{aligned} c &= 92.9 - 69.9 = 23.0 \\ d &= 95.0 - 58.3 = 36.7 \\ k &= \frac{c}{c + d} = 38.5\% \end{aligned}$$

That pair of mixes is a Nash equilibrium, named after the mathematician John Nash: a pair of plans where neither player can do better by changing his own plan while the other keeps his. A plan that picks at random, in set proportions, is called a mixed strategy. In a penalty, a mixed strategy is the only kind with no weakness to exploit.

Do the professionals play it?

This is where football becomes one of the best tests game theory has had. Laboratory experiments, with volunteers playing for small stakes, had mostly found people not mixing the way the theory says. Palacios-Huerta checked what professional takers and keepers actually did:

Within two points on both sides. He also tested the 42 players involved in at least 30 penalties each, one at a time. Game theory says each should score (or save) equally often whichever way he goes, and that held for all but three of them, about the 2.1 that luck alone would give. And he found no sign that a player's next choice could be predicted from his previous ones: they were as unpredictable as the theory says they should be.

A second study, by Pierre-André Chiappori, Steven Levitt and Tim Groseclose, looked at 459 penalties from the French and Italian top leagues and reached the same conclusion: the results were consistent with players choosing their mix as the theory says. None of these players was working it out with a formula. Practice, and opponents who study them, seem to be enough to push them to the same answer.

The surprise: a better weaker side

Here's where game theory stops being common sense. Suppose the taker works on his weaker side in training and gets more accurate with it: when the keeper goes the wrong way, he now misses 1 kick in 100 instead of 5 (made-up numbers, starting from the study's). Should he aim there more often?

Less often. The keeper now has more to fear from the weaker side, so he dives there more. That leaves the natural side more open, and the taker's best mix moves towards it. The improvement still pays: he scores more, just not by aiming at the side he improved.

It depends which way he improves, though. If instead he got better at beating the keeper when the keeper guesses right (scoring 65.0% on his weaker side when the keeper guesses right, not 58.3%), his best mix would move the other way: 43.4% to his weaker side, scoring 80.8%. What's always true is that the keeper shifts towards the side that improved. In a mixed-strategy equilibrium, your own mix is set by what keeps your opponent guessing, so a change in your ability shows up first in his behaviour.

Try it yourself in the Penalty Game Solver: change any of the four scoring chances and watch both mixes move.

Why it matters

A penalty is the cleanest example, but the same logic applies wherever two sides choose at once and each wants to surprise the other: which way to play a corner, when to press, whether to go long. The lesson is the same in each. Being predictable costs you, and the right amount of variety depends on how good each option is, including against an opponent who has read you. The rest of this series takes that idea into tactics, repeated meetings and even the rules of the league.

For the other half of the penalty question, how well a taker actually converts, see the Beta distribution, which works out how sure you can be of a taker's rate from his record.

Limitations

• Every taker and keeper is different. The four chances are averages. A specialist's table would look different, and so would his best mix.
• Two choices is a simplification. Real takers also choose height, power and the middle of the goal, and some wait for the keeper to move first, which turns it into a different game.
• The theory says how often, not which kick. It says a taker should go to his weaker side about four times in ten. It can't say which four, and that's the point: the choice has to be unpredictable.
• Pressure isn't in the table. A shootout's fifth kick may not be scored as often as an early one in a league match.

Try it yourself

Think of a taker you know well. Does he have a favourite side? If he went there every time, how quickly would keepers notice? Then open the Penalty Game Solver and try a taker whose weaker side is really weak: at some point he should stop using it at all, and the solver shows when.

Reproduce the analysis

This solves the penalty game from the four scoring chances and prints every number in the article. Nothing to download.

# Scoring chances (%) from Palacios-Huerta (2003). The first word is where the taker aims, the second
# where the keeper dives: "weak" is the taker's weaker side, "natural" his stronger side.
study = {"weak/weak": 58.30, "weak/natural": 94.97, "natural/weak": 92.91, "natural/natural": 69.92}


def solve(s):
    """The equilibrium: how often each goes to the taker's weaker side, and how often he scores."""
    ww, wn, nw, nn = s["weak/weak"], s["weak/natural"], s["natural/weak"], s["natural/natural"]
    taker = (nw - nn) / ((nw - nn) + (wn - ww))    # leaves the keeper nothing to gain either way
    keeper = (wn - nn) / ((wn - nn) + (nw - ww))   # leaves the taker nothing to gain either way
    return taker, keeper, keeper * ww + (1 - keeper) * wn


def show(label, s):
    taker, keeper, scored = solve(s)
    print(f"{label:<26} taker {taker:6.1%}  keeper {keeper:6.1%}  scored {scored:.1f}%")


print(f"Always natural side, keeper knows: {study['natural/natural']:.1f}%")
print(f"Always weaker side, keeper knows:  {study['weak/weak']:.1f}%")
print("How often each goes to the taker's weaker side:")
show("The study", study)
show("Fewer misses wide", {**study, "weak/natural": 99.0})   # made up: 1 in 100 wide, not 5
show("Beats the keeper more", {**study, "weak/weak": 65.0})  # made up: harder to save when he guesses right

It prints:

Always natural side, keeper knows: 69.9%
Always weaker side, keeper knows:  58.3%
How often each goes to the taker's weaker side:
The study                  taker  38.5%  keeper  42.0%  scored 79.6%
Fewer misses wide          taker  36.1%  keeper  45.7%  scored 80.4%
Beats the keeper more      taker  43.4%  keeper  47.3%  scored 80.8%

The formulas only apply when neither player has a single best choice. If one of the taker's sides were better whatever the keeper did, he should always go there; the Penalty Game Solver checks for that first.

Further reading

• Professionals Play Minimax, Ignacio Palacios-Huerta, Review of Economic Studies (2003). The study behind the numbers here, on the author's website.
• Testing Mixed-Strategy Equilibria When Players Are Heterogeneous, Chiappori, Levitt and Groseclose, American Economic Review (2002). The French and Italian study, on Steven Levitt's university page.
• Nash equilibrium, Wikipedia. The general idea, with more examples.
• Matching pennies, Wikipedia. The simplest game of this kind, where the answer is to go each way half the time.