Why don't keepers just stand still? The middle, the Panenka and three choices

The football question

Watch a few penalties and the keeper almost always dives. He hardly ever stays where he is. Across three big studies of professional penalties, keepers stayed put at most about 6 times in 100, and in two of them fewer than 3. Yet when they did stay, they did well: in one study they saved a third of the kicks on target, more than twice as many as when they dived.

So why don't keepers just stand still? And if they almost never do, why don't takers go down the middle every time?

Part 1 treated a penalty as two choices each: the taker's natural side or his weaker side, and a dive one way or the other. That was enough to show why both should mix it up. To answer this question we need a third choice on each side: the taker can go down the middle, and the keeper can stay put.

The concept

With three choices each, the payoff table grows to three rows and three columns. Two ideas from Part 1 carry straight over:

• It's still a zero-sum game: every goal the taker gains is one the keeper loses.
• Each still wants a mixed strategy, picking at random in set proportions, so the other has no habit to exploit.

What's new is how to find the right mix. With two choices, the formula from Part 1 does it. With three, the idea behind it does instead. The taker looks for the mix whose worst case is as good as possible, whatever the keeper does. The keeper looks for the mix that holds the taker's best case as low as possible. That's called minimax, and a famous result of the mathematician John von Neumann, from 1928, says the two always meet at the same scoring chance. That meeting point is the equilibrium.

It also brings in a question that two choices barely needed. Is every choice worth using at all? A choice can be worth having and still get none of the mix.

A football example

Pierre-André Chiappori, Steven Levitt and Tim Groseclose recorded every penalty in the French top league from 1997 to 1999 and the Italian one from 1997 to 2000: 459 kicks, with where each went and where the keeper went. They flipped the left-footers' kicks so that "natural side" means the same thing for everyone.

Here's what the players actually did with those choices:

Kicks down the middle went in most often, and keepers who stayed put let in the fewest. Both point the same way: more middle, from both sides. But the keeper's row rests on 11 kicks, far too few to prove anything by themselves, so let's see what the game says.

Is the middle worth it?

Take the middle away first. With only the two sides on this table, the game from Part 1 has the taker scoring 73.8% and the keeper never needing to stay.

Now add the middle back. While the keeper never stays, a kick down the middle only ever meets a keeper who has dived. So it's worth having only if, on average, it goes in more than 73.8% of the time when the keeper dives. Here it goes in 81% to 89% of the time, so it earns a place.

A taker whose chip down the middle goes in less than that, say 70% of the time when the keeper dives, should never use it. That's true even though, when the keeper dives to the natural side, it does better than aiming there (70 against 63.2), so it isn't worse in every case. A choice that's worse in every case is called dominated, and it's never worth using. But a choice can avoid being dominated and still get none of the mix. The equilibrium is stricter than "never pick a bad option".

The equilibrium

Say the keeper dives to the taker's natural side a share n of the time, stays put a share s, and dives to his weaker side a share w, with the three adding up to 1. Reading across each row of the table, the taker's chance of scoring for each place he can aim is

$$\begin{aligned} \text{natural:}\;\; &63.2n + 100s + 94.1w \\ \text{middle:}\;\; &81.2n + 0s + 89.3w \\ \text{weaker:}\;\; &89.5n + 100s + 44.0w \end{aligned}$$

As in Part 1, the keeper's best plan makes the lines equal for every place the taker should use, so the taker gains nothing from any of them. Three equal lines plus "the shares add up to 1" make enough equations to pin down n, s and w, and the snippet at the end solves them. The answer, with the same reasoning from the taker's side:

Why the keeper stays so rarely

The two answers look lopsided: the middle in nearly a quarter of kicks, the keeper staying in fewer than a tenth. Each side's mix is set by the other's numbers, just as in Part 1.

The keeper stays rarely because the middle is so fragile. A kick down the middle scores nothing against a keeper who stays. So it takes only a little staying, about 9%, to bring the middle down to the same 76.2% as the sides. Any more and the middle would be worse than the sides, takers would stop using it, and the keeper would be standing still for nothing.

The taker goes down the middle often because staying is such a gamble for the keeper. A keeper who stays concedes every kick to a side. For staying to be as good as diving, enough kicks have to come down the middle to make it pay, and that takes about a quarter of them.

Those two figures are the theory's numbers on this table. Its firmest prediction doesn't depend on the exact values, and Chiappori and colleagues proved it under a few mild conditions: takers should go down the middle more often than keepers stay. All three studies agree.

The studies count "the middle" differently: Bar-Eli and colleagues used the whole central third of the goal, which is why their middle share is highest. In each one, though, kicks down the middle outnumber keepers staying by at least four to one.

The Panenka

On 20 June 1976 in Belgrade, the European Championship final between Czechoslovakia and West Germany went to penalties at 2–2. Uli Hoeneß had just blazed West Germany's fourth over the bar when Antonín Panenka stepped up with the chance to win it. He ran up as if to shoot to the side. Sepp Maier dived, and Panenka chipped the ball gently into the middle of the net. Czechoslovakia won 5–3, and the kick has carried his name ever since.

A Panenka is the middle taken to its extreme. If the keeper dives, it almost always goes in. If he stays, it's the easiest save he'll ever make. So suppose a taker has a perfect Panenka that goes in 98 times in 100 when the keeper dives (a made-up number), and nothing else changes. Does he use it more?

Slightly less. The better chip mostly changes what the keeper does: it's now worth staying put twice as often, which makes the middle riskier, so the taker's best mix moves back towards the sides. He still scores more, 78.9% against 76.2%, because the threat of the chip leaves the sides more open. It's the same surprise as Part 1's better weaker side: the keeper moves towards whatever improved.

It also explains why a Panenka works best as a surprise. Its value comes from keepers who dive, and they keep diving only while staying put is a bad bet for them.

So should keepers stand still?

In 2007 Michael Bar-Eli and colleagues looked at 286 penalties on target from top leagues and championships. Keepers who stayed put stopped 33.3% of them (6 of 18). Keepers who dived stopped 14.2% diving left and 12.6% diving right. Yet keepers stayed put only 6.3% of the time. The authors' explanation is action bias: a goal conceded while standing still feels worse than one conceded after a dive, because the keeper looks as if he did nothing.

Game theory adds a warning. Staying put works so well because keepers so rarely do it, and takers don't expect it. In the equilibrium, staying and diving concede exactly the same, so neither is better than the other. A keeper who stood still every time would soon face takers who simply put it to one side, and on the table above all 8 such kicks went in.

The two views still meet somewhere useful. On the 459 penalties, the theory says keepers should stay put about 9% of the time and they actually did 2.4%. Takers should go down the middle about 24% of the time and actually did 17.2%. Both played the middle less than the theory says they should. That fits Bar-Eli's finding. Keepers should probably stay put a bit more often, around one penalty in ten, not every time.

Try it in the Penalty Game Solver, which opens with the middle included: try the Panenka example and watch the keeper's share of staying put rise.

Why it matters

Most real decisions have more than two options, and an opponent who adapts. The lessons from the middle carry over. An option can be worth having without being used often, because the threat of it changes what the other side does. A decent option can still deserve none of the mix. And improving an option often changes your opponent's plan more than your own. The rest of the series moves away from the penalty spot, starting with two teams who both settle for a result that suits them.

Limitations

• The middle rests on few kicks. Keepers stayed put for only 11 of the 459 penalties, and the "0%" for a kick down the middle against a keeper who stays comes from 3 kicks. If staying instead saved four in five such kicks (a made-up figure), the equilibrium would have the keeper staying 11.3% of the time and the taker going down the middle 29.0%. The figures move, but the order holds: the middle is kicked more often than the keeper stays.
• The three choices are still a simplification. Height, power and placement within each third all matter, and some takers wait for the keeper to move first, which turns it into a different game.
• The table is an average. Each taker has his own numbers, and a Panenka specialist's middle row looks nothing like an average one.
• Different studies, different definitions. "Middle" and "stayed put" were judged from video, and each study drew the line differently.

Try it yourself

Think of a keeper you know who always dives. If a taker knew that for certain, where should he aim? Then open the Penalty Game Solver and lower "Aims down the middle, keeper dives to his natural side" until the taker stops using the middle at all. How far down does it have to go?

Reproduce the analysis

This solves the three-choice penalty game from the study's table and prints every number in the article. Nothing to download.

# Scoring chances (%) from Chiappori, Levitt and Groseclose (2002), 459 penalties in the French and Italian top
# leagues. Rows: where the taker aims. Columns: where the keeper goes. Both in the order natural side, middle, weaker side
# ("middle" for the keeper means he stays put). Shots by left-footers are flipped, so "natural" means the same for everyone.
study = [[63.2, 100.0, 94.1],
         [81.2,   0.0, 89.3],
         [89.5, 100.0, 44.0]]
kicks = [[117, 4, 85],            # how many penalties went each way, from the same study
         [48, 3, 28],
         [95, 4, 75]]

flip = lambda t: [list(col) for col in zip(*t)]   # columns as rows: by where the keeper went


def best_mix(table):
    """The mix of rows whose worst case (over the columns) is highest, with that worst case.
    The answer sits where two of these lines cross: an edge of the set of possible mixes, or two columns paying
    the same. So try every crossing and keep the best."""
    lines = [(1, 0, 0), (0, 1, 0), (1, 1, 1)]          # a*x + b*y = c, with x, y the shares of the first two rows
    for j in range(3):
        for k in range(j + 1, 3):
            lines.append((table[0][j] - table[2][j] - table[0][k] + table[2][k],
                          table[1][j] - table[2][j] - table[1][k] + table[2][k], table[2][k] - table[2][j]))
    best, best_value = None, float("-inf")
    for i, (a1, b1, c1) in enumerate(lines):
        for a2, b2, c2 in lines[i + 1:]:
            det = a1 * b2 - a2 * b1
            if abs(det) < 1e-12:
                continue
            x, y = (c1 * b2 - c2 * b1) / det, (a1 * c2 - a2 * c1) / det
            if x < -1e-9 or y < -1e-9 or x + y > 1 + 1e-9:
                continue
            mix = [max(v, 0) + 0.0 for v in (x, y, 1 - x - y)]
            value = min(sum(mix[r] * table[r][c] for r in range(3)) for c in range(3))
            if value > best_value + 1e-9:
                best, best_value = mix, value
    return best, best_value


def solve(table):
    taker, scored = best_mix(table)
    keeper, _ = best_mix([[-v for v in col] for col in flip(table)])   # the keeper wants it low
    return taker, keeper, scored


def show(label, table):
    taker, keeper, scored = solve(table)
    print(f"{label:<24}taker {'/'.join(f'{s:.1%}' for s in taker)}  keeper {'/'.join(f'{s:.1%}' for s in keeper)}"
          f"  scored {scored:.1f}%")


def chip(dives, stays):
    """The study's table with a different kick down the middle: scored if the keeper dives, and if he stays."""
    return [study[0], [dives, stays, dives], study[2]]


def rate(n, p):
    """Share scored over a set of cells: n kicks in each, scoring p% of them."""
    return sum(a * b for a, b in zip(n, p)) / sum(n)


total = sum(map(sum, kicks))
print(f"{total} penalties. Natural / middle / weaker:")
print("Scored, by where he aimed:   ", "  ".join(f"{rate(n, p):.1f}%" for n, p in zip(kicks, study)))
print("Scored, by where keeper went:", "  ".join(f"{rate(n, p):.1f}%" for n, p in zip(flip(kicks), flip(study))))
print(f"Scored overall: {rate(sum(kicks, []), sum(study, [])):.1f}%")
print()
show("The study", study)
print(f"{'What they actually did':<24}taker {'/'.join(f'{sum(n) / total:.1%}' for n in kicks)}  "
      f"keeper {'/'.join(f'{sum(n) / total:.1%}' for n in flip(kicks))}")
show("No middle at all", chip(0, 0))
show("A poor chip", chip(70, 0))                    # made up: goes in 70 times in 100 when the keeper dives
show("A Panenka", chip(98, 0))                      # made up: goes in 98 times in 100 when the keeper dives
show("Staying saves 4 in 5", [study[0], [81.2, 20.0, 89.3], study[2]])   # made up: the middle row rests on 11 kicks

It prints:

459 penalties. Natural / middle / weaker:
Scored, by where he aimed:    76.7%  81.0%  70.1%
Scored, by where keeper went: 76.1%  72.7%  73.4%
Scored overall: 74.9%

The study               taker 42.9%/23.8%/33.4%  keeper 59.5%/9.2%/31.3%  scored 76.2%
What they actually did  taker 44.9%/17.2%/37.9%  keeper 56.6%/2.4%/41.0%
No middle at all        taker 59.6%/0.0%/40.4%  keeper 65.6%/0.0%/34.4%  scored 73.8%
A poor chip             taker 59.6%/0.0%/40.4%  keeper 65.6%/0.0%/34.4%  scored 73.8%
A Panenka               taker 47.0%/21.1%/31.9%  keeper 52.8%/19.5%/27.7%  scored 78.9%
Staying saves 4 in 5    taker 39.2%/29.0%/31.8%  keeper 58.2%/11.3%/30.5%  scored 76.8%

The search works because the best mix always sits at a corner: a point where two of the lines meet, or where a line meets the edge of the possible mixes. It tries every one. The Penalty Game Solver does the same, and its tests check the answer against a search over every mix in steps of 0.25%.

Further reading

• Testing Mixed-Strategy Equilibria When Players Are Heterogeneous, Chiappori, Levitt and Groseclose, American Economic Review (2002). The 459 penalties and the three-choice table, on Steven Levitt's university page.
• Action bias among elite soccer goalkeepers: the case of penalty kicks, Bar-Eli, Azar, Ritov, Keidar-Levin and Schein. The working-paper version, on the Munich Personal RePEc Archive; the final version is in the Journal of Economic Psychology (2007).
• Professionals Play Minimax, Ignacio Palacios-Huerta, Review of Economic Studies (2003). The 1,417 penalties behind Part 1, including how often they went down the middle.
• Minimax theorem, Wikipedia. Von Neumann's 1928 result, and why the two sides' answers always meet.
• Panenka (penalty kick), Wikipedia. The 1976 kick and the famous ones since.