Why must the whole back four step up together? The stag hunt and the offside trap

The football question

A through ball is coming. The back four step up as one, the flag goes up, and the striker trudges back. Get it wrong, and one defender holds while the others step, the striker is played onside, and he's clean through. The offside trap is all or nothing. So why do some teams live by it and others never try it?

The concept

The philosopher Jean-Jacques Rousseau told a story about hunters in his Discourse on Inequality. Together they can catch a stag, if every one of them keeps to his post. But if a hare runs past one hunter, he may grab it, and the stag escapes. Game theorists call this the stag hunt: a game where everyone does best by cooperating, but only if they're sure the others will.

It looks like the prisoner's dilemma in part 3, but there's a big difference. In the prisoner's dilemma, each player gains by breaking ranks, whatever the others do. In the stag hunt, nobody gains by breaking ranks if the others hold. The only danger is doubt. It's a coordination game: there are two Nash equilibria, everyone hunting the stag and everyone settling for hares, and the question is which one a team ends up in.

The trap in numbers

The numbers are made up, in goals conceded for every through ball the defence faces:

• The whole line steps up together: 0.04. The striker is offside, or rushed if he isn't.
• The whole line drops off: 0.10. The striker is onside, but there are defenders around him.
• The line splits: 0.30. One defender holds, plays the striker onside, and he's through with the others stranded.

Take two centre-backs. If your partner steps up, you should too (0.04 beats 0.30). If he drops off, you should too (0.10 beats 0.30). Both "everyone steps up" and "everyone drops off" are stable: nobody wants to be the one who breaks the line. Stepping up together is better for everyone, but it needs trust.

How much? It depends on how likely you think it is that your partner will step up:

$$\text{trust needed}$$ $$= \frac{0.30 - 0.10}{(0.30 - 0.10) + (0.30 - 0.04)} = 43\%$$

Four is a different game

With two defenders, trust is easy. With four, the trap only comes off if every defender reads it right: one late step and the striker is onside. Now the question isn't what you think one partner will do, but how often the whole line gets it right.

$$\text{must come off}$$ $$= \frac{0.30 - 0.10}{0.30 - 0.04} = 77\%$$

That's the whole case for a settled back four:

• A back four that reads it 95% of the time: the trap comes off 81% of the time and concedes 0.088, just better than dropping off.
• The same four at 90%: it comes off 66% of the time and concedes 0.129. Worse than sitting deep.
• Three at 95% and a new signing at 80%: it comes off 69% of the time, 0.122. One player who doesn't know the calls sinks the whole line.
• Two centre-backs at 90% do as well as four at 95% (0.089): fewer people to coordinate.

The real game has more to it: the trap isn't the only way to defend a through ball, and defenders who drop off can still be beaten. But the shape holds. A coordinated plan that needs everyone is only as good as its weakest reader.

What happens in practice

Experiments show the same thing. In a famous study by Van Huyck, Battalio and Beil (1990), groups of players each picked an effort level, and everyone's reward depended on the lowest effort in the group, like a defensive line that's only as high as its deepest man. Groups of 14 to 16 quickly slid to the lowest effort: nobody trusted everyone else to keep up. Pairs did much better, especially pairs who stayed together.

Football history fits. The trap is credited to Notts County early in the 20th century, and Arrigo Sacchi's Milan of the late 1980s made a high line famous, keeping defence and midfield no more than 25 to 30 metres apart. A line like that has to be drilled. That's how the trust gets built: in a repeated game, the same four players learn to rely on each other, and a captain's shout or step is a signal everyone can see at once.

Why it matters

• Some plans only work if everyone's in. The offside trap, a coordinated press, a set-piece routine: each is a stag hunt.
• Doubt, not selfishness, is what breaks them. Nobody gains by playing the striker onside. They hold because they're not sure the others will step.
• More people means more trust. A plan that's easy for two can be reckless for four.
• Changing personnel has a hidden cost. A new defender may be better on his own and still make the line worse, until he knows the calls.

Limitations

• The numbers are made up. The data can't see through balls, defensive lines or who stepped when.
• Each defender is treated as getting it right or wrong on his own. In reality, mistakes come together: a bad bounce or a blocked view can catch the whole line at once.
• Strikers adapt. A team known for the trap invites runs that start onside and late passes, which changes the numbers.
• Video review changed the margins. Close offside calls are now checked on video, which changes how fine a trap can be timed.

Try it yourself

Change the numbers in the snippet. Make a split cost 0.40 and see how much more often the line must get it right. Or watch a match and count how often the back line steps as one. The theory says a team that's just changed its defence should drop off more.

Reproduce the analysis

This needs nothing but Python. All the numbers are made up.

# From Football Data Science by Bryan McGuire. Free to use with credit.
# https://www.footballdatascience.co.uk/learn/stag-hunt-offside-trap
# The offside trap as a stag hunt. All the numbers are made up: goals conceded per through ball.
ALL_UP = 0.04   # the whole line steps up together: the striker is offside, or rushed if he isn't
DEEP = 0.10     # the whole line drops off: the striker is onside but has defenders around him
SPLIT = 0.30    # the line splits: one defender holds, plays the striker onside, and he's through


def step_up(q):
    """Goals conceded if you step up, when your partner steps up with chance q."""
    return q * ALL_UP + (1 - q) * SPLIT


def drop_off(q):
    """Goals conceded if you drop off, when your partner steps up with chance q."""
    return q * SPLIT + (1 - q) * DEEP


print("two centre-backs: chance your partner steps up -> conceded if you step up / if you drop off")
for q in (0, 0.25, 0.5, 0.75, 1):
    print(f"  {q:.0%}: {step_up(q):.3f} / {drop_off(q):.3f}")
trust = (SPLIT - DEEP) / ((SPLIT - DEEP) + (SPLIT - ALL_UP))
print(f"step up only if you think your partner will more than {trust:.0%} of the time")

# A back four: the trap only works if every defender reads the call right.
works = (SPLIT - DEEP) / (SPLIT - ALL_UP)   # how often the trap must come off to beat dropping off
print(f"\nthe trap beats a deep line only if it comes off more than {works:.0%} of the time")
for n in (2, 4):
    print(f"  {n} defenders: each must read it right {works ** (1 / n):.1%} of the time")
for label, reads in (("back four, each 95%", [0.95] * 4), ("back four, each 90%", [0.90] * 4),
                     ("three at 95% and a new signing at 80%", [0.95] * 3 + [0.80]),
                     ("two centre-backs, each 90%", [0.90] * 2)):
    p = 1.0
    for r in reads:
        p *= r
    print(f"  {label}: comes off {p:.0%}, concedes {p * ALL_UP + (1 - p) * SPLIT:.3f} v {DEEP:.3f} deep")

Further reading

• Stag hunt, Wikipedia: Rousseau's story, the two equilibria, and why the stag hunt is often a better model of cooperation than the prisoner's dilemma.
• Offside trap, Wikipedia's offside article: how the trap works, its risks, and its history from Notts County to Sacchi's Milan and Klopp's Liverpool.
• Van Huyck, Battalio and Beil, "Tacit Coordination Games, Strategic Uncertainty, and Coordination Failure", American Economic Review (1990): the experiment where large groups slid to the lowest effort and pairs did not.
• Brian Skyrms, The Stag Hunt and the Evolution of Social Structure (Cambridge University Press, 2004): a short book on how trust and cooperation emerge in stag-hunt games.