Is he really hurt? Signals, types and the rule that keeps players honest

The football question

Late on, a player from the side that's winning goes down holding his ankle. Is he hurt, or is he running down the clock? The referee has a decision to make, and so do the other players: stop, or play on? None of them can see inside his ankle. All they can see is that he went down.

The concept

In every game so far in this series, both sides knew what kind of game they were playing. Here one side knows something the other doesn't: the player knows whether he's hurt. Game theorists call this a game of incomplete information, and the hidden thing his type: hurt, or fine. John Harsanyi worked out how to analyse such games in three papers in 1967 and 1968, and they're now called Bayesian games, because the side in the dark has to update its beliefs with Bayes' theorem.

Going down is a signal: something the informed side does that the other side can see. The question is when a signal tells you anything. Michael Spence's 1973 paper on why employers value degrees gave the answer: a signal is only worth reading if it costs more to fake than faking is worth.

• If going down is cheap, players who aren't hurt go down too whenever it suits them. Everyone sends the same signal, so it tells you very little. That's a pooling equilibrium.
• If going down is costly enough, only the genuinely hurt go down. The signal sorts the types, and it's a separating equilibrium.

The numbers

The numbers here are made up, to show the idea:

• When a player might go down, he's genuinely hurt 10% of the time. A hurt player always goes down.
• A player who isn't hurt would gain from wasting time 40% of the time, when his team is ahead. That's worth 2 to him (in made-up units of match advantage).
• He fakes only when that's worth more than going down costs him.

If going down costs little, a few seconds and a telling-off, say 0.5, every player who's ahead and not hurt goes down. Bayes' theorem then says: of all the players who go down, only 22% are really hurt. Nearly four in five are wasting time.

The rule that changes the cost

Law 5 of the Laws of the Game says an injured player who's treated after play is stopped must leave the field, and may only come back on one minute after play restarts. For a team protecting a lead, a minute with ten men is a real price. Say it's worth 3, more than the 2 that wasting time is worth. Now a player who isn't hurt has no reason to go down, and the only players who do are the ones who are really hurt:

Two rows show the limits of the rule:

• The prize can outgrow the price. In a cup final's last minutes, wasting time might be worth 4, more than the minute off costs, and faking comes back. A rule raises the price of a fake; it can't make it too expensive when the stakes are high enough.
• The exceptions are where to look. Law 5 lets goalkeepers, among others, be treated without leaving the field. For a keeper, going down still costs little, and the theory says that's where the fakes will be.

Who pays for the doubt

The faker isn't the only one affected. Say a referee stops play straight away only when he's at least 50% sure a player is really hurt, and otherwise waits for the ball to go out.

• When going down is cheap, he's only 22% sure, so he waits. The players who are genuinely hurt wait too.
• When going down is costly, he's 100% sure, and stops play at once.

That's the hidden cost of a cheap signal: the honest players pay for the fakers, because nobody can tell them apart. A rule that makes faking expensive helps the genuinely hurt most of all.

Signals all over the pitch

The same logic reads other signals. A full-back pushing high before kick-off might tell the opposition how his side means to play, or might be a bluff. The test is the same: what would it cost to fake? A position that leaves your team exposed if it's a bluff is expensive to fake, so it's worth reading. A shout, a gesture or a word to the press costs nothing, so a clever opponent discounts it. Bluffing itself, when it pays to send a false signal and how often, is part 10.

Why it matters

• Ask what a signal costs to fake. Signals that cost nothing tell you little; signals that cost more to fake than faking is worth tell you a lot.
• Beliefs should follow Bayes. The right question isn't "is he hurt?" but "of the players who go down in this situation, how many are hurt?"
• Rules work by changing prices. The leave-the-field rule doesn't catch fakers; it makes faking not worth it, so honest signals can be trusted again.
• Doubt has victims. When signals can't be trusted, the honest pay for the dishonest.

Limitations

• The numbers are made up. Nobody knows how often players fake; the data can't see inside an ankle.
• Two types is a simplification. Real players are a little hurt, very hurt, or not at all, and referees can see more than whether a player went down.
• Players and referees learn. A player known for diving is believed less, which turns it into a repeated game.
• The Laws have other exceptions besides goalkeepers, such as a player injured by a foul that gets a card, and they can change from season to season.

Try it yourself

Change the numbers in the snippet: make a minute off cost 1.5 instead of 3, and see faking return. Or make 30% of players genuinely hurt, and see how much more a referee can trust the signal even when faking is cheap. At the next match, count who goes down and who goes off: the theory says you'll see more keepers than you'd expect.

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/signalling-is-he-really-hurt
# "Is he really hurt?" as a signalling game. All the numbers are made up.
HURT = 0.10    # of the moments a player might go down, how often he's genuinely hurt
AHEAD = 0.40   # of the rest, how often his team is ahead and would gain from the clock running


def believe(benefit, cost):
    """A player who isn't hurt goes down only if wasting time is worth more to him than going down costs.
    A hurt player always goes down. Bayes' theorem: of the players who go down, how many are really hurt?"""
    fakers = (1 - HURT) * AHEAD if benefit > cost else 0
    return HURT / (HURT + fakers)


print("of the players who go down, the share who are really hurt")
for why, benefit, cost in (("going down costs little (a few seconds, a telling-off)", 2, 0.5),
                           ("treated players must leave the field for a minute", 2, 3),
                           ("the same rule, but a cup final's last minutes", 4, 3),
                           ("a goalkeeper, who doesn't have to leave", 2, 0.5)):
    print(f"  {why}: {believe(benefit, cost):.0%}")

# What the doubt costs the honest: say a referee stops play at once only when he's at least 50% sure
for why, benefit, cost in (("cheap to go down", 2, 0.5), ("costly to go down", 2, 3)):
    sure = believe(benefit, cost)
    print(f"{why}: {sure:.0%} sure, so the referee {'stops play at once' if sure >= 0.5 else 'waits for the ball to go out'}")

Further reading

• Law 5, The Referee, IFAB Laws of the Game: the rule on injured players leaving the field, with its full list of exceptions.
• Signaling game, Wikipedia: types, beliefs, and pooling and separating equilibria, with Spence's 1973 education example.
• Bayesian game, Wikipedia: games where one side knows something the other doesn't, as Harsanyi set them out in 1967 and 1968.