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

Source: https://www.footballdatascience.co.uk/learn/signalling-is-he-really-hurt
Published: 2026-10-04

> A player goes down. The referee can't see whether he's hurt or wasting time, only that he went down. Game theory shows when a signal like that tells you anything, and why one rule in the Laws of the Game makes it honest.

**On the terraces:** When a player goes down, half the ground shouts that he's faking. This piece shows why that doubt hurts the honest players most, and how making a player leave the pitch after treatment turns going down from a trick into a fairly reliable sign he's really hurt.

## 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](/learn/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](https://www.theifab.com/laws/latest/the-referee/), 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:

<div class="bars" markdown="1">

| Of the players who go down | Really hurt |
|---|---|
| Going down costs little | 22% |
| Off for a minute after treatment | 100% |
| Off for a minute, but in a cup final's last minutes | 22% |
| A goalkeeper, who doesn't have to leave | 22% |

</div>

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](/learn/bluffing-dummy-runs).

## 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](/learn/repeated-games-ball-back).
- **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.

```python
# 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](https://www.theifab.com/laws/latest/the-referee/), IFAB Laws of the Game: the rule on injured players leaving the field, with its full list of exceptions.
- [Signaling game](https://en.wikipedia.org/wiki/Signaling_game), Wikipedia: types, beliefs, and pooling and separating equilibria, with Spence's 1973 education example.
- [Bayesian game](https://en.wikipedia.org/wiki/Bayesian_game), Wikipedia: games where one side knows something the other doesn't, as Harsanyi set them out in 1967 and 1968.