The rules of football stipulate that if a foul is committed inside a defending player’s own penalty area, a penalty kick is awarded to the attacking team. A high proportion of penalty kicks are converted to goals so penalties can be decisive, particularly in low scoring games. It is therefore especially important for referees’ decisions about penalty kicks to be accurate. However recent evidence  shows that these decisions are subject to subtle bias effects. A study of a large number of penalty decisions in the German Bundesliga concluded that referees tend to ‘even out’ penalty decisions between opposing teams.
The study analysed data from 12902 Bundesliga matches over 40 years, in which 3723 penalties were awarded. The results showed that more matches than would be expected on statistical grounds involved two penalties. The second penalty in the match tended to be awarded to the team that was not awarded the first penalty, indicative of a ‘compensation bias’, a tendency for referees to balance out their decisions. A reasonable alternative explanation for the excess of second penalties is that after a successful penalty conversion the offending team goes on the attack to rectify the score, and is therefore more likely to be awarded the second penalty. But the study found that the bias to award the second penalty to the offending team only occurred when the first penalty was successfully converted rather than missed.
What could cause such a bias? Research on human perception has revealed that sensory information is generally much more ambiguous than we appreciate, so decisions about what we ‘see’ are more difficult than they might appear. We unconsciously resolve ambiguity by putting together incoming sensory data with stored knowledge in order to reach a decision about what we see .
Decisions made by sports officials are subject to the same limitations as other decisions in perception, so they often have to be made on the basis of partial or ambiguous sensory data. For instance, a football referee may have to decide whether a particular offence occurred just inside the penalty area or just outside it. For very close calls of this kind the visual evidence can be quite ambiguous (see  for an analysis of tennis line calls) so even the best referees are liable to make mistakes on occasion.
The problems of ambiguity and uncertainty are not confined to decisions about the location of a foul. It can be unclear, for example, whether the tackling defender made illegal bodily contact with the attacker. Attacking players are very skilful at manufacturing visual evidence consistent with contact (diving), and defenders are skillful at concealing evidence of contact (shirt pulls).
So penalty decisions depend on a complex process of combining uncertain sensory evidence with decision-making factors such as previous experience, rule interpretation and knowledge of the teams and players. The referee must maintain a criterion for deciding whether the evidence is sufficient to award a penalty. The criterion may well shift unconsciously during a game on the basis of, for example, which players are involved. The compensation bias found in the Bundesliga study may reflect a shift in the referee’s criterion in favour of a more lenient evidence threshold for penalties awarded to a previously offending team.
On average one penalty is awarded every 3.48 Bundesliga games. In other words there are about 0.29 penalties per game. How does this rate of penalty awards compare with major international competitions involving European teams, namely the World Cup and the Euros?
The recent history of both competitions shows some interesting trends. In World Cup competitions since 2000 there has been a decline in penalty awards from 0.28 per game in 2002 (similar to the Bundesliga) to 0.21 per game in 2014 (a 25% fall). In the Euros over the same period there has been a much more marked decline from 0.36 per game in 2000 to 0.13 in 2012 (a 66% fall). The large fall in Euro penalties suggests that referees may be more reluctant to award a penalty than they used to be; compared to the Bundesliga they award less than half the number of penalties per game. This shift could reflect the adoption of a much more strict evidence criterion for the award of a penalty in the Euro competition.
Among the alternative explanations are that Euro games have became ‘cleaner’, or that players are now more skilled at hiding their offences, and these changes have led to a decline in penalty awards. However the number of cautions shown in both the World Cup and the Euros does not support this interpretation: caution rates are in decline in the World Cup but not in the Euros, and 25% more yellow cards were shown during each game in the most recent Euro competition (2012) compared to the most recent World Cup (2014) despite the fact only a third as many penalties were awarded in Euro 2012 [4, 5].
By the end of the Euro 2016 competition, 11 penalties had been awarded in 51 games (not including shoot-outs). This represents an average of one penalty every 4.64 games, or about 0.22 penalties per game, quite similar to recent World Cups but lower than in the Bundesliga. Interestingly, success rate is on the decline and is at the lowest level it has ever been, at 64% of penalties converted. But that is another story.
Reading and sources
1. Schwarz, W. (2011) Compensating tendencies in penalty kick decisions of referees in professional football: Evidence from the German Bundesliga 1963–2006, Journal of Sports Sciences, 29:5, 441-447.
2. Mather, G., Sharman, R.J. (2015) Decision-level adaptation in motion perception. Royal Society Open Science, 2 (12), 150418.
3. Mather G (2008) Perceptual uncertainty and line-call challenges in professional tennis. Proceedings of the Royal Society Series B, 275, 1645-1651.