How easy would it be for a referee to fix the result of a football match, as the (former) Chairman of the Football Association and of England’s 2018 bid, Lord Triesman, has alleged?

Referees are human, and not particularly rich. The chances are that despite FIFA’s best endeavours some may be open to nobbling, either now or in the past. But would mere observation of the games they refereed provide evidence worth anything?

Research by three British academics, recently published in the RSS journal Statistics in Society, throws a bit of light on the subject. Babatunde Buraimo, David Forrest and Robert Simmons of the Universities of Central Lancashire, Salford and Lancaster respectively were not looking specifically at referee nobbling, but for any evidence of referee bias on the field of play, which is how nobbling would manifest itself.

They looked at the numbers of yellow and red cards given by referees in the English Premier League and the German Bundesliga over six seasons, 2000-01 to 2005-06. Home teams get fewer yellow and red cards than away teams (see Tables, where yellow and red cards issued in each 15 minutes of the matches are tabulated) and previous research has also shown the referees tend to allow more added time in matches in which the home team is losing. Home teams are also more likely to be the beneficiaries of disputable decisions on penalties and goals.

But previous research has not accounted for events that have taken place during the match, such as goals scored and cards issued. “What appears as referee favouritism may actually be excessive and illegal aggressive behaviour by players in teams that are behind in score” the authors of the new paper say. They deal with the issue by using a minute-by-minute bivariate probit analysis of yellow and red cards issued in games over the six seasons in both leagues.

The variables include goal difference, the minute of the match, whether either team has had a yellow or red card in the match or in the previous three minutes, the match attendance, whether it is a derby game, which team is favourite or underdog (using bookmakers’ odds) and, interestingly, whether the pitch has a running track around it. None in England now do, but several in the Bundesliga do, or did in the relevant seasons.

The results of the model show that what happens during a match does indeed affect the probability of either team getting a yellow or red card, as one would expect. A goal by the home team reduces its probability of getting a yellow card: the bigger the goal difference the greater the effect. But an away team that falls a goal behind is more likely to attract a yellow card as it struggles to get back into the game. An away team that falls two goals behind, however, is less likely to be yellow-carded: as the home team’s lead becomes decisive, neither team takes the kinds of risks that could lead to cards.

Where the bookies’ odds favour the home team (as they usually do) the less likely that team is to get a yellow card. The greater the superiority of the home team, the lower their chances of a yellow card and the higher the chances of the away team getting one. (This may explain why visitors to Manchester United often complain that it is impossible to win a penalty there. United start as favourites and stand a lower chance of sanctions than their visitors.)

The influence of a running track around the pitch is striking. Home teams are more likely to get a yellow card where there is a track, and marginally more likely to get a red. Two effects are in play here: the crowd is further from the play, so is less intimidating to the referee but also less able to rouse its team. If the second effect were the greater, home teams in grounds with tracks would get fewer yellow cards, not more. So that suggests, as previous research has done, that the greater distance from the crowd allows referees to operate under less intimidation, and show less bias towards the home team.

It is significant, maybe, that modern English grounds do not have running tracks around them, and that two German teams (Schalke and Bayern Munich) moved to new stadia without tracks during the period of the research, while another (Hannover)  modernised its stadium and eliminated the track.

When underdogs are playing at home, both leagues show a strong positive tendency for the visiting team - favourites to win the match - to get a yellow or red card.  So the conclusion is that referees are biased in favour of home teams, however much they may try to officiate fairly.

If such bias cannot be eliminated, the chances of detecting nobbling through simple observation are negligible. A referee may make a bizarre decision without it being evidence of improper behaviour. A history of bizarre decisions would certainly have him eliminated from FIFA’s panel. And the difficulty of proving what all fans already suspected – that referees favour the home team – shows how impossible it would be to identify nobbling in a handful of games at the World Cup finals.

But the paper does show how important it is for England to win the 2018 bid if they are ever to regain the World Cup, last won on home soil in 1966. English teams need every positive they can find, and home advantage could make the difference.  South Africa, who have home advantage in this summer’s World Cup, are probably too much the underdogs to make anything of it.  But of 18 World Cups played so far, six have been won by home teams.
 
Reference: Journal of the Royal Statistical Society, Series A, Statistics in Society, vol 173, part 2, April 2010, p 431-449