An article from Norwegian School of Economics (NHH)

Combating match-fixing with maths

Manual distribution of fixture dates and referees leads to suspicions of corruption and match-fixing in football. A mathematical model makes the process more transparent.

Norwegian School of Economics (NHH)

NHH was founded in 1936. With its affiliated institutes SNF and AFF, NHH constitutes the oldest and largest centre for research and study in the fields of economics and business administration in Norway.

Football is the world's biggest sport, but unfortunately it is also the world's most corrupt. In this respect, the distribution of referees and fixture dates plays a role - a process that in many places is still done by hand by a select group.

Survey of 2013

Chilean Mario Guajardo, from the Norwegian School of Economics' Department of Business and Management Science wants to change this.

Together with, among others, a professor at Universidad de Chile, Guajardo has addressed the allocation of fixture dates and referees in the Chilean first and second divisions.

He believes the study can help to eliminate opportunities for cheating and corruption.

Skewed distribution

A survey of 2013 shows that some referees officiated at far more matches than others, the difference between the most and least frequently used referees was twelve matches.

In addition, there is a big difference between which referees get to officiate certain matches. For example, a referee officiated a one particular team seven times, while another did not officiate the same team once. Guajardo emphasises that it is impossible to prove any cheating took place.

“In an ideal world, where neither referees nor anyone else makes mistakes, this shouldn't really matter. But we know that in practice referees make mistakes, and that's why we want to do something about the system,” says Guajardo.

It was also not the case that the referees who officiated the same team several times lived close by and did not have to come as far, the researcher explains. All the referees, apart from one, lived in the capital city of Santiago, and were thus almost equally far away from all matches.


“One hypothesis is that the football association simply does not care about skewed distribution. Another is that they are not aware of it - that can happen when you do things like that by hand. A third possibility is that they wanted a particular referee for a particular match, but I'll never be able prove that,” he says.

There have been controversies surrounding the allocation of referees in Chile, and the manual distribution scheme has received a lot of criticism in the Chilean press,' says Guajardo.

“It may be that they want to use the better referees more often, or that they wanted to be nicer to some than others. It is probably financially lucrative to referee lots of matches too,” he says.


Guajardo's master's dissertation in Chile was about planning models for companies in the copper industry. He used optimisation tools, adapted software, and mathematical programming to solve the planning challenges of the companies. One of his professors became temporarily involved in a project to create predictable fixture and referee allocations, and they needed a student to help them with the easier work.

“I'm a Chilean and naturally very interested in football. In addition, applied optimisation is one of my fields of expertise, so this was perfect for me,” he says.

Tools from within applied optimisation can be used in many fields, such as natural resources and transport planning, something which the Norwegian School of Economics researcher has recently started work on.

Still sceptical

Even though the researchers' contribution to the league's scheduling has been highly successful, the implementation of referee allocation has been a more variegated story.

“When we sold the model for referee allocation to the Chilean football association the first time, they were pretty positive, and they began to use the system. But one person on the committee gradually became more sceptical. When you use mathematical models in reality, people like to think they're losing control,” says Guajardo.

“With the fixture date models they understood they would have better overall control if they began using these tools, but this insight was not transferred to the model for referee allocation. The person on the referee committee was sceptical, and that stopped the process,” Guajardo explains.

It also didn't help that there was a huge scandal in which referee distribution played an important role.

This led to the association temporarily drawing referees at random.

“This is a pretty poor solution because a referee can, in principle, get all the matches,” he says.

Wants to expand

So far, Guajardo and his research colleagues have modelled the fixture schedule for both the first and second divisions in Chile, as well as for one of the country's junior leagues. In addition there is the allocation of referees, which is their final task, and that is discussed in his most recent article.

“We managed to create a kind of alliance with the Chilean football association. We started in 2005 and have created the match schedule for each season since then. We have also done tournaments for the youth league - all in all around 50 tournaments in the last ten years.”

He stresses that helping the youth league is no less important than the professional tournaments - 'possibly more important, since the levels of professionalism there are lower.'

FIFA's ranking system

“We would also like to expand to creating the fixture schedules for the World Cup. There are several models for local cups, in countries such as Norway and Belgium, but not yet for the World Cup. For example, the qualifying round in South America has used exactly the same order for many years now - a schedule originally created by Peru.”

He is also critical of the FIFA ranking system.

“In the rankings, you get, for example, more points for winning away against Malta than you do for drawing against Spain, which is a far better team. I would love to contribute some new and better methodologies to FIFA, but it's questionable whether that will ever happen,” says Guajardo.


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