Can you spot a match fix by looking at the numbers?

Aces high. Tennis by Shutterstock

Match fixing is unthinkable for most sports professionals whose very persons, careers and reputations are built on a single-minded focus on winning. However, allegations over a number of years, across a range of sports, have suggested that for some the ability to lose convincingly gains the greatest income.

First there were the match fixing bans given to some cricketers in the early 2000s, and again in 2011. Now there are allegations that officials ignored red flags on as many as 16 tennis players ranked in the top 50 who may have thrown matches – including at Wimbledon – accusations they deny.

Throughout most jurisdictions match fixing is a criminal offence, or at least a very serious professional offence which undermines any value a game, or sport, may have. After all, who wants to play against someone who is trying to lose the game. The problem for both police and the sporting bodies is how compelling the evidence is.

Losing is part of the game

Evidence can come from several quarters: confessions and recordings of conversations, as in the 2010-2011 set of cricketing scandals, and suspicious betting patterns around particular matches, as is alleged in the tennis case. Most contentious, however, is evidence from the actual sporting outcomes themselves and whether these can be used to verify an accusation. Is a world-class player losing to another who isn’t considered that good really proof that the former is losing deliberately?

If there was no uncertainty in the outcomes of sporting events then it wouldn’t be much of a game. A certain amount of upset is integral to any sport. Just last year South Africa suffered a shock defeat to Japan in the Rugby World Cup, Holly Holm defeated Ronda Rousey in the Ultimate Fighting Championship 193, and Dustin Brown, ranked 102, beat two-time former champion Rafael Nadal in the second round at Wimbledon. Examples further back include the US ice-hockey team beating the Soviet Union’s Red Army in the 1980 Winter Olympics. All were classic events – and there is, of course, no suggestion that any of them were fixed.

So, how can we go about using the evidence of unexpected losses of games as evidence that those games were in some way deliberately fixed?

Crunching the numbers

This area of statistical evidence has close structural similarities to what could be called “the nurse problem”. This is where a medical professional has been suspected of harming those in their charge, and the only evidence is a series of adverse outcomes in the patients. This may seem more serious than match fixing but as a problem it has exactly the same features, and exactly the same reasons as to why as yet we have no fully satisfactory solution to it.

It isn’t possible to directly calculate probabilities for hypotheses or propositions. However, many think that it is possible, which can muddle understanding. For instance, we cannot calculate the probability that a nurse has harmed a patient, or, that a tennis player has deliberately lost a game. What we can do is calculate the probability of seeing a given outcome were it true that the loss of the match was entirely accidental – and that is what statistics are quite good at.

When calculating probabilities isn’t that simple. Shutterstock

We can look at the probabilities concerning two competing propositions: that a match or game was deliberately lost; and that it was lost entirely by accident. We then consider the ratio of the two probabilities calculated under the two propositions. This ratio, sometimes termed a likelihood ratio, is a number which represents the force of the evidence to influence us one way or another.

As an example: one tennis player might, when playing another specific tennis player, have played four matches, only winning one of these, and we are sure none of those matches was fixed. Then, let us say that our more able tennis player goes on to lose a match against the lesser player. Were it true that the loss was purely involuntary, then an estimate of the probability of that particular outcome might be one in four.

If the better of the two players is very good, and were it considered inevitable that they would lose the match if they tried to do so, then the probability of a lost match given they were trying to lose might be one. In this case the value of the evidence that the match has been unexpectedly lost might be one, divided by one in four; which is four. This means that a lost match is four times more likely were match fixing going on, than if no match fixing is taking place. This, by itself, is not a very compelling figure in favour of match fixing.

The real problem here is the value of the probability of seeing a lost match were the better player actually trying to lose. In the example above we assumed a value of one, but there is no justification – neither theoretical or historical – for this value. This is the drawback with this sort of evidential problem, either in the context of sporting events or with medical and caring professions.

There can usually be an appeal to past data for an estimate of the probability of the unexpected adverse outcome were it true that no wrongdoing had taken place. In tennis most players have played most other players many times, and in a medical context hospitals keep meticulous records of serious incidents. Both these sources can provide perfectly legitimate estimates for probabilities. There is however no such data on which to base estimates for the probabilities of an adverse outcome in a hospital were any of the medical staff actually trying to harm the patients. It is an experiment which cannot be run. One cannot simply let a known murderer loose among the high dependency units to see how many patients are harmed.

Follow the betting … Shutterstock

In the 2010 Pakistan cricket case, which involved spot betting on step-over no-balls we can make some estimates as some players later confessed to deliberately stepping over the crease, and as their entire playing history is known, and they weren’t trying to fix games for most of it, decent estimates of the probabilities can be made. Mostly though, there is very little data for deliberately thrown games.

It is disappointing that statisticians have no foolproof way of evaluating these two disparate sounding, but structurally similar types of evidence. As statisticians we must continue to work on nurse and carer problems, but if people are concerned that their sports and games are being corrupted by deliberate fixing, then lessening the reliance upon betting, or somehow changing the nature of betting, may offer some way forward as all match fixing appears related to it and the vast sums of money linked to it.